Nondense Orbits for Anosov Diffeomorphisms of the $2$-Torus

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 of Tseng (2009) for $C^2$-expanding maps of the circle.