零熵下由×a不变集与测度生成的稠密×b轨道
Dense times-b orbits from times-a invariant sets and measures, in zero entropy
AI总结：
本文证明在零熵条件下，乘法独立整数对应的不变测度几乎处处具有稠密轨道，并推广至环面自同态，得出复杂度极值无理数的存在性。
AI中文摘要：
设$a,b$为乘法独立的整数。我们证明：若$\mu$是$[0,1]$上的零熵非原子$\times a$不变测度，则对$\mu$-几乎所有的$x$，其$\times b$轨道是稠密的。相关结论对许多闭的$\times a$不变集也成立，特别地，我们由此得出存在无理数，其在基数$a$下复杂度最小，在基数$b$下复杂度最大。我们将这些结果推广到环面在不可约且双曲条件下的交换自同态。
英文摘要：
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.
