Nondense orbits for Anosov diffeomorphisms of the $2$-torus

TL;DR

This paper proves that for certain smooth Anosov diffeomorphisms on the 2-torus, the set of points with nondense orbits is large in a measure-theoretic sense, extending previous results from circle maps.

Contribution

It establishes that the set of nondense orbits is hyperplane absolute winning for measure-preserving $C^2$-Anosov diffeomorphisms on the 2-torus, generalizing earlier circle map results.

Findings

Set of nondense orbits is hyperplane absolute winning.

Results apply to measure-preserving $C^2$-Anosov diffeomorphisms.

Extends previous circle map theorems to toral diffeomorphisms.

Abstract

Let $\lambda$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $\lambda$ with measure-theoretic entropy equal to topological entropy, we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of the circle.