Description:

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

Abstract:

Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugated by a homeomorphism $h$. It was proved by de la Llave in 1992 that the conjugacy $h$ is automatically $C^{1+}$ if and only if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all periodic orbits. We prove that if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$, then $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ that is exponentially close to $h$ in the $C^0$ norm, and such that $f$ and $f_N:=\overline{h}_N^{-1}\circ g\circ \overline{h}_N$ is exponentially close to $f$ in the $C^1$ norm.