Dieter Mayer

Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_b^{}\})$, $\varepsilon>0$, be an orientation preserving circle homeomorphism with rotation number $\rho_T^{}=[k_1^{},\,k_2^{},\,\ldots,\,k_m^{},\,1,\,1,\,\ldots]$, $m\ge1$, 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 $L_n^{}(z_0^{})= \xi_n^{}+\sum\limits_{k=1}^{n-1}\xi_k^{}\prod\limits_{j=k}^{n-1} T'(z_j^{})$, ${z_0^{}\in S^1}$ and a higher order term, which is possible in a neighbourhood $A_k^n$ of the points $z_k^{}$, ${k\le n-1}$, 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.

M. Herman showed that the invariant measure $\mu_h$ of a piecewise linear (PL) circle homeomorphism $h$ with two break points and an irrational rotation number $\rho_{h}$ is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise $ C^{2+\varepsilon} $-circle maps $f$ with an irrational rotation number $\rho_f$ and two break points $ a_{0}, c_{0}$, which do not lie on the same orbit and whose total jump ratio is $\sigma_f=1$, as follows: if $\mu_f$ denotes the invariant measure of the $P$-homeomorphism $f$, then for Lebesgue almost all values of $\mu_f([a_0, c_{0}])$ the measure $\mu_f$ is singular with respect to Lebesgue measure.