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Non-monotonicity of entropy of interval maps
3 July 199.5
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 202 (1995) 359-362
Non-monotonicity
of entropy of interval maps Henk Bruin
’
Mathematisches Institut, Universitdt Erlangen, BismarckstraJe I i, D-91054 Erlangen, Germany Received 27 March
1995; revised manuscript received 9 May 1995; accepted Communicated by A.P. Fordy
for publication
11 May 1995
Abstract We prove that the topological entropy of a unimodal map (with a unique maximum) can increase if the map is perturbed to a map whose graph lies below the original map. The perturbed map can be symmetric, concave, smooth and/or have negative Schwarzian derivative, if the original map has these properties.
Let f : I + I, I = [O, 1] be a unimodal map. This means that f has a unique turning point, c. Assume that f(c) is the global maximum. The question of increasing topological entropy concerns the generally observed phenomenon that unimodal maps become more complicated topologically if the map increases. Topological complexity may be measured by the topological entropy hior,. Then the question reads: Does f(x)
< g(x)
Often this question
for all x imply &r(f)
< hi,,(g)?
(1)
is put in a stronger form:
Is the map a H hior
increasing?
(2)
where f is some fixed unimodal map. For tent-maps, fa = Here f<,(x) = af(x) or J%(X) = a + f(x), u(i - Ix - iI), (1) is true. Indeed htDp(fa) = max(O,loga) in this case. If fa(x) = a - x1, I an even integer, (2) was answered affirmatively in Ref. [ 11. For trapezoidal maps, there are results by Brucks et al. [ 21. In general (2) is not true; several counter-examples have been found [3-51. These are not pathological; the scenario described in Ref. [ 41 is encountered in physical experiments [ 61. Yet these examples involve asymmetric maps or attracting fixed points, which restricts their generality. They do not explain why the entropy decreases in the example below. Exunzple. To the unimodal map fa(x) = 1 - ux2 we add a symmetric C3 perturbation eg, where g(x) = (x - l/J;;)3]x= (x + l/fi)3[X
(l/!& + (l/J;;
= 0, ’ Supported
by the Netherlands
&J13, - $j)13,
ifxE(l/&-+j,l/JZ;), if x E <-l/J;;,-l/fi+ otherwise.
Organization
for Scientific Research
(NWO)
037%9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00362-2
h),
H. Bruin/Physics LettersA 202 (1995) 359-362
360 Table 1 a = 1.72980800
-400 -300 -200 -100 0 100 200 300 400
0.47755558644103 0.47755558173050 0.47755556806528 0.47755555329306 0.4775555475762 0.47755554565188 0.47755554203778 0.47755553490973 0.47755551882449
-0.04 -0.03 -0.02 1
0.47755554758150 0.47755554758036 0.47755554757877 0.47755554757708 0.47755554757621 0.47755554757532 0.47755554757500 0.47755554757491 0.47755554757457
-0.01 0.00 0.01 0.02 0.03 0.04
Note that f;‘(O) = {-l/G, l/&}. If E is not too large, then fa +eg is concave and has negative derivative. (The Schwarzian derivative of f is defined as f”‘/f’ - i( f”/f’)2. Maps with negative derivative are very frequently studied in the theory of iterated interval maps.) Table 1 shows that values of a, the entropy of fa + eg decreases if E increases. Our main theorem explains this example, and shows that the answer to question (1) is “no” general set of maps.
Schwarzian Schwarzian for certain for a very
A unimodal map f is called long-branched if there exists K > 0 such that for every n and every maximal interval T on which f” is monotonic, If”(T) 1 > K. Here f” denotes the n-fold composition. For example, f is long-branched if c is not recurrent, i.e. if the forward orbit of c does not accumulate on c. But there are also long-branched maps with a recurrent turning point, see Ref. [ 71. Theorem 1. (Main theorem). Let f be a long-branched unimodal map with a recurrent point. Then for each E > 0 and r E N, there exists a map g such that (i) g(x) < f(x)for all x E I,
(ii) &r(g) > htop(f)3 (iii) ]j g - f Ilr< E,where II llr denotes the C’ norm, (iv) if f is symmetric, concave and/or has negative Schwarzian Remarks.
The set of long-branched
unimodal
maps is densely
derivative,
non-periodic
turning
g has the same properties.
uncountable
in the following
sense:
Let
{fa}o,R be a one-parameter family of unimodal maps, and let (ua, at) be an interval on which a H htop( fa) is not constant. Then f. is long-branched and has a recurrent non-periodic turning point for uncountably many parameters a E (uo,ut). The main theorem can be seen as a partial answer to the C’-closing lemma, as it asserts that f cannot be C’-structurally stable. The theorem extends a result of Gambaudo and Tresser [8], indicating an uncountable set of parameters for which the map fa is not C-structurally stable. Lemma 2. Let f. be a family of unimodal maps such that fa exhibits a saddle node bifurcation is such that this saddle node bifurcation has not taken place yet, then htop( fa)< htop( f,).
at ua. If a
Sketch of the proofi The proof relies on the fact that each unimodal map is semi-conjugate to a tent-map with the same topological entropy [9,10]. fd, is semi-conjugate to a tent-map with a periodic turning point. The topological type of this periodic point is not exhibited by the tent-map to which f0 is semi-conjugate. The •1 latter tent-map therefore has smaller entropy than the first one.
H. Bruin/Physics Letters A 202 (19951359462
361
As we said before, there are long-branched maps with a recurrent turning point. Yet the returns of c are somewhat special:
Lemma 3. If f is long-branched, then there exists a neighbourh~d then If”(x) - cl has a local minimum in c.
U 3 c such that whenever f”(c)
E fJ,
Pro@ Let U 3 c be such that /U/, 1f(U) j < 151.Clearly c is a turning point of t cf If”(r) - ~1. Suppose f”(c) E U. Let y be maximal such that f”/ (c, y) is monotonic. By long-branchedness, f”(y) 4 U. If f”(c) is a local maximum of t H If”(t) - cl, then f”(x) = c for some x E (c, y). So (c, x) is a maximal interval Cl such that fn+tl(c,.r) is monotonic. But then ~f”+‘((c,x))~ < If(U)1 < K. Lemma 4. Let f be long-branched and have a recurrent non-periodic turning point. Then for every E > 0 there exists j: such that (i) f”(n) b f(x) for all x f I, (ii) ~~~~~~) > &r(f) and (iii) (17 - f Ilr< E.
Proo$ Choose z such that fN(z > = c and f’(z) $! (z, 2) for i < N. Here i Z z is the point such that f(2) = f(z ).-Let U = (z, z^) and assume that z is so close to c that U satisfies the properties of Lemma 3. We construct f by perturbing f in U in the following way: (i) f(x) 3 f(x) for x E U and f(n) = f(x) for x $ U, but f # f. (ii) II j; - f /I& E, (iii) c is also the unique turning point of f”. Assume that there exists a conjugacy h such that h o j: = f o h. We write jj for h(y) for any point y and we use the same notation for sets. By definition of z, z = 2, and therefore 2 E 6 = U whenever x E U. Let - p+‘(c). no 2 0 be the smallest integer such that f”“+’ (c) > fm+‘(c). Clearly f”” f U. Let S= p+‘(c) Let (ni}i>a be the sequence of subsequent return iterates of c to U. So ni+l = min{n > rri 1f”(c) E U>. Let K 3 fnl+’ (c) be the maximal neighbourhood such that ~+l-“I~ is monotonic, fj (V;:) n U = 0 for j = 0, .‘( Ili+, _ ni - 2 and fn,+M--l (K) C U. The boundary points of V are preimages of the turning point, possibly preimages of z or 2. Therefore j’j(q) =fj(&) forj=O,...,ni+i -ni1. By Lemma 3, f”‘+l-‘l IF preserves orientation. Therefore if x E F$ and _F2 X, then also f
-““I-“‘(k)
2 fn,+l-n,(X)*
Let V = (f”o+‘(c),y) (i) P+‘(c)
(3)
be such that
< Y,
(ii) (VI < 6, and (iii) There exists M such that f”( y) = c and f”jV is monotonic. By definition of &:, p-“O( V) E Q whenever ni - no < M. Applying (3) repeatedly, we obtain that 9 < y. On the other hand, p+’ (c) = fno+’ (c) + 6, but then _?“O+l(c) 2 y, a contradiction. So f and f are not conjugate. By continuity, we can choose f” such that f”+no+’ (c) = c, while t +-+ loss of generality, this new M + no + l-periodic point can If M+rwi’ (t) - c/ has a local minimum in c, ~thout E be taken to emerge in a saddle node bifurcation. Hence by Lemma 2, htop( f) > htop( f) .
Proof of the main theorem. Let f be long-branched and have a recurrent non-periodic turning point. By Lemma 4, there exists 1 ~bitr~ily close to f such that htop(f) > htop(f). Without loss of generality we may assume that f(x) = f(x)for all x < c. Let U as in Lemma 4, and let g(x) = f(x),
if x 4 f-‘(Un
(c, l)),
H. Bruin/Physics
362
=
f-1
0
f2(x),
Letters A 202 (1995) 359-362
ifxEf_‘(Un(c,l)).
or g*(x) = f(x) . So g and f have the same topological entropy. Because reverses orientation, g(x) < f(x). Finally, because topological entropy is f(f(x)) 2 f2(x) and fl(c,l) lower semi-continuous [ 111, there exists 77 > 0 such that h,,(g - r]) > htop( f). Clearly g(x) - 7 < f(x) for all x, and it is not difficult to see that g - r] can be symmetric, concave and have negative Schwarzian 0 derivative, if f has these properties. By construction
either g(x)
= f(x)
References [ 1 ] A. Douady and J.H. Hubbard,
Publ. Math. Orsay (19x4-85).
[ 21 K. Brucks, M. Misiurewicz and C. Tresser, Commun. Math. Phys. 137 ( 1991) I. 131 SE Koljada, Ukr. Mat. Z. 41 (1989) 258. ]4] H.E. Nusse and J.A. Yorke, Phys. Len. A 127 (1988) 328. 15 1 A. Zdunik, Fund. Math. 124 (1984) 235. 161 M. Bier and T. Bountis, Phys. Lett. A 104 ( 1984) 239. [ 7 1 H. Bruin, Erg. Th. Dyn. Sys. 14 ( 1994) 433. 181 J.-M. Gambaudo and C. Tresser, Contemp. Math. 135 ( 1992) 213. 191 L. Alseda, J. Llibre and M. Misiurewicz, in: Advanced series in non-linear dynamics, Vol. 5. Combinatorial dynamics and entropy in one dimension (World Scientific, Singapore, 1993). I 10 1 J. Milnor and W. Thurston, in: Lecture notes in mathematics, Vol. 1342. On iterated maps of the interval, ed. J.C. Alexander (Springer, Berlin, 1988) pp. 465-563. I 11 I M. Misiurewicz and W. Szlenk, Studia Math. 67 (1980) 45.
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 202 (1995) 359-362
Non-monotonicity
of entropy of interval maps Henk Bruin
’
Mathematisches Institut, Universitdt Erlangen, BismarckstraJe I i, D-91054 Erlangen, Germany Received 27 March
1995; revised manuscript received 9 May 1995; accepted Communicated by A.P. Fordy
for publication
11 May 1995
Abstract We prove that the topological entropy of a unimodal map (with a unique maximum) can increase if the map is perturbed to a map whose graph lies below the original map. The perturbed map can be symmetric, concave, smooth and/or have negative Schwarzian derivative, if the original map has these properties.
Let f : I + I, I = [O, 1] be a unimodal map. This means that f has a unique turning point, c. Assume that f(c) is the global maximum. The question of increasing topological entropy concerns the generally observed phenomenon that unimodal maps become more complicated topologically if the map increases. Topological complexity may be measured by the topological entropy hior,. Then the question reads: Does f(x)
< g(x)
Often this question
for all x imply &r(f)
< hi,,(g)?
(1)
is put in a stronger form:
Is the map a H hior
increasing?
(2)
where f is some fixed unimodal map. For tent-maps, fa = Here f<,(x) = af(x) or J%(X) = a + f(x), u(i - Ix - iI), (1) is true. Indeed htDp(fa) = max(O,loga) in this case. If fa(x) = a - x1, I an even integer, (2) was answered affirmatively in Ref. [ 11. For trapezoidal maps, there are results by Brucks et al. [ 21. In general (2) is not true; several counter-examples have been found [3-51. These are not pathological; the scenario described in Ref. [ 41 is encountered in physical experiments [ 61. Yet these examples involve asymmetric maps or attracting fixed points, which restricts their generality. They do not explain why the entropy decreases in the example below. Exunzple. To the unimodal map fa(x) = 1 - ux2 we add a symmetric C3 perturbation eg, where g(x) = (x - l/J;;)3]x= (x + l/fi)3[X
(l/!& + (l/J;;
= 0, ’ Supported
by the Netherlands
&J13, - $j)13,
ifxE(l/&-+j,l/JZ;), if x E <-l/J;;,-l/fi+ otherwise.
Organization
for Scientific Research
(NWO)
037%9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00362-2
h),
H. Bruin/Physics LettersA 202 (1995) 359-362
360 Table 1 a = 1.72980800
-400 -300 -200 -100 0 100 200 300 400
0.47755558644103 0.47755558173050 0.47755556806528 0.47755555329306 0.4775555475762 0.47755554565188 0.47755554203778 0.47755553490973 0.47755551882449
-0.04 -0.03 -0.02 1
0.47755554758150 0.47755554758036 0.47755554757877 0.47755554757708 0.47755554757621 0.47755554757532 0.47755554757500 0.47755554757491 0.47755554757457
-0.01 0.00 0.01 0.02 0.03 0.04
Note that f;‘(O) = {-l/G, l/&}. If E is not too large, then fa +eg is concave and has negative derivative. (The Schwarzian derivative of f is defined as f”‘/f’ - i( f”/f’)2. Maps with negative derivative are very frequently studied in the theory of iterated interval maps.) Table 1 shows that values of a, the entropy of fa + eg decreases if E increases. Our main theorem explains this example, and shows that the answer to question (1) is “no” general set of maps.
Schwarzian Schwarzian for certain for a very
A unimodal map f is called long-branched if there exists K > 0 such that for every n and every maximal interval T on which f” is monotonic, If”(T) 1 > K. Here f” denotes the n-fold composition. For example, f is long-branched if c is not recurrent, i.e. if the forward orbit of c does not accumulate on c. But there are also long-branched maps with a recurrent turning point, see Ref. [ 71. Theorem 1. (Main theorem). Let f be a long-branched unimodal map with a recurrent point. Then for each E > 0 and r E N, there exists a map g such that (i) g(x) < f(x)for all x E I,
(ii) &r(g) > htop(f)3 (iii) ]j g - f Ilr< E,where II llr denotes the C’ norm, (iv) if f is symmetric, concave and/or has negative Schwarzian Remarks.
The set of long-branched
unimodal
maps is densely
derivative,
non-periodic
turning
g has the same properties.
uncountable
in the following
sense:
Let
{fa}o,R be a one-parameter family of unimodal maps, and let (ua, at) be an interval on which a H htop( fa) is not constant. Then f. is long-branched and has a recurrent non-periodic turning point for uncountably many parameters a E (uo,ut). The main theorem can be seen as a partial answer to the C’-closing lemma, as it asserts that f cannot be C’-structurally stable. The theorem extends a result of Gambaudo and Tresser [8], indicating an uncountable set of parameters for which the map fa is not C-structurally stable. Lemma 2. Let f. be a family of unimodal maps such that fa exhibits a saddle node bifurcation is such that this saddle node bifurcation has not taken place yet, then htop( fa)< htop( f,).
at ua. If a
Sketch of the proofi The proof relies on the fact that each unimodal map is semi-conjugate to a tent-map with the same topological entropy [9,10]. fd, is semi-conjugate to a tent-map with a periodic turning point. The topological type of this periodic point is not exhibited by the tent-map to which f0 is semi-conjugate. The •1 latter tent-map therefore has smaller entropy than the first one.
H. Bruin/Physics Letters A 202 (19951359462
361
As we said before, there are long-branched maps with a recurrent turning point. Yet the returns of c are somewhat special:
Lemma 3. If f is long-branched, then there exists a neighbourh~d then If”(x) - cl has a local minimum in c.
U 3 c such that whenever f”(c)
E fJ,
Pro@ Let U 3 c be such that /U/, 1f(U) j < 151.Clearly c is a turning point of t cf If”(r) - ~1. Suppose f”(c) E U. Let y be maximal such that f”/ (c, y) is monotonic. By long-branchedness, f”(y) 4 U. If f”(c) is a local maximum of t H If”(t) - cl, then f”(x) = c for some x E (c, y). So (c, x) is a maximal interval Cl such that fn+tl(c,.r) is monotonic. But then ~f”+‘((c,x))~ < If(U)1 < K. Lemma 4. Let f be long-branched and have a recurrent non-periodic turning point. Then for every E > 0 there exists j: such that (i) f”(n) b f(x) for all x f I, (ii) ~~~~~~) > &r(f) and (iii) (17 - f Ilr< E.
Proo$ Choose z such that fN(z > = c and f’(z) $! (z, 2) for i < N. Here i Z z is the point such that f(2) = f(z ).-Let U = (z, z^) and assume that z is so close to c that U satisfies the properties of Lemma 3. We construct f by perturbing f in U in the following way: (i) f(x) 3 f(x) for x E U and f(n) = f(x) for x $ U, but f # f. (ii) II j; - f /I& E, (iii) c is also the unique turning point of f”. Assume that there exists a conjugacy h such that h o j: = f o h. We write jj for h(y) for any point y and we use the same notation for sets. By definition of z, z = 2, and therefore 2 E 6 = U whenever x E U. Let - p+‘(c). no 2 0 be the smallest integer such that f”“+’ (c) > fm+‘(c). Clearly f”” f U. Let S= p+‘(c) Let (ni}i>a be the sequence of subsequent return iterates of c to U. So ni+l = min{n > rri 1f”(c) E U>. Let K 3 fnl+’ (c) be the maximal neighbourhood such that ~+l-“I~ is monotonic, fj (V;:) n U = 0 for j = 0, .‘( Ili+, _ ni - 2 and fn,+M--l (K) C U. The boundary points of V are preimages of the turning point, possibly preimages of z or 2. Therefore j’j(q) =fj(&) forj=O,...,ni+i -ni1. By Lemma 3, f”‘+l-‘l IF preserves orientation. Therefore if x E F$ and _F2 X, then also f
-““I-“‘(k)
2 fn,+l-n,(X)*
Let V = (f”o+‘(c),y) (i) P+‘(c)
(3)
be such that
< Y,
(ii) (VI < 6, and (iii) There exists M such that f”( y) = c and f”jV is monotonic. By definition of &:, p-“O( V) E Q whenever ni - no < M. Applying (3) repeatedly, we obtain that 9 < y. On the other hand, p+’ (c) = fno+’ (c) + 6, but then _?“O+l(c) 2 y, a contradiction. So f and f are not conjugate. By continuity, we can choose f” such that f”+no+’ (c) = c, while t +-+ loss of generality, this new M + no + l-periodic point can If M+rwi’ (t) - c/ has a local minimum in c, ~thout E be taken to emerge in a saddle node bifurcation. Hence by Lemma 2, htop( f) > htop( f) .
Proof of the main theorem. Let f be long-branched and have a recurrent non-periodic turning point. By Lemma 4, there exists 1 ~bitr~ily close to f such that htop(f) > htop(f). Without loss of generality we may assume that f(x) = f(x)for all x < c. Let U as in Lemma 4, and let g(x) = f(x),
if x 4 f-‘(Un
(c, l)),
H. Bruin/Physics
362
=
f-1
0
f2(x),
Letters A 202 (1995) 359-362
ifxEf_‘(Un(c,l)).
or g*(x) = f(x) . So g and f have the same topological entropy. Because reverses orientation, g(x) < f(x). Finally, because topological entropy is f(f(x)) 2 f2(x) and fl(c,l) lower semi-continuous [ 111, there exists 77 > 0 such that h,,(g - r]) > htop( f). Clearly g(x) - 7 < f(x) for all x, and it is not difficult to see that g - r] can be symmetric, concave and have negative Schwarzian 0 derivative, if f has these properties. By construction
either g(x)
= f(x)
References [ 1 ] A. Douady and J.H. Hubbard,
Publ. Math. Orsay (19x4-85).
[ 21 K. Brucks, M. Misiurewicz and C. Tresser, Commun. Math. Phys. 137 ( 1991) I. 131 SE Koljada, Ukr. Mat. Z. 41 (1989) 258. ]4] H.E. Nusse and J.A. Yorke, Phys. Len. A 127 (1988) 328. 15 1 A. Zdunik, Fund. Math. 124 (1984) 235. 161 M. Bier and T. Bountis, Phys. Lett. A 104 ( 1984) 239. [ 7 1 H. Bruin, Erg. Th. Dyn. Sys. 14 ( 1994) 433. 181 J.-M. Gambaudo and C. Tresser, Contemp. Math. 135 ( 1992) 213. 191 L. Alseda, J. Llibre and M. Misiurewicz, in: Advanced series in non-linear dynamics, Vol. 5. Combinatorial dynamics and entropy in one dimension (World Scientific, Singapore, 1993). I 10 1 J. Milnor and W. Thurston, in: Lecture notes in mathematics, Vol. 1342. On iterated maps of the interval, ed. J.C. Alexander (Springer, Berlin, 1988) pp. 465-563. I 11 I M. Misiurewicz and W. Szlenk, Studia Math. 67 (1980) 45.