Nondense orbits for Anosov diffeomorphisms of the 2-torus

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Let λ denote the probability Lebesgue measure on T². For any C²-Anosov diffeomorphism of the 2-torus preserving λ 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 [Theorem 1.4]T4 for C²-expanding maps of the circle.