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

Abstract

Let $T\in C^{2+\epsilon }(S^1\setminus \{x_b\}),\epsilon >0,$ be an orientation preserving circle homeomorphism with rotation number ${\rho }_T=[k_1,k_2,..,k_m,1,1,...],m\ge 1$, 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 $z_k=T^k{z}_0$ 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..

34 pages