Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets

TL;DR

This paper investigates the complex dynamics near a universal area-preserving map related to period doubling, revealing hyperbolic sets, heteroclinic tangles, and horseshoes, with bounds on the Hausdorff dimension of invariant sets.

Contribution

It demonstrates the existence of hyperbolic structures and chaotic dynamics near the universal map, extending understanding of area-preserving bifurcations with rigorous bounds.

Findings

Existence of a bi-infinite heteroclinic tangle with transversally intersecting manifolds.

Presence of topological Markov chains on invariant sets.

Horseshoe dynamics for the third iterate near the universal map.

Abstract

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. 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 $\{z_k\}$, $k \in \fZ$, |z_k| \converge{{k \to \infty}} 0, \quad |z_k| \converge{{k \to -\infty}} \infty, whose stable and unstable manifolds intersect transversally; and, for any $N \in \fN$, 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 {\it unbounded} and {\it 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: $$ 0.7673 \ge {\rm dim}_H(\cC_F) \ge \varepsilon \approx 0.00044 e^{-1797}.$$