Schmidt's game, fractals, and orbits of toral endomorphisms

Details Schmidt's game, fractals, and orbits of toral endomorphisms Schmidt's game, fractals, and orbits of toral endomorphisms

2010-01-02

arxiv.org/abs/1001.0318v2

Mathematics

Dynamical Systems

13 pages; a slightly modified version, to appear in Ergodic Theory Dynamical Systems

Scientific paper

10.1017/S0143385710000374

Given an integer nonsingular $n\times bn$ matrix $M$ and a point $y \in \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod \Z^n: k\in\N\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y \in \mathbb{R}^n/\mathbb{Z}^n$ and $M\in \GL_n(\R) \cap \M_{n\times n}(\Z)$, and in fact replacing the sequence of powers of $M$ by any lacunary sequence of (not necessarily integer) $m\times n$ matrices. Furthermore, we show that sets of the form $\tilde E(M,y)$ and their generalizations always intersect with `sufficiently regular' fractal subsets of $\mathbb{R}^n$. As an application we give an alternative proof of a recent result of Einsiedler and Tseng on badly approximable systems of affine forms.

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