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arXiv · 1407.4014

Simultaneous dense and nondense orbits for noncommuting toral endomorphisms

Beverly Lytle·Alex Maier
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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 $T$.

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Beverly Lytle, Alex Maier. 2014-07-15. Simultaneous dense and nondense orbits for noncommuting toral endomorphisms. https://arxiv.org/abs/1407.4014

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