Ergodic Theory and Dynamical Systems

Research Article

On orbits of endomorphisms of tori and the Schmidt game

S. G. Dania1

a1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Abstract

We show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any xF and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.

(Received October 05 1987)