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

Abstract

Let $a,b$ be multiplicatively independent integers. We prove that if $\mu$ is a zero-entropy non-atomic $\times a$-invariant measure on $[0,1]$, then $\mu$-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.

27 pages