- Email: [email protected]
Persistence of order and structure in chaos
Physica 20D (1986) 374-386 North-Holland, Amsterdam
PERSISTENCE
OF ORDER AND STRUCTURE
IN CHAOS
Received 26 January 1985 Revised manuscript received 18 November 1985
H.E. NUSSE University
of Groningen, Institute of Econometrics,
P.O. Box 800, 9700 A V Groningen, The Netherlands
In this paper some results are presented concerning one-dimensional chaotic maps with arbitrarily many critical points. Let f be a chaotic map belonging to some suitable class of C’ maps from a nontrivial interval X into itself. Assuming that f is of class C 1+a for some a > 0, we have that the set of aperiodic points for f has Lebesgue measure zero: further, if f(X) is bounded then there exists a positive integer p such that almost every point in the interval is asymptotically periodic with period p. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics off is persistent under small smooth perturbations. The topological structure of the nonwandering set of f will be described, and this structure is invariant under small C’ perturbations of the map f. Assuming that f is of class C2, the map f is C2 structurally stable provided that f satisfies some suitable conditions. Finally, it will turn out that maps with a negative Schwarzian derivative belong to the suitable class of maps mentioned
1. Introduction
The aim of this paper is to present the reformulated main results obtained in [26]. The proofs of the theorems, properties and facts can be found in the mentioned reference. It is a pleasure to recall the celebrated “Period three implies chaos” result due to Li and Yorke [18]: Fix some interval J and let F: J + J be a continuous mapping. Assume that there is a point ae.I such that b=F(a), c=F*(a) and d= F3(a) satisfy dsab>c). Then we have: (i) for every positive integer k there is a periodic point in J having period k; (ii) there is an uncountable set S c J (containing no periodic points) that satisfies the following two conditions: (a) for every p, q E S with p f q, lim ,I_P,supIF’l(~)-F’f(q)I>lim,,_,inf) (p)-F”(q)1 =O; (b) for every FES
periodicpoint > 0.
qEJlim.,,sup]F”(p)-F”(q)]
F”
and any
Li and Yorke observed that the set S occurring above contains no asymptotically periodic points. Further they mentioned the question, whether the existence of an asymptotically stable periodic point implies that almost every point is asymptotically periodic. There exists a lot of numerical investigations dealing with iteration of mappings, see e.g. Stein and Ulam [33], Metropolis, Stem and Stein [21], Hoppensteadt and Hyman [14], Gumowski and Mira [12], and Coste [lo]. The numerical results of density functions suggest, for some examples, that almost every point in the interval approach to an asymptotically stable periodic orbit. Many established results in the study of iteration of mappings initiated by Lorenz [19], Li and Yorke [18] and May [20] can be found in the monographs by Collet and Eckmann [9] and Preston [27], and in the article by Nitecki [25]. The question posed by Li and Yorke has been answered affirmatively in a rather general manner
0167-2789/86/$03.50 Q Else& Science Publishers B.V. (North-Holland Physics Publishing Division)
H. E. Nose/
Persistence of order and structure in chaos
by the “No chance to get lost” theorem in section 3. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics of a chaotic map f, belonging to some suitable class of maps, is persistent under small smooth perturbations. We note that such a map f may have many critical points (i.e. points at which the derivative of f vanishes). Besides we give a description of the topological structure of the nonwandering set of a chaotic map f, we also consider the set of periodic points for f. The question: “Given that a continuous mapping f from the real line into itself has a periodic orbit with period n, for some fixed positive integer n, which other positive integers must occur as periods of the periodic orbits of f ?”has been discussed and has been answered by Li and Yorke [18], Sharkovsky [28], Stefan [32], Strafhn [34] and Block [4]. The main result is due to Sharkovsky, see Stefan [32]. Let the positive integers lexicographically ordered in the following way: 3 i 5 -I 7 -I . . . -I 2 .3i2.5-12.7-I +.. -I2*~3-t2*.5-12*.7-i . - . -I 23 -I 2* i 2 i 1. Sharkovsky’s theorem says the following. Let f be a continuous mapping from W into itself, which has a periodic orbit of period n. Then f has a periodic orbit of period m for every m E N for which n -I m holds. One may ask the following question: “Given any continuous mapping f from a nontrivial interval into itself. Fix any positive integer n. How many periodic points with period n does f have?“. Smale and Williams [31] answered this question for the mapping f: [0, l] + [0, l] defined by f(x) = 3.83x(1 - x). We will answer this question for a map f, which may have many critical points, belonging to some suitable class of functions. It holds that the structure of the nonwandering set of such a map f, and also the number of periodic points with period n, n E N, for f doesn’t change under small smooth perturbations. The condition negative Schwarzian derivative is invariant under composition, but it is not invariant under conjugation with a diffeomorphism. Conse-
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quently, this condition is not persistent under smooth perturbations. Therefore we looked for a new condition. We refer the reader to section two for some preliminary definitions. Inspired by Smale’s definition of the Axiom A property for diffeomorphisms [30], we will introduce the Axiom A property for one-dimensional noninvertible mappings. To give an idea of the Axiom A property (the precise definition can be found in section three) we consider the following. Let f be a differentiable map from a non-trivial interval X into itself. Assume that the closure of the set of periodic points for f can be split up into a set S consisting of at most finitely many asymptotically stable periodic points, and a compact positively f-invariant set U (i.e. f(U) c U) with I(f”)‘(x)l >lforeachx~U,foreveryn>N,for some fixed NE N. (Note that the set S is a compact positively f-invariant set, and the intersection of S with U is empty; further S may be empty.) Then: f is an Axiom A mapping. For clarity, we will restrict our attention for a moment to a chaotic polynomial map P: X+ X from a bounded real interval X into itself. Assume that P satisfies the following conditions: (1) each critical point of P is real; (2) the orbit of each critical point of P converges to an asymptotically stable periodic orbit of P or to a subset of the possible present absorbing boundary for P; (3) P is contracting on the set of asymptotically stable periodic points for P, provided that this set is nonempty. Then we have: P is a nonsingular Axiom A map. The obtained results for Axiom A maps imply that for such a polynomial map P the following properties hold: (i) almost every point in X, in the sense of periodic Lebesgue measure, is asymptotically for P. (ii) the nonwandering set of P can be decomposed into finitely many basic sets and P has a dense orbit on each of these basic sets. (iii) the map P is C’+a-persistent for any (Y> 0, i.e. the asymptotically periodic behaviour in the
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dynamics of P and the structure of the nonwandering set don’t change under small smooth perturbations. (iv) the number of periodic points for P with period n can be computed explicitly, for each positive integer n. (v) the map P is C2-structurally stable, provided that some additional conditions are satisfied (P is C 2 structurally stable means that there exists some neighbourhood U of P in C2( X, X) such that g and P are topologically conjugate for each g E U). The organization of this paper is as follows. In section 2.1 we give some simple examples of Axiom A maps and we state their properties; in section 2.2 we give some preliminary definitions. In section 3 we will present the results for Axiom A maps. Let f~ C’( X, X) be a fixed chaotic nonsingular Axiom A mapping from a nontrivial interval X into itself. The most interesting results are the following. There exists a positive integer p such that almost every point in X, in the sense of Lebesgue measure, is asymptotically periodic with period p; this phenomenon concerning the asymptotically periodic behaviour in the complicated dynamics of f doesn’t change under small smooth perturbations. The nonwandering set of f (i.e. the closure of the set of periodic points) can be decomposed into finitely many compact positively f-invariant subsets, and f has a dense orbit in each of these basic sets; this structure of the nonwandering set is invariant under small smooth perturbations. The topological entropy of f can be determined, further the number of periodic points for f with period n can be computed for each positive integer n; Finally we will see that, under some additional conditions, f is C2-structurally stable. It is worthwhile to note that the Axiom A property is invariant under conjugation with a diffeomorphism, and it is also invariant under small smooth perturbations. In section 4 we consider maps with a negative Schwarzian derivative. We note that this condition
is not invariant under conjugation with a diffeomorphism. It will turn out, that these maps are, under reasonable conditions, Axiom A maps. In particular, the well-studied maps with one critical point and negative Schwarzian derivative are Axiom A maps, provided that the orbit of the critical point converges to an asymptotically stable periodic orbit.
Remarks
There are several published results related to the obtained results presented in this paper. (1) The following results are related to theorem 3c: Jonker and Rand [17] studied mappings of a compact interval into itself with one critical point. Their main result concerns a canonical decomposition of the nonwandering set in terms of the iterated topological entropy of the map in question. The proof of this result is based on the kneading theory developed by Milnor and Thurston. Van Strien [35] obtained for a subclass of the mappings studied by Jonker and Rand, namely those mappings with a negative Schwarzian derivative that the sets in the decomposition have, except at most one set, a hyperbolic structure. (2) The following reformulated result due to Guckenheimer [ll], Misiurewicz [23] and Van Strien [35], see also Collet and Eckmann [9] and Preston [27], is a special case of theorem 4a. Let f be a chaotic C3-mapping from a compact interval [a, b] into itself. Assume that f satisfies the following conditions: (1) f has one critical point c which is nondegenerate, f is increasing on [a, c] and f is decreasing on [c, b]; (2) f(a) = f(b) = a; (3) f has a negative Schwarzian derivative (i.e. f”‘(x)/f ‘(x) - ~[f”]’ (0 forall xE[a, b]\(c)); (4) f has an asymptotically stable periodic orbit; (5) f’(a) > 1. Then the set of points, whose orbits don’t converge to the asymptotically stable periodic orbit, has Lebesgue measure zero. (3) If a map f has an invariant measure absolutely continuous with respect to Lebesgue mea-
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sure, see Misiurewicz [23] and Jakobson [16], then
f is no Axiom A map. (4) After writing the paper the author learned from the review by Keller [37] that results due to Jakobson [15] are related to the stability results in this paper. 2. Preliminaries 2.1. Some very simple examples Example 1.1. Fix any real number A > 4. Let the map f: R --)R be defined by f(x) = hx(1 - x). Then: - f is an Axiom A mapping. -The set of points, whose orbits don’t converge to the absorbing boundary point {-cc}, has Lebesgue measure zero, see also Chaundy and Phillips [S], Brolin [7] and Henry [13]. In fact, this set of points whose orbits will stay in the unit interval forever, is the nonwandering set. - The nonwandering set of f consists of one basic set, see also Jonker and Rand [17]. -The qualitative behaviour for the dynamics of f doesn’t change under small smooth perturbations of f, the same holds for the structure of the nonwandering set. -f is C*-structurally stable. It follows from this, that if we vary the parameter a little bit, then the qualitative properties don’t change. Example 1.2. Let f: [0, l] + [0, l] be defined by f(x) = 3.83x(1 - x). This example has already attracted some fame. Smale and Williams [31] showed the existence of an asymptotically stable periodic orbit with period 3. The graph of f3 is giveninfig.l.WewriteO=x, 1, we obtain from a result due to
00
X-3 Fig. 1. The graph of f3, for f(x) =3.83x(1 -x).
Singer [21]: f has precisely one asymptotically stable periodic orbit. The direct domain of attraction of the asymptotically stable periodic orbit of period three is the union of the intervals
1.~~~ ~2Wl~4~xWIX~~ Y& whh y2 = mm {x El% x21; f 3(x) = x2}, y4= ma {x El+, x4L f3(x)=x4} and y,=min{xE]xg,l[; x6 }. See the figure.
f3(x)=
We have: -f is an Axiom A mapping. -Almost every point in the unit interval (in the sense of Lebesgue measure) is asymptotically periodic with period 3, see also Guckenheimer [ll] and Van Strien [35]. -The nonwandering set of f consists of three basic sets, see also Jonker and Rand [17]. -The asymptotically periodic behaviour in the complicated dynamics of f doesn’t change under small smooth perturbations of f; the same holds for the structure of the nonwandering set. -f is C*-structurally stable. It follows from this fact that a small change in the parameter h = 3.83 has no influence on the qualitative behaviour of the dynamics. Example 1.3. Let f: [0, l] + [0, l] be defined by f has f(x) = X,x(1 -x)with3
stable periodic orbit with small-
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Persistence of order and structure in chaos
est period p, and (fP)‘(x,) # - 1, (fP)‘(x,) # 0 with x,, an asymptotically stable periodic point for f. Then: -f is an Axiom A mapping. -Almost every point in the unit interval is asymptotically periodic with period p, see also Guckenheimer [ll] and Van Strien [35]; this behaviour in the dynamics doesn’t change under small smooth perturbations of f. -The nonwandering set of f consists of at least three basic sets, see also Jonker and Rand [17]; the structure of the nonwandering set is invariant under small smooth perturbations of f. -f is C*-structurally stable; consequently the qualitative behaviour of the dynamics are stable under small variations of the parameter X,. (Note that f satisfies condition (iii) of theorem 3e because the critical point of f is in the direct domain of attraction of the asymptotically stable periodic orbit and the critical point is assumed to be not periodic because ( f p)‘( f) = 0). Example 2.1. Fix any real number X > 4. The map f: W + W defined by f(x) = Ax3 + (1 - X)x
has the following properties: -f is an Axiom A map. -The orbit of almost every point on the real axis converges to either the absorbing point - cc or to the absorbing point + cc. -The nonwandering set of f consists of one basic set. -f is C1+a -persistent for some positive real number OL,i.e. the qualitative behaviour of the dynamics as well as the structure of the nonwandering set doesn’t change under small smooth perturbations of f. -f is C*-structurally stable. Example 2.2. Define the map f: [ - 1, l] + [ - 1, l] by f(x) = 3.701x3 - 2.701x. The map f has two asymptotically stable periodic orbits with period three. We have: -f is an Axiom A map. - Almost every point in [ - 1, l] is asymptotically periodic with period three.
-The nonwandering set of f consists of five basic sets namely the isolated fixed points - 1, + 1, the two asymptotically stable periodic orbits and the rest of the nonwandering set. -f is Cl+“-persistent for some positive real number (Y. -f is C*-structurally stable. Example
2.3. Let f: [ - 1, l] --) [ - 1, l] be defined
by f(x) = X,x3 + (1 - X,)x with 2 < A, < 4 such that f has an asymptotically stable periodic orbit with smallest period p for which (f J’)‘(x,) f - 1 and ( f P)‘( xp) # 0, with xP an asymptotically stable periodic point for f. If p is odd, then f has two asymptotically stable periodic orbits; if p is even, then f has one or f has two asymptotically stable periodic orbits, both cases can occur. Then we have: -f is an Axiom A map. -Almost every point in the interval [ - l,l] is asymptotically periodic with period p. -The nonwandering set consists of at least four basic sets. -f is Cl +“-persistent, for some positive real number (Y. -f is C*-structurally stable. 2.2. Some preliminary dejinitions and notations Throughout the paper XC W will be a nontrivial interval. Let D and Y be nonempty Lebesgue measurable subsets of X such that D C Y. We denote the complement of D in Y by Y\D. We write p(Y) for the Lebesgue measure of Y. Let f: X + X be a differentiable (noninvertible) mapping. For any positive integer n, the n th iterate of f, denoted by f “, is inductively defined by f”=f Ofn-1, with f” the identity mapping. For each point x E X, the orbit of x under f is the set { f O(x); n E N u (0)). The set f”(D) is defined x E D}, and the set f-“(D) byf”(D)= {f”(x); is defined by f-“(D)={xEX, f”(x)ED} for each n E N U (0). The set D is called positiuely f-invariant if f(D) c D, D is called negatiuely f-inuariant if f-‘(D)
H. E. Nuwe/ Persistence of order and structure in chaos
c D. The set D is a component of Y if it is a
maximally connected subset of Y. A component is called trioial if it consists of one point. A point x E X is called a periodic point for f with period p if f r(x) = x for some p E N. A periodic point x for f with period p is called asymptotically stable if there exists an open neighbourhood U of x in X such that lim n _ m f “r( y) =x for all y E U. A point x E X is called an asymptotically periodic point for f if lim n _oo f”“(x) exists for some m E N; a point x E X is called an aperiodic point for f if it is not an asymptotically periodic point and if the orbit of x is bounded. A point x E X is called a nonwandering point for f, if for every neighbourhood U of x there exists n E N such that f”(U) n U # 0. A point x E X is called a criticalpoint for f if
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property if the following two conditions are satisfied: (a) the nonwandering set P(g) is hyperbolic; (b) the periodic points of g are dense in
{y~X;lim,,,f”r(y)=q}iscalledthedomain
O(g). For a continuous one-dimensional (noninvertible) mapping g the set of periodic points for g and the nonwandering set of g are positively ginvariant sets, but they are not necessarily negatively g-invariant sets. Now we will formulate the Axiom A property for mappings from a nontrivial interval X into itself. A mapping g E C’( X, X) is called an Axiom A mapping if the following conditions are satisfied: (a) the nonwandering set s2(g) can be split up into at most two, but at least one, nonempty positively g-invariant compact subsets say O(g) = In,(g) U Q,(g) such that g is contracting on G,(g) provided that 9,(g) is nonempty, and g is expanding on 6?,(g) provided that an,(g) is nonempty; (b) the periodic points for g are dense in G(g). (g is contracting on Q,(g) (expanding on L?,(g)) means there are constants C > 0 and 0 < K < 1 such that ](g”)‘(x)] 6 CK” for each x E i2n,(g)(l(g”);(x)l 2 C[l/K]” for each x E Q,(g)) for every n E N.) We note that the Axiom A property for onedimensional noninvertible maps is invariant under conjugation with a diff’morphism (i.e. if g E C’( X, X) is an Axiom A mapping and h is a diffeomorphism defined on X, then the mapping h 0 g 0 h-’ is an Axiom A map). First, we state the result that an Axiom A map has the so called “No chance to get lost” property. Let (Ybe a positive real number.
of attraction of q. The component of the domain of attraction of q containing q as a boundary point, is called the direct domain of attraction of q.
Theorem 3.1. Let f E C1+a(X, X) be a nonsingu-
f’(x)
= 0.
The map f is called chaotic, if there exists at least one aperiodic point. The map f is called nonsingular if p(E) = 0 implies p( f-‘( E)) = 0 with E measurable subset of X. Let q E X be an asymptotically stable periodic point with period p. The domain of attraction of q is the set of points {x=X, lim,,, f”r(x)=q}. The direct domain of attraction of q is the component of the domain of attraction of q containing q. A point x, x e X, is called an absorbing boundaty point of X for f with period p, if there exists an open set U in X such that lim, _ m f “r( y) = x for each y E U. Assume that q is an absorbing boundary point of X for f with period p. The set
3. Main results For clarity of exposition we recall Smale’s Axiom A property for diffeomorphisms [30]: Let M be a compact manifold and let g: M -+ M be a diffeomorphism. The mapping g satisfies the Axiom A
lar Axiom A mapping from a nontrivial interval X into itself. Then we have: The set of points, whose orbits don’t converge to an asymptotically stable periodic orbit of f (or to the possible present absorbing boundary of X), has Lebesgue measure zero. Furthermore, there is a positive integer p such that almost every point x in X is asymptotically periodic with period p, provided that f(X) is bounded.
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It is due to the phenomenon that the orbits of almost all x in X in the sense of Lebesgue measure, converge to an asymptotically stable periodic orbit of f, or to an (or to the) possible present absorbing boundary point(s) of X for f, that we say that f has the “No chance to get lost” property. The above result implies that the set of aperiodic points for an Axiom A map f, or equivalently the set on which the dynamical behaviour of f is chaotic, has Lebesgue measure zero. A natural question to ask is the following: “Does the asymptotically periodic behaviour in the dynamics of a chaotic Axiom A map persist under a small perturbation?’ If we allow continuous per-
turbations then the answer is negative because the following argument. Fact: for each E > 0, for each chaotic Axiom A map f E C1( X, X) we can find a map g E C’( X, X) with the properties: (i) ]g(x) f(x) 1 -z E for every x E X, (ii) g has at least one absolutely continuous invariant measure. Such a map g can be “constructed” as follows: Let f E C’( X, X) be a chaotic Axiom A map, let E > 0 be given. We write x* for a fixed point of f in the interior of X with f ‘(x*) < - 1. (Such an unstable fixed point exists, since f is chaotic.) Fix a real number 6 with 0 < 6 < s/2, such that f’(x) # 0 for all x in the open interval lx* - 6, x* + 6[; pick a positive number 7) with 11~ 6. Let h be a mapping of class C3 from the interval [x * - r), x* + 71 into itself with the following properties: (i) h has a negative Schwarzian derivative (i.e. h”‘(x)/h’(x) - $[h”(x)/h’(x)]’ c 0 for each x with h’(x) f 0), (ii) h(x* - 7) = x* + 9, h(x* + q) = x* - 7, Ih’(x* - q)l > 1, Jh’(x* + v)I > 1, and (iii) h has two critical points say c1 and c2 (i.e. h’(q) = h’(cZ) = 0 and h’(x) # 0 for x # ci, x # c2) with x* - n < ci < c2 < x* + n, h(c,) = x* - q, h(cZ) = x* + 7). From a result due to Misiurewicz [23] it follows that h has an invariant measure absolutely continuous with respect to the Lebesgue measure. For any map gECl(X,X) with g(x)=f(x) for x E X\]x* -6, x* + S[, g(x) = h(x) for x E [x* - 9, x* + n] and g’(x) # 0 for x E]X* - 6, x* - q[ U]X* + 7, x* + 6[ we have: (1) (g(x) - f(x)1
< E for every x E X, and (2) g has an absolutely continuous invariant measure. W The persistence of the asymptotically periodic behaviour in the dynamics of a chaotic Axiom A map under smooth perturbations follow from the following result: Theorem 3.2. Let f be an Axiom A map from an open interval X into itself. Then there exists an open neighbourhood U of f in C’( X, X) such that for each map g E U we have: (1) g is an Axiom A map; and (2) there exists a homeomorphism $J: Q(g)-,Wf)
with~“gln(g)=flO(,)o~.
In other words, the above result says the following. A map g, which is obtained by a small smooth perturbation of an Axiom A map f, is again an Axiom A map whose nonwandering set has the same topological structure as the nonwandering set of the original map f. Hence, the qualitative behaviour of the dynamics of an Axiom A map doesn’t change under small smooth perturbations. Consequently, combining the two results above, we have that the asymptotically periodic behaviour in the dynamics of an Axiom A map persists under small C * perturbations. One can ask “How chaotic is chaos?’ In the foregoing we have already seen that there is order in the chaos (the set of aperiodic points has Lebesgue measure zero) for the class of nonsingular Axiom A maps. First, we will restrict our attention to the nonwandering set. The following question arises: What can be said about the structure of the nonwandering set of an Axiom A map? Recall that the nonwandering set of an Axiom A map f is equal to the closure of the set of periodic points for f. We state the following fact: The subset In,( f ) of the nonwandering set of an Axiom A map f (i.e. the maximally positively f-invariant subset of the nonwandering set on which f is contracting) is equal to the set of asymptotically stable periodic points for f. Similar as Smale did for the Axiom A diffeomorphisms, we will present the “Spectral decomposition theorem” for Axiom A mappings.
H. E. Nurse/ Persistence of order and structure in chaos
Theorem 3.3. Let f be an Axiom A mapping from a nontrivial interval into itself. The nonwandering set of f can be decomposed into finitely many positively f-invariant sets, called basic sets, and f is one-sided topologically transitive (i.e. f has a dense orbit) on each of these basic sets.
We know that the periodic points are contained in the nonwandering set. If a basic set of the nonwandering set doesn’t contain an aperiodic point, then it consists of one single periodic orbit. On the other hand, if a basic set of the nonwandering set contains an aperiodic point, then it contains many periodic points. So we can ask: Is’ it possible to estimate the number of periodic points with period n, for any positive integer n? We will see that we can compute the number of periodic points with period n for an Axiom A map exactly. In order to do so, we will introduce the following sets. For any map g E C’( X, X) we define: D,(g) is the union of the direct domains of attraction of the asymptotically stable periodic points for g and the absorbing boundary points of X for g (i.e. union of intervals containing either an asymptotically stable periodic point for g in the interior or an absorbing boundary point of X for g in the boundary), A,(g) is the complement of D,(g) in X; by induction, for each positive integer k, Ak+Jg) = ix E A,(g); g“(x) E A,(g)) and D,(g) = {x EA,(g); gk(x) E D,(g)}. Further we
define A,(g) = fl r_iAk( g). We will refer to the sets Dk(g) and A,(g) as filtering sets: For each positive integer k we have A,, i( g) = {x E A,(g); g’(x) EAl(g); 0 sjs k} and D,(g) = {x E A,(g); g’(x) 4 D,(g), 0 rj s k - 1, g“(x) E D&)1, hence A,(g) = Ak+i(g) U D,(g), Ak+i(g) n D,(g) = 0 From now on we will assume that f E C’( X, X) is an Axiom A map for which the following properties hold: (i) A,(f) is compact; (ii) f is an expanding map on A,(f) (this implies that no critical point c of f (i.e. f’(c) = 0) can eventually mapped into 52,( f )).‘From the fact: there exist positive integers N and M such that 1(f N)‘( x) 1 >
381
1 for all xEAM, it follows that all the periodic points in A,(f) are unstable. Let D,* c D,(f) be a closed set in X with the following properties: (1) D,* has a nonempty intersection with each component of D,( f ); (2) the asymptotically stable periodic points for f are contained in the interior of D,*, the absorbing boundary points of X for f are contained in the boundary of D& (3)D,* is mapped into its interior by applying the map f 7 (4) if a critical point c of f satisfies f”(c) is in D,,(f) for some nonnegative integer n, then f”(c) is contained in the interior of D,*; (5) the number of components of the set D$ is equal to the number of components of the set Ddf
)-
The minimum of the nonnegative integers n for which f”(c) is contained in D,,( f ) for each critical point c off, is a well-defined number; call it m. In other words, the set of critical points of f is contained in &&Dk( f) and D,,,(f) contains at least one critical point of f. We define B = {x E Ur_,,DJf); f”(x)ED$} i.e. B is the set of points which are mapped into D,* under f m, and further we denote by C the complement of B in X. Then we have that the set C is open in X and for which the following properties hold: (1) C is a negatively f-invariant set i.e. f-‘(C) c C, (2) the restriction of f to f-‘(C) is locally a homeomorphism; (3) f-‘(C) has finitely many components. We denote by L the number of disjoint components of the set C; further we write C= U { Ik; l=
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sume that the set of points A,(f), whose orbits don’t converge to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f, is a compact set, and f is expanding on this set. Then we have: The number of unstable periodic points for f with period n is equal to the trace of the matrix [AC; f]n (with A,,, as introduced above) for each positive integer n. Following Artin and Mazur [3], the zeta function {, of f is defined by
S,(z) =exp
i
O” NPer(f; c n n-1
n)> Zn i
with z a complex variable and N(Per (f; n)) the number of periodic points for f with period n. From the decomposition theorem we know that there are at most finitely many asymptotically stable periodic points. We also note that if f has no asymptotically stable periodic points, then there exists at least one (but at most two, of course) absorbing boundary point of X for f. Theorem 3.4.2. Let f be as in 3.4.1. Then, for some well-defined square matrix A, with entries either one or zero, we have: (1) the number of periodic points for f with period n is equal to the trace of the matrix [A, 1” for each positive integer n. (2) the zeta function {, of f is a rational function with the property that [, has a pole in z = X-’ of order the multiplicity of A where A is an eigenvalue of the matrix A,. A quantity, which is assumed to be a measure for the complicated dynamics, is topological entropy. For completeness, we give a definition of toposee Adler, Konheim and logical entropy, McAndrew [l]. Let Y be a compact topological space. Assume that F: Y + Y is a continuous mapping. The join V ,y__l91i of any finite collection of open covers 91i of Y, 15 i s n, is the open
in chaos
cover consisting of all sets of the form fli”,lAi, with A,E%~, lsisn. If 8 is an open cover of Y, let N( 9l) denote the number of sets in a finite subcover of % with smallest cardinality. Then the topological entropy of F, denoted by h(F), is given by
where the supremum is taken over all open covers 8 of Y. Theorem 3.4.3. Let f be as in 3.4.1. Then we have: (1) the topological entropy of f is equal to the logarithm of the inverse of the radius of convergence of the zeta function of f. (2) the topological entropy of f is equal to the logarithm of the spectral radius of the matrix A, (which is equal to the spectral radius of the matrix A,,,), with A, as in 3.4.2. Summarizing, we have obtained the result that the phenomenon “order and structure in chaos” in the dynamics of a chaotic nonsingular Axiom A map is persistent under small smooth perturbations. In fact, this result is a kind of stability property. But what about structural stability? A mapping g E C“( X, X) is called C“-structurally stable if there exists an open neighbourhood U of g in C“( X, X) such that each mapping h in U is topologically conjugate with g (i.e. there exists a homeomorphisms +: X + X such that g 0 + = + 0 h), with k any nonnegative integer. It is obvious that there exist Axiom A maps of class Ck which are not Ck-structurally stable for any nonnegative integer k. For example, assume that f~ Ck( X, X) is an Axiom A mapping such that at least one of the critical points of f is periodic, then f is not Ck-structurally stable, with k 2 1. We will consider a subset of the class of Axiom A mappings satisfying “reasonable” conditions. It will turn out that these mappings are C 2-structurally stable.
H. E. Nusse/ Persistence of order and structure in chaos
Theorem 3.5. Let f be an Axiom A mapping, of class C2, from an open interval X into itself. Assume that f has the following properties: (i) f has N critical points for some fixed positive integer Iv, (ii) each critical point of f is nondegenerate, and the critical values are all different; (iii) for each critical point c of f, the intersection of the set consisting of the forward iterates of c with the set consisting of the union of the critical points of f and the nonwandering points of f, is empty; (iv) the set of points whose orbits don’t converge to an asymptotically stable periodic orbit of f or to an (the) absorbing boundary point(s) of X for f, is compact. Then we have: f is C2-structurally stable. Let’s end this section by stating some remarks. Remark 3.1. The condition Axiom A(a) implies Axiom A(b), consequently if a map f satisfies condition (a) of the conditions for an Axiom A map, then f is an Axiom A map. Hence, the condition (b) in the definition for Axiom A map could be skipped. Remark 3.2. If an Axiom A map f has finitely many critical points, then f is nonsingular. Remark 3.3. The matrix A, in theorem 3.4.2 is associated with the concept of Markov partition of a map; the definition of Markov partition was inspired by the work of Bowen [6] on Axiom A diffeomorphisms. The matrix A,; f in theorem 3.4.1 is not necessarily a matrix whose entries are either zero or one. Remark 3.4. Let the map f be as in theorem 3.4.1. In [26] p. 207 we obtained the result that the restriction of f to A,(f) is topologically conjugate to a one-sided shift of finite type, namely the one-sided shift induced by the matrix A,- ICcj;,. Using this’ fact and recalling that G(f) has at most finitely many points outside A,(f), the ob-
383
tained results in theorem 3.4 would also follow from some established results concerning the calculation of topological entropy and the zeta function from the transition matrix A,-I~~,;,, see e.g. Bowen and Lanford [5], Adler Konheim and McAndrew [l], and Walters [36] for these results for one-sided shifts of finite type. Remark 3.5. It also has been proved in [26] that the restriction of an Axiom A map f to the subset a,(f) of its nonwandering set is topologically conjugate with a one-sided subshift of finite type. Using the results for one-sided shifts of finite type, see remark 3.4, and recalling that a(f) has at most finitely many points outside Q,(f) the same conclusions as in theorem 3.4 hold for f.
4. Schwarzian derivative and Axiom A Singer [29] introduced the Schwarzian derivative for real valued mappings; see also Allwright [2]. For any mapping I, of class C3, the Schwarzian derivative of f at a point x with f’(x) # 0, denoted by Sf(x), is defined by f”‘( x)/f’(x) - t[f”(x)/f’(~)]~. Singer showed that the class of mappings with negative Schwarzian derivative is closed under composition; consequently any iterate of a mapping with negative Schwarzian derivative, has a negative Schwarzian derivative. Singer’s main result can be formulated as follows (see also Misiurewicz [22]): “Let f be any mapping, of class C 3, from the unit interval [O,l] into itself for which the following conditions hold: (i) f has finitely many critical points, and (ii) f has a negative Schwarzian derivative. Assume that f has an asymptotically stable periodic point q, with primitive (i.e. smallest) period p, such that the direct domain of attraction of f’(q), 0 4 i $ p - 1, doesn’t contain a boundary point of the unit interval. Then there is a critical point c of f such that the orbit of c converges to the orbit of q.” Consequently, for the well-studied one-parameter family of maps {f,} with f, defined by f,(x)
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= a~(1 - x) on the unit interval and parameter a in the interval [2, 41, the condition (i) “f, has an asymptotically stable periodic orbit” and (ii) “ the orbit of the critical point of f, converges to an asymptotically stable periodic orbit of f,” are equivalent. Now we will state our result that a map with negative Schwarrian derivative is an Axiom A map, provided that some reasonable conditions are satisfied. Theorem 4.1. Let f be a chaotic C3-mapping from a nontrivial interval X into itself. Assume that f satisfies that following conditions: (i) f has a negative Schwarzian derivative; (ii) the set A,(f) of points, whose orbits don’t converge to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f, has the following properties: (a) A,(f) is a compact set and it is contained in the interior of X, (b) A,(f) doesn’t contain critical points of f; (iii) f is contracting on the set of asymptotically stable periodic points for f, provided that this set is nonempty. Then we have: f is a nonsingular Axiom A mapping. We note that the condition “A,(f) doesn’t contain critical points of f ” is equivalent with the condition “the orbit of each critical point of f converges to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f “. From this condition it follows that j cannot have so called one-sided asymptotically stable periodic points. In section 2.1 we gave some examples of quadratic maps and cubic maps which satisfy the conditions of theorem 4.1. Under suitable assumptions on general polynomial maps, we will see that they satisfy the conditions of the above theorem. Theorem 4.2. Assume that f: R + W is a polynomial mapping satisfying the conditions: (i) each critical point of f is real; (ii) the orbit of each critical point of f converges to an asymptotically stable periodic orbit of f or to a subset of the
possible present absorbing boundary of X for f; (iii) f is contracting on the set of asymptotically stable periodic points for f, provided that this set is nonempty. Then f satisfies the conditions of theorem 4.1. Consequently, j is a nonsingular Axiom A mapping.
From now on (in this section) let f E C3( X, X) be fixed, and assume that f satisfies the conditions of theorem 4.1. Then, for clarity of exposition, we will enumerate some properties. Property 4.1. Almost every point in X, in the sense of Lebesgue measure, is asymptotically periodic, provided that f(X) is bounded. Property 4.2. The nonwandering set of f decomposed into finitely many basic sets.
can be
Property 4.3. The map f is Cl+“-persistent for some a! > 0, i.e. the asymptotically periodic behaviour in the dynamics of f and the structure of the nonwandering set don’t change under small smooth perturbations. Property 4.4. There exists a well-defined square matrix A,, with entries zero or one, for which we have: (a) the number of periodic points for j with period n is equal to the trace of the matrix [A,]“, for each positive integer n; and (b) the topological entropy of f is equal to the logarithm of the spectral radius of A,. Property 4.5. The map f is C2-structurally stable, provided that some additional conditions are satisfied.
We will conclude this section by making a remark concerning the advantage of the Axiom A condition for the stability properties of maps, e.g. persistence of asymptotically periodic behaviour in the complicated dynamics of chaotic maps and structural stability.
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Persistence of order and structure in chaos
Remark 4.1. Let g E C3(X, X) be C’-close to f with f as in theorem 4.1. Then we know by the theorems 4.1 and 3.2 that g is an Axiom A map which has the “same” qualitative behaviour in the dynamics as f has. On the other hand, the Schwarzian derivative of g is not a priori negative; still worse, the Schwarzian derivative of g may be positive on some suitable (small) intervals! This phenomenon also may occur, even if f and g are topologically conjugate. Remark 4.2. Let f satisfy the conditions of theorem 4.1. Assume that f has one critical point c, and that the interval is compact. It is no restriction to assume that f is increasing on {x E X, x _I c}, and f is decreasing on { x E X, x 2 c }. Since f can have an asymptotically stable fixed point in { x E X; x < f2( c)}, see e.g. Collet and Eckmann [9] p. 95, the following three cases can occur: (a) f has an asymptotically stable fixed point in {x E X; x
This paper is based on the results of the author’s thesis [26] at the University of Utrecht supervised by Hans Duistermaat. I would like to
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thank him for the stimulating discussions and critical comments. It is a pleasure to thank Wilma van Nieuwamerongen for the excellent typing of this manuscript. The referee’s comments are gratefully acknowledged.
References [l] R.L. Adler, A.G. Konheim and M.H. McAndrew, Topological entropy, Trans. Amer. Math. Sot. 114 (1965) 309-319. [2] D.J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34 (1978) 687-691. [3] M. Artin and B. Mazur, On periodic points, Ann. Math. 81 (1965) 82-99. [4] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Sot. 254 (1979) 391-398. [5] R. Bowen and O.E. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Symp. Pure Math. Vol 14; 43-49 (Amer. Math. Sot., Providence, 1970). [6] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970) 725-747. [7] H. Brolin, Invariant sets under iteration of rational functions, Arkiv. Mat. 6 (1965) 103-144. [8] T.W. Chaundy and E. Phillips, The convergence of sequences defined by quadratic recurrence-formulae, Quart. J. Math. Oxford 7 (1936) 74-80. [9] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhiiuser, Boston, 1980). [lo] J. Coste, Iterations of transformations of the unit interval: approach to an attractor, J. Stat. Phys. 23 (1980) 521-536. [ll] J. Guckenheimer, Sensitivity dependence to initial cdnditions for one dimensional maps, Comm. Math. Phys. 70 (1979) 133-160. [12] I. Gumowski and C. Mira, Dynamique Chaotique (Cepadues Editions, Toulouse, 1980). [13] B.R. Henry, Escape from the unit interval under the transformation x -) Xx(1 - x), Proc. Amer. Math. Sot. 41 (1973) 1466150. [14] F.C. Hoppensteadt and J. Hyman, Periodic solutions of a logistic difference equation, SIAM J. Appl. Math. 32 (1977) 73-81. 1151 M.V. Jakobson, On smooth mappings of the circle into itself. Math USSR Sbomik 14 (1971) 161-185. M.V. Jakobson, Absolutely continuous invariant measures [W for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981) 39-88. P71 L. Jonker and D. Rand, Bifurcations in one dimension I: The non-wandering set, Invent. Math. 62 (1981) 347-365. 1181T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 II (1975) 985-992. 1191 E.N. Lorenz, Deterministic nonperiodic flow, J. Atm. Sc. 20 (1963) 130-141. WI R.M. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459-467.
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[21] N. Metropolis, M.L. Stein and P.R. Stein, On finite limit
sets for transformations of the unit interval, J. Combin. Theory A 15 (1973) 25-44. [22] M. Misiurewia, Structure of mappings of an interval with zero entropy, IHES Publ. 53 (1981) 5-16. [23] M. Misiurewia. Absolutely continuous measures for certain mappings of an interval, IHES Publ. 53 (1981) 17-51. (241 P.J. Myrberg, Iteration der reellen Polynome zweiten Grades III, Ann. Acad. SC. Fermicae 366 (1963) 1-18. [25] Z. Nitecki, Topological dynamics on the interval, in: Ergodic Theory and Dynamical Systems II, Progress in Math., vol. 21. (Birkhguser, Boston, 1982) pp. l-73. [26] H.E. Nusse, Chaos, yet no chance to get lost. Order and structure in the chaotic dynamical behaviour of one dimensional noninvertible Axiom A mappings, arising in discrete biological models. Thesis, State University at Utrecht 1983. (271 C. Preston, Iterates of Maps on an Interval, Lect. Notes Math. 999 (Springer, Berlin, 1983). [28] A.N. Sharkovsky, Coexistence of the cycles of a continuous mapping of the line into itself, Ukr. Mat. Z. 16 (1964) 61-71.
v91 D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978) 260-267. [301 S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Sot. 73 (1967) 747-817. [311 S. Smale and R.F. Williams, The qualitative analysis of a difference equation of population growth, .I. Math. Biol. 3 (1976) l-4. 1321 P. Stefan, A theorem of Sharkovsky on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977) 237-248. ]331 P.R. Stein and S. Ulam, Nonlinear transformation studies on electronic computers, Rozprawy Matematyczne 39 (1964) l-66. 1341 P.D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (1978) 99-105. ]351 S.J. van Strien, On the bifurcation creating horseshoes. Proc. Dynamical Systems and Turbulence, Warwick 1980; 316-351, Lect. Notes Math. 898 (Springer, Berlin, 1981). ]361 P. Walters, An introduction to ergodic theory, G.T.M. 79, (Springer, Berlin, 1982). 1371 Zentralblatt fur Mathematik 521 (1984) D58040.
PERSISTENCE
OF ORDER AND STRUCTURE
IN CHAOS
Received 26 January 1985 Revised manuscript received 18 November 1985
H.E. NUSSE University
of Groningen, Institute of Econometrics,
P.O. Box 800, 9700 A V Groningen, The Netherlands
In this paper some results are presented concerning one-dimensional chaotic maps with arbitrarily many critical points. Let f be a chaotic map belonging to some suitable class of C’ maps from a nontrivial interval X into itself. Assuming that f is of class C 1+a for some a > 0, we have that the set of aperiodic points for f has Lebesgue measure zero: further, if f(X) is bounded then there exists a positive integer p such that almost every point in the interval is asymptotically periodic with period p. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics off is persistent under small smooth perturbations. The topological structure of the nonwandering set of f will be described, and this structure is invariant under small C’ perturbations of the map f. Assuming that f is of class C2, the map f is C2 structurally stable provided that f satisfies some suitable conditions. Finally, it will turn out that maps with a negative Schwarzian derivative belong to the suitable class of maps mentioned
1. Introduction
The aim of this paper is to present the reformulated main results obtained in [26]. The proofs of the theorems, properties and facts can be found in the mentioned reference. It is a pleasure to recall the celebrated “Period three implies chaos” result due to Li and Yorke [18]: Fix some interval J and let F: J + J be a continuous mapping. Assume that there is a point ae.I such that b=F(a), c=F*(a) and d= F3(a) satisfy dsa
periodicpoint > 0.
qEJlim.,,sup]F”(p)-F”(q)]
F”
and any
Li and Yorke observed that the set S occurring above contains no asymptotically periodic points. Further they mentioned the question, whether the existence of an asymptotically stable periodic point implies that almost every point is asymptotically periodic. There exists a lot of numerical investigations dealing with iteration of mappings, see e.g. Stein and Ulam [33], Metropolis, Stem and Stein [21], Hoppensteadt and Hyman [14], Gumowski and Mira [12], and Coste [lo]. The numerical results of density functions suggest, for some examples, that almost every point in the interval approach to an asymptotically stable periodic orbit. Many established results in the study of iteration of mappings initiated by Lorenz [19], Li and Yorke [18] and May [20] can be found in the monographs by Collet and Eckmann [9] and Preston [27], and in the article by Nitecki [25]. The question posed by Li and Yorke has been answered affirmatively in a rather general manner
0167-2789/86/$03.50 Q Else& Science Publishers B.V. (North-Holland Physics Publishing Division)
H. E. Nose/
Persistence of order and structure in chaos
by the “No chance to get lost” theorem in section 3. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics of a chaotic map f, belonging to some suitable class of maps, is persistent under small smooth perturbations. We note that such a map f may have many critical points (i.e. points at which the derivative of f vanishes). Besides we give a description of the topological structure of the nonwandering set of a chaotic map f, we also consider the set of periodic points for f. The question: “Given that a continuous mapping f from the real line into itself has a periodic orbit with period n, for some fixed positive integer n, which other positive integers must occur as periods of the periodic orbits of f ?”has been discussed and has been answered by Li and Yorke [18], Sharkovsky [28], Stefan [32], Strafhn [34] and Block [4]. The main result is due to Sharkovsky, see Stefan [32]. Let the positive integers lexicographically ordered in the following way: 3 i 5 -I 7 -I . . . -I 2 .3i2.5-12.7-I +.. -I2*~3-t2*.5-12*.7-i . - . -I 23 -I 2* i 2 i 1. Sharkovsky’s theorem says the following. Let f be a continuous mapping from W into itself, which has a periodic orbit of period n. Then f has a periodic orbit of period m for every m E N for which n -I m holds. One may ask the following question: “Given any continuous mapping f from a nontrivial interval into itself. Fix any positive integer n. How many periodic points with period n does f have?“. Smale and Williams [31] answered this question for the mapping f: [0, l] + [0, l] defined by f(x) = 3.83x(1 - x). We will answer this question for a map f, which may have many critical points, belonging to some suitable class of functions. It holds that the structure of the nonwandering set of such a map f, and also the number of periodic points with period n, n E N, for f doesn’t change under small smooth perturbations. The condition negative Schwarzian derivative is invariant under composition, but it is not invariant under conjugation with a diffeomorphism. Conse-
375
quently, this condition is not persistent under smooth perturbations. Therefore we looked for a new condition. We refer the reader to section two for some preliminary definitions. Inspired by Smale’s definition of the Axiom A property for diffeomorphisms [30], we will introduce the Axiom A property for one-dimensional noninvertible mappings. To give an idea of the Axiom A property (the precise definition can be found in section three) we consider the following. Let f be a differentiable map from a non-trivial interval X into itself. Assume that the closure of the set of periodic points for f can be split up into a set S consisting of at most finitely many asymptotically stable periodic points, and a compact positively f-invariant set U (i.e. f(U) c U) with I(f”)‘(x)l >lforeachx~U,foreveryn>N,for some fixed NE N. (Note that the set S is a compact positively f-invariant set, and the intersection of S with U is empty; further S may be empty.) Then: f is an Axiom A mapping. For clarity, we will restrict our attention for a moment to a chaotic polynomial map P: X+ X from a bounded real interval X into itself. Assume that P satisfies the following conditions: (1) each critical point of P is real; (2) the orbit of each critical point of P converges to an asymptotically stable periodic orbit of P or to a subset of the possible present absorbing boundary for P; (3) P is contracting on the set of asymptotically stable periodic points for P, provided that this set is nonempty. Then we have: P is a nonsingular Axiom A map. The obtained results for Axiom A maps imply that for such a polynomial map P the following properties hold: (i) almost every point in X, in the sense of periodic Lebesgue measure, is asymptotically for P. (ii) the nonwandering set of P can be decomposed into finitely many basic sets and P has a dense orbit on each of these basic sets. (iii) the map P is C’+a-persistent for any (Y> 0, i.e. the asymptotically periodic behaviour in the
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dynamics of P and the structure of the nonwandering set don’t change under small smooth perturbations. (iv) the number of periodic points for P with period n can be computed explicitly, for each positive integer n. (v) the map P is C2-structurally stable, provided that some additional conditions are satisfied (P is C 2 structurally stable means that there exists some neighbourhood U of P in C2( X, X) such that g and P are topologically conjugate for each g E U). The organization of this paper is as follows. In section 2.1 we give some simple examples of Axiom A maps and we state their properties; in section 2.2 we give some preliminary definitions. In section 3 we will present the results for Axiom A maps. Let f~ C’( X, X) be a fixed chaotic nonsingular Axiom A mapping from a nontrivial interval X into itself. The most interesting results are the following. There exists a positive integer p such that almost every point in X, in the sense of Lebesgue measure, is asymptotically periodic with period p; this phenomenon concerning the asymptotically periodic behaviour in the complicated dynamics of f doesn’t change under small smooth perturbations. The nonwandering set of f (i.e. the closure of the set of periodic points) can be decomposed into finitely many compact positively f-invariant subsets, and f has a dense orbit in each of these basic sets; this structure of the nonwandering set is invariant under small smooth perturbations. The topological entropy of f can be determined, further the number of periodic points for f with period n can be computed for each positive integer n; Finally we will see that, under some additional conditions, f is C2-structurally stable. It is worthwhile to note that the Axiom A property is invariant under conjugation with a diffeomorphism, and it is also invariant under small smooth perturbations. In section 4 we consider maps with a negative Schwarzian derivative. We note that this condition
is not invariant under conjugation with a diffeomorphism. It will turn out, that these maps are, under reasonable conditions, Axiom A maps. In particular, the well-studied maps with one critical point and negative Schwarzian derivative are Axiom A maps, provided that the orbit of the critical point converges to an asymptotically stable periodic orbit.
Remarks
There are several published results related to the obtained results presented in this paper. (1) The following results are related to theorem 3c: Jonker and Rand [17] studied mappings of a compact interval into itself with one critical point. Their main result concerns a canonical decomposition of the nonwandering set in terms of the iterated topological entropy of the map in question. The proof of this result is based on the kneading theory developed by Milnor and Thurston. Van Strien [35] obtained for a subclass of the mappings studied by Jonker and Rand, namely those mappings with a negative Schwarzian derivative that the sets in the decomposition have, except at most one set, a hyperbolic structure. (2) The following reformulated result due to Guckenheimer [ll], Misiurewicz [23] and Van Strien [35], see also Collet and Eckmann [9] and Preston [27], is a special case of theorem 4a. Let f be a chaotic C3-mapping from a compact interval [a, b] into itself. Assume that f satisfies the following conditions: (1) f has one critical point c which is nondegenerate, f is increasing on [a, c] and f is decreasing on [c, b]; (2) f(a) = f(b) = a; (3) f has a negative Schwarzian derivative (i.e. f”‘(x)/f ‘(x) - ~[f”
H. E. Nusse/ Persistence of order and structure in chaas
371
sure, see Misiurewicz [23] and Jakobson [16], then
f is no Axiom A map. (4) After writing the paper the author learned from the review by Keller [37] that results due to Jakobson [15] are related to the stability results in this paper. 2. Preliminaries 2.1. Some very simple examples Example 1.1. Fix any real number A > 4. Let the map f: R --)R be defined by f(x) = hx(1 - x). Then: - f is an Axiom A mapping. -The set of points, whose orbits don’t converge to the absorbing boundary point {-cc}, has Lebesgue measure zero, see also Chaundy and Phillips [S], Brolin [7] and Henry [13]. In fact, this set of points whose orbits will stay in the unit interval forever, is the nonwandering set. - The nonwandering set of f consists of one basic set, see also Jonker and Rand [17]. -The qualitative behaviour for the dynamics of f doesn’t change under small smooth perturbations of f, the same holds for the structure of the nonwandering set. -f is C*-structurally stable. It follows from this, that if we vary the parameter a little bit, then the qualitative properties don’t change. Example 1.2. Let f: [0, l] + [0, l] be defined by f(x) = 3.83x(1 - x). This example has already attracted some fame. Smale and Williams [31] showed the existence of an asymptotically stable periodic orbit with period 3. The graph of f3 is giveninfig.l.WewriteO=x,
00
X-3 Fig. 1. The graph of f3, for f(x) =3.83x(1 -x).
Singer [21]: f has precisely one asymptotically stable periodic orbit. The direct domain of attraction of the asymptotically stable periodic orbit of period three is the union of the intervals
1.~~~ ~2Wl~4~xWIX~~ Y& whh y2 = mm {x El% x21; f 3(x) = x2}, y4= ma {x El+, x4L f3(x)=x4} and y,=min{xE]xg,l[; x6 }. See the figure.
f3(x)=
We have: -f is an Axiom A mapping. -Almost every point in the unit interval (in the sense of Lebesgue measure) is asymptotically periodic with period 3, see also Guckenheimer [ll] and Van Strien [35]. -The nonwandering set of f consists of three basic sets, see also Jonker and Rand [17]. -The asymptotically periodic behaviour in the complicated dynamics of f doesn’t change under small smooth perturbations of f; the same holds for the structure of the nonwandering set. -f is C*-structurally stable. It follows from this fact that a small change in the parameter h = 3.83 has no influence on the qualitative behaviour of the dynamics. Example 1.3. Let f: [0, l] + [0, l] be defined by f has f(x) = X,x(1 -x)with3
stable periodic orbit with small-
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est period p, and (fP)‘(x,) # - 1, (fP)‘(x,) # 0 with x,, an asymptotically stable periodic point for f. Then: -f is an Axiom A mapping. -Almost every point in the unit interval is asymptotically periodic with period p, see also Guckenheimer [ll] and Van Strien [35]; this behaviour in the dynamics doesn’t change under small smooth perturbations of f. -The nonwandering set of f consists of at least three basic sets, see also Jonker and Rand [17]; the structure of the nonwandering set is invariant under small smooth perturbations of f. -f is C*-structurally stable; consequently the qualitative behaviour of the dynamics are stable under small variations of the parameter X,. (Note that f satisfies condition (iii) of theorem 3e because the critical point of f is in the direct domain of attraction of the asymptotically stable periodic orbit and the critical point is assumed to be not periodic because ( f p)‘( f) = 0). Example 2.1. Fix any real number X > 4. The map f: W + W defined by f(x) = Ax3 + (1 - X)x
has the following properties: -f is an Axiom A map. -The orbit of almost every point on the real axis converges to either the absorbing point - cc or to the absorbing point + cc. -The nonwandering set of f consists of one basic set. -f is C1+a -persistent for some positive real number OL,i.e. the qualitative behaviour of the dynamics as well as the structure of the nonwandering set doesn’t change under small smooth perturbations of f. -f is C*-structurally stable. Example 2.2. Define the map f: [ - 1, l] + [ - 1, l] by f(x) = 3.701x3 - 2.701x. The map f has two asymptotically stable periodic orbits with period three. We have: -f is an Axiom A map. - Almost every point in [ - 1, l] is asymptotically periodic with period three.
-The nonwandering set of f consists of five basic sets namely the isolated fixed points - 1, + 1, the two asymptotically stable periodic orbits and the rest of the nonwandering set. -f is Cl+“-persistent for some positive real number (Y. -f is C*-structurally stable. Example
2.3. Let f: [ - 1, l] --) [ - 1, l] be defined
by f(x) = X,x3 + (1 - X,)x with 2 < A, < 4 such that f has an asymptotically stable periodic orbit with smallest period p for which (f J’)‘(x,) f - 1 and ( f P)‘( xp) # 0, with xP an asymptotically stable periodic point for f. If p is odd, then f has two asymptotically stable periodic orbits; if p is even, then f has one or f has two asymptotically stable periodic orbits, both cases can occur. Then we have: -f is an Axiom A map. -Almost every point in the interval [ - l,l] is asymptotically periodic with period p. -The nonwandering set consists of at least four basic sets. -f is Cl +“-persistent, for some positive real number (Y. -f is C*-structurally stable. 2.2. Some preliminary dejinitions and notations Throughout the paper XC W will be a nontrivial interval. Let D and Y be nonempty Lebesgue measurable subsets of X such that D C Y. We denote the complement of D in Y by Y\D. We write p(Y) for the Lebesgue measure of Y. Let f: X + X be a differentiable (noninvertible) mapping. For any positive integer n, the n th iterate of f, denoted by f “, is inductively defined by f”=f Ofn-1, with f” the identity mapping. For each point x E X, the orbit of x under f is the set { f O(x); n E N u (0)). The set f”(D) is defined x E D}, and the set f-“(D) byf”(D)= {f”(x); is defined by f-“(D)={xEX, f”(x)ED} for each n E N U (0). The set D is called positiuely f-invariant if f(D) c D, D is called negatiuely f-inuariant if f-‘(D)
H. E. Nuwe/ Persistence of order and structure in chaos
c D. The set D is a component of Y if it is a
maximally connected subset of Y. A component is called trioial if it consists of one point. A point x E X is called a periodic point for f with period p if f r(x) = x for some p E N. A periodic point x for f with period p is called asymptotically stable if there exists an open neighbourhood U of x in X such that lim n _ m f “r( y) =x for all y E U. A point x E X is called an asymptotically periodic point for f if lim n _oo f”“(x) exists for some m E N; a point x E X is called an aperiodic point for f if it is not an asymptotically periodic point and if the orbit of x is bounded. A point x E X is called a nonwandering point for f, if for every neighbourhood U of x there exists n E N such that f”(U) n U # 0. A point x E X is called a criticalpoint for f if
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property if the following two conditions are satisfied: (a) the nonwandering set P(g) is hyperbolic; (b) the periodic points of g are dense in
{y~X;lim,,,f”r(y)=q}iscalledthedomain
O(g). For a continuous one-dimensional (noninvertible) mapping g the set of periodic points for g and the nonwandering set of g are positively ginvariant sets, but they are not necessarily negatively g-invariant sets. Now we will formulate the Axiom A property for mappings from a nontrivial interval X into itself. A mapping g E C’( X, X) is called an Axiom A mapping if the following conditions are satisfied: (a) the nonwandering set s2(g) can be split up into at most two, but at least one, nonempty positively g-invariant compact subsets say O(g) = In,(g) U Q,(g) such that g is contracting on G,(g) provided that 9,(g) is nonempty, and g is expanding on 6?,(g) provided that an,(g) is nonempty; (b) the periodic points for g are dense in G(g). (g is contracting on Q,(g) (expanding on L?,(g)) means there are constants C > 0 and 0 < K < 1 such that ](g”)‘(x)] 6 CK” for each x E i2n,(g)(l(g”);(x)l 2 C[l/K]” for each x E Q,(g)) for every n E N.) We note that the Axiom A property for onedimensional noninvertible maps is invariant under conjugation with a diff’morphism (i.e. if g E C’( X, X) is an Axiom A mapping and h is a diffeomorphism defined on X, then the mapping h 0 g 0 h-’ is an Axiom A map). First, we state the result that an Axiom A map has the so called “No chance to get lost” property. Let (Ybe a positive real number.
of attraction of q. The component of the domain of attraction of q containing q as a boundary point, is called the direct domain of attraction of q.
Theorem 3.1. Let f E C1+a(X, X) be a nonsingu-
f’(x)
= 0.
The map f is called chaotic, if there exists at least one aperiodic point. The map f is called nonsingular if p(E) = 0 implies p( f-‘( E)) = 0 with E measurable subset of X. Let q E X be an asymptotically stable periodic point with period p. The domain of attraction of q is the set of points {x=X, lim,,, f”r(x)=q}. The direct domain of attraction of q is the component of the domain of attraction of q containing q. A point x, x e X, is called an absorbing boundaty point of X for f with period p, if there exists an open set U in X such that lim, _ m f “r( y) = x for each y E U. Assume that q is an absorbing boundary point of X for f with period p. The set
3. Main results For clarity of exposition we recall Smale’s Axiom A property for diffeomorphisms [30]: Let M be a compact manifold and let g: M -+ M be a diffeomorphism. The mapping g satisfies the Axiom A
lar Axiom A mapping from a nontrivial interval X into itself. Then we have: The set of points, whose orbits don’t converge to an asymptotically stable periodic orbit of f (or to the possible present absorbing boundary of X), has Lebesgue measure zero. Furthermore, there is a positive integer p such that almost every point x in X is asymptotically periodic with period p, provided that f(X) is bounded.
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It is due to the phenomenon that the orbits of almost all x in X in the sense of Lebesgue measure, converge to an asymptotically stable periodic orbit of f, or to an (or to the) possible present absorbing boundary point(s) of X for f, that we say that f has the “No chance to get lost” property. The above result implies that the set of aperiodic points for an Axiom A map f, or equivalently the set on which the dynamical behaviour of f is chaotic, has Lebesgue measure zero. A natural question to ask is the following: “Does the asymptotically periodic behaviour in the dynamics of a chaotic Axiom A map persist under a small perturbation?’ If we allow continuous per-
turbations then the answer is negative because the following argument. Fact: for each E > 0, for each chaotic Axiom A map f E C1( X, X) we can find a map g E C’( X, X) with the properties: (i) ]g(x) f(x) 1 -z E for every x E X, (ii) g has at least one absolutely continuous invariant measure. Such a map g can be “constructed” as follows: Let f E C’( X, X) be a chaotic Axiom A map, let E > 0 be given. We write x* for a fixed point of f in the interior of X with f ‘(x*) < - 1. (Such an unstable fixed point exists, since f is chaotic.) Fix a real number 6 with 0 < 6 < s/2, such that f’(x) # 0 for all x in the open interval lx* - 6, x* + 6[; pick a positive number 7) with 11~ 6. Let h be a mapping of class C3 from the interval [x * - r), x* + 71 into itself with the following properties: (i) h has a negative Schwarzian derivative (i.e. h”‘(x)/h’(x) - $[h”(x)/h’(x)]’ c 0 for each x with h’(x) f 0), (ii) h(x* - 7) = x* + 9, h(x* + q) = x* - 7, Ih’(x* - q)l > 1, Jh’(x* + v)I > 1, and (iii) h has two critical points say c1 and c2 (i.e. h’(q) = h’(cZ) = 0 and h’(x) # 0 for x # ci, x # c2) with x* - n < ci < c2 < x* + n, h(c,) = x* - q, h(cZ) = x* + 7). From a result due to Misiurewicz [23] it follows that h has an invariant measure absolutely continuous with respect to the Lebesgue measure. For any map gECl(X,X) with g(x)=f(x) for x E X\]x* -6, x* + S[, g(x) = h(x) for x E [x* - 9, x* + n] and g’(x) # 0 for x E]X* - 6, x* - q[ U]X* + 7, x* + 6[ we have: (1) (g(x) - f(x)1
< E for every x E X, and (2) g has an absolutely continuous invariant measure. W The persistence of the asymptotically periodic behaviour in the dynamics of a chaotic Axiom A map under smooth perturbations follow from the following result: Theorem 3.2. Let f be an Axiom A map from an open interval X into itself. Then there exists an open neighbourhood U of f in C’( X, X) such that for each map g E U we have: (1) g is an Axiom A map; and (2) there exists a homeomorphism $J: Q(g)-,Wf)
with~“gln(g)=flO(,)o~.
In other words, the above result says the following. A map g, which is obtained by a small smooth perturbation of an Axiom A map f, is again an Axiom A map whose nonwandering set has the same topological structure as the nonwandering set of the original map f. Hence, the qualitative behaviour of the dynamics of an Axiom A map doesn’t change under small smooth perturbations. Consequently, combining the two results above, we have that the asymptotically periodic behaviour in the dynamics of an Axiom A map persists under small C * perturbations. One can ask “How chaotic is chaos?’ In the foregoing we have already seen that there is order in the chaos (the set of aperiodic points has Lebesgue measure zero) for the class of nonsingular Axiom A maps. First, we will restrict our attention to the nonwandering set. The following question arises: What can be said about the structure of the nonwandering set of an Axiom A map? Recall that the nonwandering set of an Axiom A map f is equal to the closure of the set of periodic points for f. We state the following fact: The subset In,( f ) of the nonwandering set of an Axiom A map f (i.e. the maximally positively f-invariant subset of the nonwandering set on which f is contracting) is equal to the set of asymptotically stable periodic points for f. Similar as Smale did for the Axiom A diffeomorphisms, we will present the “Spectral decomposition theorem” for Axiom A mappings.
H. E. Nurse/ Persistence of order and structure in chaos
Theorem 3.3. Let f be an Axiom A mapping from a nontrivial interval into itself. The nonwandering set of f can be decomposed into finitely many positively f-invariant sets, called basic sets, and f is one-sided topologically transitive (i.e. f has a dense orbit) on each of these basic sets.
We know that the periodic points are contained in the nonwandering set. If a basic set of the nonwandering set doesn’t contain an aperiodic point, then it consists of one single periodic orbit. On the other hand, if a basic set of the nonwandering set contains an aperiodic point, then it contains many periodic points. So we can ask: Is’ it possible to estimate the number of periodic points with period n, for any positive integer n? We will see that we can compute the number of periodic points with period n for an Axiom A map exactly. In order to do so, we will introduce the following sets. For any map g E C’( X, X) we define: D,(g) is the union of the direct domains of attraction of the asymptotically stable periodic points for g and the absorbing boundary points of X for g (i.e. union of intervals containing either an asymptotically stable periodic point for g in the interior or an absorbing boundary point of X for g in the boundary), A,(g) is the complement of D,(g) in X; by induction, for each positive integer k, Ak+Jg) = ix E A,(g); g“(x) E A,(g)) and D,(g) = {x EA,(g); gk(x) E D,(g)}. Further we
define A,(g) = fl r_iAk( g). We will refer to the sets Dk(g) and A,(g) as filtering sets: For each positive integer k we have A,, i( g) = {x E A,(g); g’(x) EAl(g); 0 sjs k} and D,(g) = {x E A,(g); g’(x) 4 D,(g), 0 rj s k - 1, g“(x) E D&)1, hence A,(g) = Ak+i(g) U D,(g), Ak+i(g) n D,(g) = 0 From now on we will assume that f E C’( X, X) is an Axiom A map for which the following properties hold: (i) A,(f) is compact; (ii) f is an expanding map on A,(f) (this implies that no critical point c of f (i.e. f’(c) = 0) can eventually mapped into 52,( f )).‘From the fact: there exist positive integers N and M such that 1(f N)‘( x) 1 >
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1 for all xEAM, it follows that all the periodic points in A,(f) are unstable. Let D,* c D,(f) be a closed set in X with the following properties: (1) D,* has a nonempty intersection with each component of D,( f ); (2) the asymptotically stable periodic points for f are contained in the interior of D,*, the absorbing boundary points of X for f are contained in the boundary of D& (3)D,* is mapped into its interior by applying the map f 7 (4) if a critical point c of f satisfies f”(c) is in D,,(f) for some nonnegative integer n, then f”(c) is contained in the interior of D,*; (5) the number of components of the set D$ is equal to the number of components of the set Ddf
)-
The minimum of the nonnegative integers n for which f”(c) is contained in D,,( f ) for each critical point c off, is a well-defined number; call it m. In other words, the set of critical points of f is contained in &&Dk( f) and D,,,(f) contains at least one critical point of f. We define B = {x E Ur_,,DJf); f”(x)ED$} i.e. B is the set of points which are mapped into D,* under f m, and further we denote by C the complement of B in X. Then we have that the set C is open in X and for which the following properties hold: (1) C is a negatively f-invariant set i.e. f-‘(C) c C, (2) the restriction of f to f-‘(C) is locally a homeomorphism; (3) f-‘(C) has finitely many components. We denote by L the number of disjoint components of the set C; further we write C= U { Ik; l=
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sume that the set of points A,(f), whose orbits don’t converge to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f, is a compact set, and f is expanding on this set. Then we have: The number of unstable periodic points for f with period n is equal to the trace of the matrix [AC; f]n (with A,,, as introduced above) for each positive integer n. Following Artin and Mazur [3], the zeta function {, of f is defined by
S,(z) =exp
i
O” NPer(f; c n n-1
n)> Zn i
with z a complex variable and N(Per (f; n)) the number of periodic points for f with period n. From the decomposition theorem we know that there are at most finitely many asymptotically stable periodic points. We also note that if f has no asymptotically stable periodic points, then there exists at least one (but at most two, of course) absorbing boundary point of X for f. Theorem 3.4.2. Let f be as in 3.4.1. Then, for some well-defined square matrix A, with entries either one or zero, we have: (1) the number of periodic points for f with period n is equal to the trace of the matrix [A, 1” for each positive integer n. (2) the zeta function {, of f is a rational function with the property that [, has a pole in z = X-’ of order the multiplicity of A where A is an eigenvalue of the matrix A,. A quantity, which is assumed to be a measure for the complicated dynamics, is topological entropy. For completeness, we give a definition of toposee Adler, Konheim and logical entropy, McAndrew [l]. Let Y be a compact topological space. Assume that F: Y + Y is a continuous mapping. The join V ,y__l91i of any finite collection of open covers 91i of Y, 15 i s n, is the open
in chaos
cover consisting of all sets of the form fli”,lAi, with A,E%~, lsisn. If 8 is an open cover of Y, let N( 9l) denote the number of sets in a finite subcover of % with smallest cardinality. Then the topological entropy of F, denoted by h(F), is given by
where the supremum is taken over all open covers 8 of Y. Theorem 3.4.3. Let f be as in 3.4.1. Then we have: (1) the topological entropy of f is equal to the logarithm of the inverse of the radius of convergence of the zeta function of f. (2) the topological entropy of f is equal to the logarithm of the spectral radius of the matrix A, (which is equal to the spectral radius of the matrix A,,,), with A, as in 3.4.2. Summarizing, we have obtained the result that the phenomenon “order and structure in chaos” in the dynamics of a chaotic nonsingular Axiom A map is persistent under small smooth perturbations. In fact, this result is a kind of stability property. But what about structural stability? A mapping g E C“( X, X) is called C“-structurally stable if there exists an open neighbourhood U of g in C“( X, X) such that each mapping h in U is topologically conjugate with g (i.e. there exists a homeomorphisms +: X + X such that g 0 + = + 0 h), with k any nonnegative integer. It is obvious that there exist Axiom A maps of class Ck which are not Ck-structurally stable for any nonnegative integer k. For example, assume that f~ Ck( X, X) is an Axiom A mapping such that at least one of the critical points of f is periodic, then f is not Ck-structurally stable, with k 2 1. We will consider a subset of the class of Axiom A mappings satisfying “reasonable” conditions. It will turn out that these mappings are C 2-structurally stable.
H. E. Nusse/ Persistence of order and structure in chaos
Theorem 3.5. Let f be an Axiom A mapping, of class C2, from an open interval X into itself. Assume that f has the following properties: (i) f has N critical points for some fixed positive integer Iv, (ii) each critical point of f is nondegenerate, and the critical values are all different; (iii) for each critical point c of f, the intersection of the set consisting of the forward iterates of c with the set consisting of the union of the critical points of f and the nonwandering points of f, is empty; (iv) the set of points whose orbits don’t converge to an asymptotically stable periodic orbit of f or to an (the) absorbing boundary point(s) of X for f, is compact. Then we have: f is C2-structurally stable. Let’s end this section by stating some remarks. Remark 3.1. The condition Axiom A(a) implies Axiom A(b), consequently if a map f satisfies condition (a) of the conditions for an Axiom A map, then f is an Axiom A map. Hence, the condition (b) in the definition for Axiom A map could be skipped. Remark 3.2. If an Axiom A map f has finitely many critical points, then f is nonsingular. Remark 3.3. The matrix A, in theorem 3.4.2 is associated with the concept of Markov partition of a map; the definition of Markov partition was inspired by the work of Bowen [6] on Axiom A diffeomorphisms. The matrix A,; f in theorem 3.4.1 is not necessarily a matrix whose entries are either zero or one. Remark 3.4. Let the map f be as in theorem 3.4.1. In [26] p. 207 we obtained the result that the restriction of f to A,(f) is topologically conjugate to a one-sided shift of finite type, namely the one-sided shift induced by the matrix A,- ICcj;,. Using this’ fact and recalling that G(f) has at most finitely many points outside A,(f), the ob-
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tained results in theorem 3.4 would also follow from some established results concerning the calculation of topological entropy and the zeta function from the transition matrix A,-I~~,;,, see e.g. Bowen and Lanford [5], Adler Konheim and McAndrew [l], and Walters [36] for these results for one-sided shifts of finite type. Remark 3.5. It also has been proved in [26] that the restriction of an Axiom A map f to the subset a,(f) of its nonwandering set is topologically conjugate with a one-sided subshift of finite type. Using the results for one-sided shifts of finite type, see remark 3.4, and recalling that a(f) has at most finitely many points outside Q,(f) the same conclusions as in theorem 3.4 hold for f.
4. Schwarzian derivative and Axiom A Singer [29] introduced the Schwarzian derivative for real valued mappings; see also Allwright [2]. For any mapping I, of class C3, the Schwarzian derivative of f at a point x with f’(x) # 0, denoted by Sf(x), is defined by f”‘( x)/f’(x) - t[f”(x)/f’(~)]~. Singer showed that the class of mappings with negative Schwarzian derivative is closed under composition; consequently any iterate of a mapping with negative Schwarzian derivative, has a negative Schwarzian derivative. Singer’s main result can be formulated as follows (see also Misiurewicz [22]): “Let f be any mapping, of class C 3, from the unit interval [O,l] into itself for which the following conditions hold: (i) f has finitely many critical points, and (ii) f has a negative Schwarzian derivative. Assume that f has an asymptotically stable periodic point q, with primitive (i.e. smallest) period p, such that the direct domain of attraction of f’(q), 0 4 i $ p - 1, doesn’t contain a boundary point of the unit interval. Then there is a critical point c of f such that the orbit of c converges to the orbit of q.” Consequently, for the well-studied one-parameter family of maps {f,} with f, defined by f,(x)
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= a~(1 - x) on the unit interval and parameter a in the interval [2, 41, the condition (i) “f, has an asymptotically stable periodic orbit” and (ii) “ the orbit of the critical point of f, converges to an asymptotically stable periodic orbit of f,” are equivalent. Now we will state our result that a map with negative Schwarrian derivative is an Axiom A map, provided that some reasonable conditions are satisfied. Theorem 4.1. Let f be a chaotic C3-mapping from a nontrivial interval X into itself. Assume that f satisfies that following conditions: (i) f has a negative Schwarzian derivative; (ii) the set A,(f) of points, whose orbits don’t converge to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f, has the following properties: (a) A,(f) is a compact set and it is contained in the interior of X, (b) A,(f) doesn’t contain critical points of f; (iii) f is contracting on the set of asymptotically stable periodic points for f, provided that this set is nonempty. Then we have: f is a nonsingular Axiom A mapping. We note that the condition “A,(f) doesn’t contain critical points of f ” is equivalent with the condition “the orbit of each critical point of f converges to an asymptotically stable periodic orbit of f or to a subset of the possible present absorbing boundary of X for f “. From this condition it follows that j cannot have so called one-sided asymptotically stable periodic points. In section 2.1 we gave some examples of quadratic maps and cubic maps which satisfy the conditions of theorem 4.1. Under suitable assumptions on general polynomial maps, we will see that they satisfy the conditions of the above theorem. Theorem 4.2. Assume that f: R + W is a polynomial mapping satisfying the conditions: (i) each critical point of f is real; (ii) the orbit of each critical point of f converges to an asymptotically stable periodic orbit of f or to a subset of the
possible present absorbing boundary of X for f; (iii) f is contracting on the set of asymptotically stable periodic points for f, provided that this set is nonempty. Then f satisfies the conditions of theorem 4.1. Consequently, j is a nonsingular Axiom A mapping.
From now on (in this section) let f E C3( X, X) be fixed, and assume that f satisfies the conditions of theorem 4.1. Then, for clarity of exposition, we will enumerate some properties. Property 4.1. Almost every point in X, in the sense of Lebesgue measure, is asymptotically periodic, provided that f(X) is bounded. Property 4.2. The nonwandering set of f decomposed into finitely many basic sets.
can be
Property 4.3. The map f is Cl+“-persistent for some a! > 0, i.e. the asymptotically periodic behaviour in the dynamics of f and the structure of the nonwandering set don’t change under small smooth perturbations. Property 4.4. There exists a well-defined square matrix A,, with entries zero or one, for which we have: (a) the number of periodic points for j with period n is equal to the trace of the matrix [A,]“, for each positive integer n; and (b) the topological entropy of f is equal to the logarithm of the spectral radius of A,. Property 4.5. The map f is C2-structurally stable, provided that some additional conditions are satisfied.
We will conclude this section by making a remark concerning the advantage of the Axiom A condition for the stability properties of maps, e.g. persistence of asymptotically periodic behaviour in the complicated dynamics of chaotic maps and structural stability.
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Persistence of order and structure in chaos
Remark 4.1. Let g E C3(X, X) be C’-close to f with f as in theorem 4.1. Then we know by the theorems 4.1 and 3.2 that g is an Axiom A map which has the “same” qualitative behaviour in the dynamics as f has. On the other hand, the Schwarzian derivative of g is not a priori negative; still worse, the Schwarzian derivative of g may be positive on some suitable (small) intervals! This phenomenon also may occur, even if f and g are topologically conjugate. Remark 4.2. Let f satisfy the conditions of theorem 4.1. Assume that f has one critical point c, and that the interval is compact. It is no restriction to assume that f is increasing on {x E X, x _I c}, and f is decreasing on { x E X, x 2 c }. Since f can have an asymptotically stable fixed point in { x E X; x < f2( c)}, see e.g. Collet and Eckmann [9] p. 95, the following three cases can occur: (a) f has an asymptotically stable fixed point in {x E X; x
This paper is based on the results of the author’s thesis [26] at the University of Utrecht supervised by Hans Duistermaat. I would like to
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thank him for the stimulating discussions and critical comments. It is a pleasure to thank Wilma van Nieuwamerongen for the excellent typing of this manuscript. The referee’s comments are gratefully acknowledged.
References [l] R.L. Adler, A.G. Konheim and M.H. McAndrew, Topological entropy, Trans. Amer. Math. Sot. 114 (1965) 309-319. [2] D.J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34 (1978) 687-691. [3] M. Artin and B. Mazur, On periodic points, Ann. Math. 81 (1965) 82-99. [4] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Sot. 254 (1979) 391-398. [5] R. Bowen and O.E. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Symp. Pure Math. Vol 14; 43-49 (Amer. Math. Sot., Providence, 1970). [6] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970) 725-747. [7] H. Brolin, Invariant sets under iteration of rational functions, Arkiv. Mat. 6 (1965) 103-144. [8] T.W. Chaundy and E. Phillips, The convergence of sequences defined by quadratic recurrence-formulae, Quart. J. Math. Oxford 7 (1936) 74-80. [9] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhiiuser, Boston, 1980). [lo] J. Coste, Iterations of transformations of the unit interval: approach to an attractor, J. Stat. Phys. 23 (1980) 521-536. [ll] J. Guckenheimer, Sensitivity dependence to initial cdnditions for one dimensional maps, Comm. Math. Phys. 70 (1979) 133-160. [12] I. Gumowski and C. Mira, Dynamique Chaotique (Cepadues Editions, Toulouse, 1980). [13] B.R. Henry, Escape from the unit interval under the transformation x -) Xx(1 - x), Proc. Amer. Math. Sot. 41 (1973) 1466150. [14] F.C. Hoppensteadt and J. Hyman, Periodic solutions of a logistic difference equation, SIAM J. Appl. Math. 32 (1977) 73-81. 1151 M.V. Jakobson, On smooth mappings of the circle into itself. Math USSR Sbomik 14 (1971) 161-185. M.V. Jakobson, Absolutely continuous invariant measures [W for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981) 39-88. P71 L. Jonker and D. Rand, Bifurcations in one dimension I: The non-wandering set, Invent. Math. 62 (1981) 347-365. 1181T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 II (1975) 985-992. 1191 E.N. Lorenz, Deterministic nonperiodic flow, J. Atm. Sc. 20 (1963) 130-141. WI R.M. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459-467.
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[21] N. Metropolis, M.L. Stein and P.R. Stein, On finite limit
sets for transformations of the unit interval, J. Combin. Theory A 15 (1973) 25-44. [22] M. Misiurewia, Structure of mappings of an interval with zero entropy, IHES Publ. 53 (1981) 5-16. [23] M. Misiurewia. Absolutely continuous measures for certain mappings of an interval, IHES Publ. 53 (1981) 17-51. (241 P.J. Myrberg, Iteration der reellen Polynome zweiten Grades III, Ann. Acad. SC. Fermicae 366 (1963) 1-18. [25] Z. Nitecki, Topological dynamics on the interval, in: Ergodic Theory and Dynamical Systems II, Progress in Math., vol. 21. (Birkhguser, Boston, 1982) pp. l-73. [26] H.E. Nusse, Chaos, yet no chance to get lost. Order and structure in the chaotic dynamical behaviour of one dimensional noninvertible Axiom A mappings, arising in discrete biological models. Thesis, State University at Utrecht 1983. (271 C. Preston, Iterates of Maps on an Interval, Lect. Notes Math. 999 (Springer, Berlin, 1983). [28] A.N. Sharkovsky, Coexistence of the cycles of a continuous mapping of the line into itself, Ukr. Mat. Z. 16 (1964) 61-71.
v91 D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978) 260-267. [301 S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Sot. 73 (1967) 747-817. [311 S. Smale and R.F. Williams, The qualitative analysis of a difference equation of population growth, .I. Math. Biol. 3 (1976) l-4. 1321 P. Stefan, A theorem of Sharkovsky on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977) 237-248. ]331 P.R. Stein and S. Ulam, Nonlinear transformation studies on electronic computers, Rozprawy Matematyczne 39 (1964) l-66. 1341 P.D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (1978) 99-105. ]351 S.J. van Strien, On the bifurcation creating horseshoes. Proc. Dynamical Systems and Turbulence, Warwick 1980; 316-351, Lect. Notes Math. 898 (Springer, Berlin, 1981). ]361 P. Walters, An introduction to ergodic theory, G.T.M. 79, (Springer, Berlin, 1982). 1371 Zentralblatt fur Mathematik 521 (1984) D58040.