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PreprintarXiv licencearXiv

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 × a-invariant measure on [0,1], then μ-a.e.~x has dense × b-orbit. Related statements follow for many closed × 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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Cite

BibTeXRIS
BibTeX
@misc{hochman2026dense,
  title = {{Dense times-b orbits from times-a invariant sets and measures, in zero entropy}},
  author = {Michael Hochman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21481},
  eprint = {2609.21481},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21481},
}
RIS
TY  - UNPB
TI  - Dense times-b orbits from times-a invariant sets and measures, in zero entropy
AU  - Michael Hochman
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21481
UR  - https://arxiv.org/abs/2609.21481
AB  - Let a,b be multiplicatively independent integers. We prove that if μ is a zero-entropy non-atomic × a-invariant measure on [0,1], then μ-a.e.~x has dense × b-orbit. Related statements follow for many closed × 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.
ER  -