Pointwise perturbations of countable Markov maps

TL;DR

This paper investigates how the Hausdorff dimension of certain conjugacy sets between countable Markov maps behaves under pointwise perturbations, revealing a form of continuity in the Hausdorff dimension despite other singularity measures failing to be continuous.

Contribution

The authors establish a continuity theorem for Hausdorff dimensions of conjugacy-related sets under pointwise convergence of inverse branches of countable Markov maps, with applications to intermittent dynamical systems.

Findings

Hausdorff dimension of conjugacy sets converges to 1 under perturbations

Other measures like Hölder exponent are not continuous under the same conditions

Application to perturbations of Manneville-Pomeau maps

Abstract

We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let $T_k$ and $T$ be expanding countable Markov maps such that the inverse branches of $T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then under suitable regularity assumptions on the maps $T_k$ and $T$ the following limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : \theta_k'(x) \neq 0\} = 1,$$ where $\theta_k$ is the topological conjugacy between $T_k$ and $T$ and $\dim_\mathrm{H}$ stands for the Hausdorff dimension. This is in contrast with the fact that other natural quantities measuring the singularity of $\theta_k$ fail to be continuous in this manner under pointwise convergence such as the H\"older exponent of $\theta_k$ or the Hausdorff dimension $\dim_\mathrm{H} (\mu \circ \theta_k)$ for the preimage of the absolutely continuous invariant measure $\mu$ for $T$. As an application we obtain a perturbation theorem in non-uniformly hyperbolic dynamics for conjugacies between intermittent Manneville-Pomeau maps $x \mapsto x + x^{1+\alpha} \mod 1$ when varying the parameter $\alpha$.