The Thermodynamic Formalism and the 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 $\rho_{T}=[k_{1},k_{2},\ldots,k_{m},1,1,\ldots]$, $m\geq 1$, and a single break point $x_{b}$. Stochastic perturbations $\overline{z}_{n+1}=T(\overline{z}_{n})+\sigma\xi_{n+1}$, $\overline{z}_{0}:=z\in S^{1}$ of critical circle maps have been studied some time ago by Diaz-Espinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai’s et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point ${z\in S^{1}}$ and to define a transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables $\xi_{i}$ into the linear term $L_{n}(z_{0})=\xi_{n}+\sum\limits_{k=1}^{n-1}\xi_{k}\prod\limits_{j=k}^{n-1}T^{% \prime}(z_{j})$, ${z_{0}\in S^{1}}$ and a higher order term, which is possible in a neighbourhood $A_{k}^{n}$ of the points $z_{k}$, ${k\leq n-1}$, not containing the break points of $T^{n}$. For this we construct for a certain sequence $\{n_{m}\}$ a series of neighbourhoods $A_{k}^{n_{m}}$ of the points $z_{k}$ which do not contain any break point of the map $T^{q_{n_{m}}}$, $q_{n_{m}}$ the first return times of $T$. The proof of our results follows from the proof of the central limit theorem for the linearized process.