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arXiv · 2609.21481

Dense times-b orbits from times-a invariant sets and measures, in zero entropy

Michael Hochman
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Abstract

Let $a,b$ be multiplicatively independent integers. We prove that if $μ$ is a zero-entropy non-atomic $\times a$-invariant measure on $[0,1]$, then $μ$-a.e.~$x$ has dense $\times b$-orbit. Related statements follow for many closed $\times a$-invariant sets, and in particular we conclude the existence of irrational numbers with minimal complexity in base $a$ and maximal complexity in base $b$. We extend~these results to commuting endomorphisms of the torus under an irreducibility and hyperbolicity condition.

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Michael Hochman. 2026-09-18. Dense times-b orbits from times-a invariant sets and measures, in zero entropy. https://arxiv.org/abs/2609.21481

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