Global complexity and essential simplicity

Physica D 39 (1989) 163-168 North-Holland, Amsterdam

GLOBAL COMPLEXITY AND ESSENTIAL SIMPLICITY A CONJECTURAL PICTURE OF THE BOUNDARY OF CHAOS FOR SMOOTH ENDOMORFHISMS OF THE INTERVAL

M.V. OTERO-ESPINAR Departamento de Analise Matematica, Facultade de Matematicas, 15706 Santiago de Compostela, Spain

Campus Universitario, s/n,

and

C. TRESSER* T.J. Watson Research Center, IBM, P.O. Box 218, Yorktown Heights, NY 10598, USA

Received 4 April 1989 Accepted 4 May 1989 Communicated by Y. Pomeau

Mixing a symbolic approach to the dynamics of the period-doubling operators, recent results by Sullivan on the renormalization for real analytic maps, and some confidence, a global picture emerges for the structure of the boundary of positive topological entropy in spaces of smooth endomorphisms of the interval.

1. Introduction The aim of this paper is to describe the complexity of the dynamics of the renormalization group for period doubling, when one looks at general smooth maps on the interval rather than just unimodal ones. The main message will be that, although this dynamics is quite complicated, universality and other smoothness properties seem to work together to ensure that the usual universal numbers found in simple quadratic maps, have in fact much more general relevance than previous renormalization group arguments allowed to believe. Before we deal with interval maps, let us *Previous address: Department of Theoretical (CNRS), Pam Valrose, 06034 Nice Cedex, France.

Physics

recall some well known facts related to the universality theory for circle maps, which will just serve as model for the development of our discussion. To this end, we consider either the set of circle diffeomorphisms or the set of critical circle maps (i.e. smooth circle homeomorphisms with a critical point). Any map f of such a kind has a well defined rotation number mod 1, denoted hereafter p( f ), and under renormalization in one of the usual ways [6, 131, the renormalized map a,(f) has a rotation number mod 1, p( &‘,( f )), equal to (l/p ( f )) mod 1. Hence the usual renormalizations for circle maps, which act on function spaces, project to the Gauss map, R(B) = (l/Q mod 1, acting on [0, l[. It is then an arduous task to formulate how the properties of the Gauss map R should lift to properties of 2” acting on a proper

0167-2789/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

164

hf. V. Otero-Espinar,

C. Tresser/

function space, before even trying to prove anything (see e.g. ref. [9] for conjectures). In the case of unimodal maps on the interval, it is the kneading sequence which usually plays a role similar to the one the rotation number plays for circle maps, as a symbolic representation of the topological dynamics. Since there is just a single unimodal sequence (up to orientation of the interval) which describes the accumulation of period-doubling bifurcations, there is no hope of a rich symbolic dynamics for the renormalization group. On the other hand, the symbolic description becomes quite rich when one goes from unimodal maps to bimodal maps which have both endpoints of the interval as fixed points. Then one can define a kneading-invariant-dependent renormalization group which projects to an operation on symbolic space, which has a full horseshoe [7, 111. At this point, it should be noticed that the dynamics of the functional renormalization is at least as complicated as the one to which it projects on the proper symbolic space, part of the game of universality theory being to control how much more it is complicated; see the precise conjectures below. The drawback of refs. [7, 111, as well as their “internal beauty” is linked to the fact that there are lots of “2” in the problem: doubling for bimodal maps, with twp generators of the renormalization semi-group, leading to a horseshoe with topological entropy log2. The consequence is that the approach there cannot be (easily) generalized to the set of all multimodal maps. The aim of this short paper is threefold: (i) First we will present a symbolic theory for renormalization for period doubling formulated for arbitrary multimodal maps, but without proofs (which will appear elsewhere [14]) nor the mathematically precise terminology (involving inverse limits) needed for the formulation of the main theorems (see refs. [8, 141). (ii) Then we will formulate conjectures in the way this symbolic description lifts to the (physically interesting) function space of smooth en-

Global complexity and essential simplicit?,

domorphisms of the interval which are twice differentiable and have no flat critical point (i.e. no zero of infinite order for the derivative). The missing bridge from these conjectures to actual experiments is (up to philosophical considerations of the same sort as the relation between knot theory and actual knots made with ropes) an adaptation of Collet-Eckmann-Koch theory of doubling in R” that they formulated for the usual doubling in ref. [4], to the general case. (iii) At last we will present an implication of the picture in function space which would follow from our conjectures.

2. Symbolic renormalization the large

for period doubling in

Let 0, be a set of orders on the set (1,. . . ,2”} of all positive integers between 1 and 2”, to be constructed now, with 0, = ((1)) and 1 always the largest element. An element of 0, can be represented as a 2”-uple: o”=(01,02

)...) 02”=1),

where we write o,,, for o”(m), and where the string (01,02,..., 02,,) represents the fact that for the order o”, oi < o2 < . . . -c 02” = 1. Let first S, be the set of 2”-uples of + and signs which terminate with a + . The element an = {a;, U-J,. . . , u; = + } of S, transforms replacing

o” E 0, into a”( 0”) E Onsl by

-0; to oi + 2”, oi

if uin= + ,

-oitooi,oi+2”

if uin= - .

Writing Sm+i OS,,, for the set of all u~+~~u~‘s, we can then set 0 n+i = SnOSn-iO . . . “S&l,), and it occurs that 0, is precisely the set of per-

hi. V. Otero-Espinar, C. Tres.ser/Global complexity and essential simplicity

mutations on points of a 2”-cycle which can be realized by a nonchaotic continuous map on the interval [l, 21. In order to understand both this statement, and the particularity of those orders on the set {l,..., 2”) which do belong to O,, just think of how a 2”-cycle is transformed to a 2”+lcycle by a period-doubling bifurcation. Let us illustrate these definitions in the cases when n is small enough. We get Oo= {(I)),

165

tor sets at the accumulation of cascades of period-doubling bifurcations, invariant under nonchaotic maps [8, 141. Essentially, an element P of 0, can be thought of as an infinite chain: Orn= (00,01)...)

on)... ),

where o’+’ = a’(~‘) for each i2 0 and some u’ E Si. It is convenient to represent alternatively P as

So= {(+>)7

0, = S,(Oo) = ((271) = (+)(I)}, S,= {(-,+),(+,+)}, 0, = S, “So(O0)

{(2,4,3,1) =(7+)(U) = (-,+)0(+)(l), @JJJ) = (+>+>(2J)}, s,= {(-,-,-,+),(+,-,-,+), (--,+,-,+>,(+,+,-~+>9 (-,-,+,+),(+r,+,+), (-,+,+,+),(+,+,+,+>}, 0, = {(2,6,48,3,T5,1), (6,2,4,8,3,7,5,1),(2,6,8,4,3,7,5,1), (6,2,8,4,3,7,5,1>,(2,6,4,8,7,3,5,1), (6,2,4,8,7,3,5,1),(2,6,8,4,7,3,5,1), (6,2,8,4,7,3,5,1),(4,8,2,6,3,7,5,1), (8,4,2,6,3,7,5,1),(4,8,6,2,3,7,5,1), (8,4,6,2,3,7,5,1),(4,8,2,6,7,3,5,1), (8,4,2,6,7,3,5,1>,(4,8,6,2,7,3,5,1), (8,4W,T3,5J)}.

OO”= (UOJ)..‘)

cr.”)...)

or, using u(+) = 1 and u(-) number

= 0, by the dyadic

=

With, for instance (2,6,4,8,3,7,5,1) =(-,-,-,+)(2,4,&I) =(-,-,--,+)+-,+)(2,1) =(-,-,-,+)+,+)0(+)(l). There is now a proper way to define a limit 0, which will describe all possible dynamics for Can-

.

In order to define (symbolic) renormalization operators on O,, it is intuitively more transparent to go to the function space. Let then K be a period-doubling Cantor set for the smooth endomorphism f of the interval I. By this we mean that f has a nested set of periodic orbits with periods 1,2,4 ,..., and K is the corresponding set of accumulation points. K is a disjoint union K = K, u K, with f(Ki) = K,_i, and we assume that K, stays on the right of K,. Let A:, SE{+,--), i~{O,l}, be the affine map which sends I on the support of Ki, preserving or not the orientation according to whether s = + or s = - . Then the possible renormalizations are any of the following four choices: 9;(f)

= (A”i)-l+A~;.

For instance, for the quadratic map c - x2, one uses A, and A:, while A 0’ and A; are more relevant for c + x2. Now 9s projects on R;,which acts on 0,. While the discussion to follow could be done using only R, and R:, or Ri and R;, considering all four renormalization group generators is not only the most natural point of view, but also allows to ask questions such as e.g. “what are

M. V. Otero-Espinar, C. Tresser/ Global complexity and essential simplicity

166

the orders

invariant

under

both

(see ref. [8]). R: offers the particularity dyadic

number

n(

R,

R:?”

and

that its action on the

) is easily obtained

as

.

All Rf ‘s have an extremely ics. To investigate 0,

carries

a somewhat

natural

ture, with an ultrametric like representation (the inverse 0,

limits

Denoting which

complicated

this, it is useful

dynam-

to realize

that

metric space struc-

distance due to its treeformal definition using

[S] tells us that it is a Cantor

by S: and s; can be realized

set).

the only elements of by unimodal maps,

respectively, like c - x2 and c + x2 (S stands for “standard order”), the main fact for the sequel is that [8, 141: The set of preimages under iteration of any Rs, of any order in O,, R[, and in particular of s: and s i,

is dense in 0,.

[+:maps

2’

with zero entropy & all periods

Fig. 1. Maps with zero entropy

and all periods

2”.

with a finite number of periods. This is stronger than the conjecture that densely all critical points go to a stable periodic orbit. When specializing this last conjecture to the period-doubling problem: the last conjecture would prevent the pathol-

3. A conjectural picture for the dynamics of perioddoubling renormalization Let Z be the set of C2 maps on I with no flat critical point, which are at the boundary of positive topological entropy. Conjecture : (I) Z is the set of maps at the accumulation of a cascade of period-doubling bifurcations. (II) Furthermore an open dense subset Zi,B of B splits into disjoint cells ZF,Sm such that .G%‘f( E~~Sm) of the belongs to ZF,, the connected component basin for .9;, of the quadratic-like fixed point f;‘* of 9s which contains ft*. Remarks : (A) Another formulation of the first statement is that any map in the class considered which has zero entropy and all periods 2” can be approached both by maps with positive entropy and by maps

ogy of fig. la, but not those of fig. lb. (B) The second (and main) statement conjecture can be decomposed as follows:

of the

(11.1) In an open dense subset Xi of Z, the maps have a single invariant Cantor set: Z, is the subset of Z where the dynamics of 9: is unambiguously defined. Assuming I, the intuitive idea for (11.1) goes as follows: (i) The accumulation of period-doubling bifurcations is a codimension-one phenomenon. (ii) Many accumulations higher codimension.

occurring

together have

(11.2) In an open dense subset Z, of Z,, there exists a finite n 2 0 such that the symbolic dynamics on the Cantor set of (9%‘,!)“(f) is described by the order s;, i.e. (a,‘)“(f) is a unimodal map. The motivation for that is given below. That there exist orders not realizable in Z, is easily gotten by playing around at the symbolic level (see e.g. the dyadic formula for R[ given above). It is less

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M. V. Otero-Espinar, C. Tresser/ Global complexity and essential simplicity

obvious that by this way, one still generates orders realizable by maps with finitely many critical points. The simplest examples in Z,\Z, are obtained by considering all orders compatible with bimodal maps, which need two critical points in the Cantor set, none of which is at an extremity of a hole of the Cantor set. Most orders found in the bimodal case are in fact of this sort (see refs. [7, III)* (11.3) This is the easy part: an open dense subset Z, of Z, is made of maps with all critical points nondegenerate. (11.4) All of Z, is in the basin of ft*. While this is still a challenging question in the class of functions we have mentioned, it is noticeable that this statement would already be guaranteed in the quite interesting class of real analytic maps, by the recent result of D. Sullivan [15] that all real analytic unimodal maps with a quadratic-like critical point are attracted to the same tixed point f;* under iteration of 9,. (C) What we know from the symbolic dynamics is enough to tell us that the dynamics of 9: restricted to Z,\Z, is quite complicated. In fact the dyadic formula for R[ implies that R: has infinite topological entropy, and similar results hold for all Ri ‘s. Since the asymptotic dynamics of R: (for instance) restricted to Z, is quite simple by definition, Z, \ X 2 has to support all the chaotic dynamics. To get a satisfactory description at the functional level, it would remain to prove that the codimension of all indecomposable parts of the nonwandering set, and in particular the fixed points of R: in Z,\Z,, is what one guesses from the combination of the topology and the order of degeneracy of the critical points of the corresponding maps. (D) The interested reader can consult the pictures for the boundary of chaos for bimodal maps in refs. [7, 10, 111 in order to understand how Z\Z, is important in the global structure of Z: in order to go from an order o E 0, to another order o’ E O,, one has to go through the coexistence of many Cantor sets or a Cantor set containing many critical points.

About statement (II. 2): Let f be in Z,, K be its Cantor set. One can understand K as produced by a cascade of period-doubling bifurcations. The argument for (11.2) then goes as follows: (i) Consider in a one-parameter family, a 2”orbit C, just born by bifurcation, and follow it until it loses its stability to give birth to a 2”+‘cycle C,: an odd number of points of C, need to cross a critical point (if this number is even, the bifurcation involves two more periodic orbits with period 2” and one starts from there). (ii) A single critical point being crossed is the simplest event of that sort. (iii) After this last bifurcation, assuming a single critical point has been crossed, it gets even more natural that a single point of C,, 1 crosses the same critical point to get the next bifurcation (by simply considering the proximity of all points in C n+l to the nearest critical point of the map). (iv) The punchline is that with such a scenario, the symbolic theory [14] tells us that one gets a preimage of a standard order s: at the symbolic level.

4. An implication Let us consider the p-periodic perturbations of unimodal maps with p odd, i.e. the dynamics of

X’

(th+ E”(R)L(,,d&,)

=x,+1,

where q(R) fi is a perturbation of size ei( R), iE {l,..., p } of some unimodal map g, depending on the parameter R. When l&l= Ep=,le,l* = 0, one has just the traditional problem of one-dimensional mappings. A natural question is how the periodic perturbation affects the transition to positive topological entropy. The renormalization group study in ref. [S] shows that the unimodal fixed point is unstable in the new space, and Collet and Lesne asked what is going on for small (~1.

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M. V. Otero-Espinar, C. Tresser/ Global complexity and essential simplicity

The connection of this problem with what was presented in the previous sections, is that

Sullivan. We also thank Y. Pomeau for useful suggestions for the presentation of this material.

[gR+E1(R)filO[gR+&2(R)f21 0 . . . o[&+~pm.i]

References

is a multimodal map, so that in the smooth case, the transition to chaos is via a cascade of perioddoubling bifurcations [3, 121. Hence, assuming the conjecture, the picture which emerges in this context is that, although in any l&l-small neighborhood of the pth iterate of the standard fixed point, a path in Z toward this point must cross Z\Z,, most one-parameter families crossing from zero to positive entropy will exhibit a cascade of period doubling at the usual rate 4.669.. . . However, the number of iterations of the renormalization group necessary to get the unimodal order should diverge as 1~1goes to 0, so that 4.669.. . would not be observable for small 1~1.

Acknowledgements

C.T. thanks the Arizona Center for Mathematical Sciences (ACMS) for an invitation for the academic year 1988-89. The ACMS is sponsored by AFOSR contract F49620-86-CO130 with the URI program at the University of Arizona. This paper owes much to conversations with P. Collet, J.M. Gambaudo, P. Glendinning, J. Los and D.

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