On non-dense orbits of certain non-algebraic dynamical systems

In this paper, we manage to apply Schmidt games to certain non-algebraic dynamical systems. More precisely, we show that the set of points with non-dense forward orbit under a $C^2$-Anosov diffeomorphism with conformality on unstable manifolds is a winning set for Schmidt games. It is also proved that for a $C^{1+θ}$-expanding endomorphism the set of points with non-dense forward orbit is a winning set for certain variants of Schmidt games.

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