{
  "id": "2307.11284",
  "version": "v1",
  "published": "2023-07-21T01:01:26.000Z",
  "updated": "2023-07-21T01:01:26.000Z",
  "title": "Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems",
  "authors": [
    "Wenmeng Zhang",
    "Kening Lu",
    "Weinian Zhang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,\\beta}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,\\beta}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-21T01:01:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C15",
      "37H15"
    ],
    "keywords": [
      "random dynamical systems",
      "smooth invariant foliations",
      "partial second order",
      "bunching condition",
      "satisfy belitskiis partial second"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}