Simultaneous Equidistribution and Nondense Points for Noncommuting Toral Automorphisms

TL;DR

This paper proves that in prime dimensions, for two non-commuting totally irreducible toral automorphisms, the set of points that equidistribute under one automorphism but have non-dense orbits under the other has full Hausdorff dimension, highlighting a nuanced dynamical behavior.

Contribution

It establishes the full Hausdorff dimension of points with contrasting orbit properties for non-commuting automorphisms in prime dimensions, extending understanding of their dynamical complexity.

Findings

Full Hausdorff dimension set in prime dimensions

Failure of argument in non-prime dimensions with algebraic relations

Distinct orbit behaviors under non-commuting automorphisms

Abstract

We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.