Simultaneous dense and nondense orbits for noncommuting toral endomorphisms

TL;DR

This paper proves that under certain conditions on hyperbolic and quasihyperbolic toral endomorphisms, the intersection of points with equidistributing orbits and nondense orbits has full Hausdorff dimension.

Contribution

It establishes full Hausdorff dimension for intersections of specific orbit sets under noncommuting toral endomorphisms, extending previous results to broader classes.

Findings

Full Hausdorff dimension of intersection under hyperbolic endomorphisms.

Full Hausdorff dimension when including central eigenspaces in quasihyperbolic automorphisms.

Results apply to noncommuting toral endomorphisms with spanning conditions.

Abstract

Let $S$ and $T$ be hyperbolic endomorphisms of $\mathbb{T}^d$ with the property that the span of the subspace contracted by $S$ along with the subspace contracted by $T$ is $\mathbb{R}^d$. We show that the Hausdorff dimension of the intersection of the set of points with equidistributing orbits under $S$ with the set of points with nondense orbit under $T$ is full. In the case that $S$ and $T$ are quasihyperbolic automorphisms, we prove that the Hausdorff dimension of the intersection is again full when we assume that $\mathbb{R}^d$ is spanned by the subspaces contracted by $S$ and $T$ along with the central eigenspaces of $S$ and $T$.