Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms

TL;DR

This paper demonstrates that two topologically conjugate area-preserving Anosov diffeomorphisms on the torus with matching Jacobian periodic data are approximately smoothly conjugate, with convergence rates uniform across bounded sets.

Contribution

It introduces a method to construct approximate smooth conjugacies based on weighted holonomy and effective Bowen's equidistribution, under periodic data matching.

Findings

Existence of a $C^{1+eta}$ conjugacy close to the topological conjugacy.

Uniform exponential convergence rates among bounded sets of diffeomorphisms.

Application of effective Bowen's theorem to estimate convergence.

Abstract

Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$. We show that $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ such that $h$ and $\overline{h}_N$ are $C^0$ exponentially close in $N$, and $f$ and $f_N:=\overline{h}_N^{-1}g\overline{h}_N$ are $C^1$ exponentially close in $N$. Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$ bounded set of Anosov diffeomorphisms. The main idea in constructing $\overline{h}_N$ is to do a ``weighted holonomy" construction, and the main technical tool in obtaining our estimates is a uniform effective version of Bowen's equidistribution theorem of weighted discrete orbits to the SRB measure.