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Dimension theory of group actions by circle diffeomorphisms I: Stationary measures

  • Weikun He
  • Yuxiang Jiao
  • Disheng Xu
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Abstract

We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II "Dimension theory of group actions by circle diffeomorphisms II: Minimal sets". They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup T contained in Diff+1+Ξ±(π•Š1) without finite orbits has the same number of minimal sets as its inverse semigroup Tβˆ’1.

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Cite

BibTeXRIS
BibTeX
@misc{he2026dimension,
  title = {{Dimension theory of group actions by circle diffeomorphisms I: Stationary measures}},
  author = {Weikun He and Yuxiang Jiao and Disheng Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13254},
  eprint = {2610.13254},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13254},
}
RIS
TY  - UNPB
TI  - Dimension theory of group actions by circle diffeomorphisms I: Stationary measures
AU  - Weikun He
AU  - Yuxiang Jiao
AU  - Disheng Xu
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13254
UR  - https://arxiv.org/abs/2610.13254
AB  - We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II "Dimension theory of group actions by circle diffeomorphisms II: Minimal sets". They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup T contained in Diff_+^1+Ξ±(π•Š^1) without finite orbits has the same number of minimal sets as its inverse semigroup T^βˆ’1.
ER  -