Dynamics of the universal area-preserving map associated with period doubling: hyperbolic sets

Abstract

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R 2 . A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a "universal" area-preserving map F * -a map with orbits of all binary periods 2 k , k ∈ N. In this paper, we consider maps in some neighbourhood of F * and study their dynamics.We first demonstrate that the map F * admits a "bi-infinite heteroclinic tangle": a sequence of periodic points {z k }, k ∈ Z,(1) whose stable and unstable manifolds intersect transversally; and, for any N ∈ N, a compact invariant set on which F * is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of N symbols. A corollary of these results is the existence of unbounded and oscillating orbits.We also show that the third iterate for all maps close to F * admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: 0.7673 ≥ dim H (C F ) ≥ ε ≈ 0.00013 e −7499 .

A numerical study of infinitely renormalizable area-preserving maps

“…To support the conjecture, we have computed the left-and right-hand sides of Equation (16) for several monomials in cos and sin as well as for an analytic function 1/(1.5 À cos(x)) (which has an infinite number of harmonics in its Taylor series). The approximation of the density of the measure has been computed by dividing the circles into a collection of N subintervals [ i , iþ1 ] of equal length, and counting the relative number K n i of angles F Ã !…”

“…The following lemma about the existence of hyperbolic fixed points for maps in a small neighbourhood of the renormalization fixed point map F Ã is a restatement of a result from Gaidashev and Johnson [16] in the setting of the functional space A s (1.75). The proof of the lemma is computer-assisted [16].…”

“…The proof of the lemma is computer-assisted [16]. This lemma implies the existence of hyperbolic 2 n -th periodic orbits for maps in W % (s 0 ).…”

“…The relative difference of the left-and right-hand sides in(16) for the function sin 2 , M ¼ 20,000, v ¼ (1, 0).…”

On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

“…This explains previously observed universality phenomena in families of such maps. Further investigations of this renormalization have been done by Gaidashev and Johnson [8][9][10] and by Gaidashev et al [11]. In these papers they prove existence of period doubling invariant Cantor sets for all infinitely renormalizable maps and also show that they are rigid.…”