Title: Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms
Abstract: A natural question asked in nearly all branches of mathematics is "When are two objects 'equivalent'?" The notion of equivalence should reflect what type of structure within our objects we wish to compare. In dynamical systems the natural notion of equivalence comes from conjugacies: Two maps $f,g: X \nightarrow X$ are said to be conjugate if there exists an invertible mapping $h: X \to X$ such that $h\circ f=g\circ h$. The regularity of the conjugacy $h$ determines the sense in which the systems are equivalent, i.e., if $h$ and $h^{-1}$ are measurable, then $f$ and $g$ are measurably equivalent, and if $h$ is a homeomorphism then they are topologically equivalent, etc. In this talk, we will let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate 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.