The thermodynamic formalism and central limit theorem for stochastic perturbations of circle maps with a break

Abstract

Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_{b}\}),\,\,\varepsilon>0,$ be an orientation preserving circle homeomorphism with rotation number $ρ_T=[k_{1},k_{2},..,k_{m},1,1,...],\,\,m\geq1$, and a single break point $x_{b}$. We consider the stochastic sequence $ \overline{z}_{n+1}(z_0,σ) = T(\overline{z}_{n}) + σξ_{n+1},\,\overline{z}_{0}:=z_0\in S^1$, where $\{ξ_{n},\,n=1,2,...\}$ is a sequence of real valued independent mean zero random variables of comparable sizes, and $σ> 0$ 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,σ)$ in $\{ξ_i\}$ leads to the decomposition into the term $T^n(z_0)$, a linearized effective noise and higher order terms in $\{ξ_i\}$. 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..