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

TL;DR

This paper demonstrates that in specific non-algebraic dynamical systems, the set of points with non-dense orbits is large in the sense of Schmidt games, using advanced techniques to analyze their properties.

Contribution

It applies Schmidt game techniques to non-algebraic systems, showing non-dense orbit sets are winning sets under certain smoothness and conformality conditions.

Findings

Non-dense orbit sets are winning for Schmidt games in $C^2$-Anosov diffeomorphisms.

Non-dense orbit sets are winning for Schmidt games in $C^{1+ heta}$-expanding endomorphisms.

The results extend the understanding of orbit distribution in non-algebraic dynamical systems.

Abstract

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 $C^{1+\theta}$-expanding endomorphism the set of points with non-dense forward orbit is a winning set for certain variants of Schmidt games.