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

TL;DR

This paper extends the thermodynamic formalism and central limit theorem for stochastic perturbations from interval maps to circle homeomorphisms with a break point, using advanced dynamical and probabilistic techniques.

Contribution

It introduces a novel extension of the CLT and thermodynamic formalism to circle maps with a break, building on recent symbolic dynamics and transfer operator methods.

Findings

Established a CLT for stochastic perturbations of circle maps with a break.

Bounded the Lyapunov function using spectral properties of a transfer operator.

Demonstrated universal bounds on barycentric coefficients away from break points.

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..