{
  "id": "2212.07215",
  "version": "v1",
  "published": "2022-12-14T13:27:09.000Z",
  "updated": "2022-12-14T13:27:09.000Z",
  "title": "On self-affine measures associated to strongly irreducible and proximal systems",
  "authors": [
    "Ariel Rapaport"
  ],
  "comment": "89 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\mu$ be a self-affine measure on $\\mathbb{R}^{d}$ associated to an affine IFS $\\Phi$ and a positive probability vector $p$. Suppose that the maps in $\\Phi$ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that $\\dim\\mu$ is equal to the Lyapunov dimension $\\dim_{L}(\\Phi,p)$ whenever $d=3$ and $\\Phi$ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring $\\dim\\mu=\\min\\{d,\\dim_{L}(\\Phi,p)\\}$, from which earlier results in the planar case also follow. Additionally, we prove that $\\dim\\mu=d$ whenever $\\Phi$ is Diophantine (which holds e.g. when $\\Phi$ is defined by algebraic parameters) and the entropy of the random walk generated by $\\Phi$ and $p$ is at least $(\\chi_{1}-\\chi_{d})\\frac{(d-1)(d-2)}{2}-\\sum_{k=1}^{d}\\chi_{k}$, where $0>\\chi_{1}\\ge...\\ge\\chi_{d}$ are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of $\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-14T13:27:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37C45"
    ],
    "keywords": [
      "self-affine measure",
      "proximal systems",
      "strongly irreducible",
      "milder strong open set condition",
      "strong separation condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 89,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}