A numerical study of infinitely renormalizable area-preserving maps

It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Holder continuous of exponent alpha>0. In this paper we investigate numerically the specific value of alpha. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real

No elliptic islands for the universal area-preserving map

A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to prove the existence of a \textit{universal area-preserving map}, a map with hyperbolic orbits of all binary periods. The existence of a horseshoe, with positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In this paper the coexistence problem is studied, and a computer-aided proof is given that no elliptic islands with period less than 20 exist in the domain. It is also shown that less than 1.5% of the measure of the domain consists of elliptic islands. This is proven by showing that the measure of initial conditions that escape to infinity is at least 98.5% of the measure of the domain, and we conjecture that the escaping set has full measure. This is highly unexpected, since generically it is believed that for conservative systems hyperbolicity and ellipticity coexist

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

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}.$