{
  "id": "2205.09398",
  "version": "v1",
  "published": "2022-05-19T08:52:16.000Z",
  "updated": "2022-05-19T08:52:16.000Z",
  "title": "The thermodynamic formalism and central limit theorem for stochastic perturbations of circle maps with a break",
  "authors": [
    "Akhtam Dzhalilov",
    "Dieter Mayer",
    "Abdurahmon Aliyev"
  ],
  "comment": "34 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $T\\in C^{2+\\varepsilon}(S^{1}\\setminus\\{x_{b}\\}),\\,\\,\\varepsilon>0,$ be an orientation preserving circle homeomorphism with rotation number $\\rho_T=[k_{1},k_{2},..,k_{m},1,1,...],\\,\\,m\\geq1$, and a single break point $x_{b}$. We consider the stochastic sequence $ \\overline{z}_{n+1}(z_0,\\sigma) = T(\\overline{z}_{n}) + \\sigma \\xi_{n+1},\\,\\overline{z}_{0}:=z_0\\in S^1$, where $\\{\\xi_{n},\\,n=1,2,...\\}$ is a sequence of real valued independent mean zero random variables of comparable sizes, and $\\sigma > 0$ is a small parameter. Using the renormalization group technique de la Llave et al. proved for stochastic perturbations of one-dim. interval maps a central limit theorem (CLT) and the rate of convergence. In the present paper we extend their results to circle homeomorphisms with a break point by using the thermodynamic formalism constructed recently by Dzhalilov et al.. for such maps. This formalism and the dynamical partition $P_n(T,x_b)$ determined by the break point allows us, following the work of Vul et al., to establish a symbolic dynamics for any $z\\in S^1$ and to define a transfer operator whose leading eigenvalue is used to bound the Lyapunov function. For a special sequence $\\{n_m\\}, m\\to\\infty$, the barycentric coefficient of any $z_k=T^kz_0$ not intersecting the orbit of $x_b$ is universally bounded in the corresponding interval in $P_{n_m}(T,x_b)$. A Taylor expansion of $ \\overline{z}_{n}(z_0,\\sigma)$ in $\\{\\xi_i\\}$ leads to the decomposition into the term $T^n(z_0)$, a linearized effective noise and higher order terms in $\\{\\xi_i\\}$. This is possible however only in certain neighbourhoods $A_k^{n_m}$ of the points $T^k z_0$ not containing break points of $T^{q_{n_m}}$, with $q_{n}$ the first return times of $T$. Proving the CLT for the linearized process leads finally to the proof of our extension of results of de la Llave et al..",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T08:52:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C05",
      "37C15",
      "37E05",
      "37E10",
      "37E20",
      "37B10"
    ],
    "keywords": [
      "central limit theorem",
      "thermodynamic formalism",
      "stochastic perturbations",
      "break point",
      "circle maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}