{
  "id": "2304.08372",
  "version": "v1",
  "published": "2023-04-17T15:35:06.000Z",
  "updated": "2023-04-17T15:35:06.000Z",
  "title": "On dimension theory of random walks and group actions by circle diffeomorphisms",
  "authors": [
    "Weikun He",
    "Yuxiang Jiao",
    "Disheng Xu"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we establish several results on the dimensional properties of invariant measures and sets associated to random walks and group actions by circle diffeomorphisms. Our main results include the exact dimensionality and a dimension formula of stationary measures, variational principles for dimensions in various settings and estimates of the Hausdorff dimensions of exceptional minimal sets. We also prove an approximation theorem for random walks on the circle which is analogous to the results of Katok, Avila-Crovisier-Wilkinson, Morris-Shmerkin. The proofs of our results are based on a combination of techniques including a new structure theorem for smooth random walks on circle, a dynamical generalization of the critical exponent of Fuchsian groups and some novel arguments inspired by the study of fractal geometry, hyperbolic geometry and holomorphic dynamics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-17T15:35:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "group actions",
      "circle diffeomorphisms",
      "dimension theory",
      "exceptional minimal sets",
      "smooth random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}