The central limit theorem for random perturbations of rotations (Q1390755)

Let \(\{T_{\varepsilon,w},\varepsilon\in {\mathbf R}\), \(w\in {\mathbf T}={\mathbf R}/{\mathbf Z}\}\) be a family of skew-products of the two-dimensional torus \({\mathbf T}^2\) defined by \(T_{ \varepsilon,w}:(x,y) \to(S(x)\), \(y+w+\varepsilon x)\text{mod} 1\), where the transformation \(S\) of the base \({\mathbf T}\) is given by \(S(x)=2x \text{mod} 1\). The authors prove a functional central limit theorem for a stationary family \(\{T_{ \varepsilon,w}\}\) by giving an explicit bound for the \(L_2({\mathbf T}^2)\) norm of powers of the transfer operator applied to a function from \(L_2({\mathbf T}^2)\) and reducing the problem to the case of stationary ergodic martingale difference sequences with values in a separable Hilbert space of square-integrable functions, \(L_2({\mathbf T}^2)\).