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

Abstract

Given an integer nonsingular $n\times n$ 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 $\{M^{k}x\mathrm{mod}{\mathbb{Z}}^n:k\in \mathbb{N}\}$. 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 integer nonsingular $M$, 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.

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

References in corpus (3)

• Schmidt games and Markov partitions
• On shrinking targets for Z^m actions on tori
• Badly approximable systems of affine forms, fractals, and Schmidt games

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• Schmidt's game, fractals, and numbers normal to no base
• Schmidt Games and Nondense forward Orbits of certain Partially Hyperbolic Systems
• Sets of inhomogeneous linear forms can be not isotropically winning
• Cylinder absolute games on solenoids
• Diophantine approximations on random fractals
• Winning of inhomogeneous bad for curves
• Bounded Geodesics on Locally Symmetric Spaces