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 in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of the circle.

Minor typos corrected. Added more exposition