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    Dieter Mayer

    Leibnizstr. 10, D- 38678 Clausthal-Zellerfeld
    Theoretische Physik, Technische Universitat Clausthal


    Dzhalilov A., Mayer D., Aliyev A.
    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.
    Keywords: circle map, rotation number, break point, stochastic perturbation, central limit theorem, thermodynamic formalism
    Citation: Dzhalilov A., Mayer D., Aliyev A.,  The Thermodynamic Formalism and the Central Limit Theorem for Stochastic Perturbations of Circle Maps with a Break, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp.  253-287
    Dzhalilov A., Mayer D., Djalilov S., Aliyev A.
    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.
    Keywords: piecewise-smooth circle homeomorphism, break point, rotation number, invariant measure
    Citation: Dzhalilov A., Mayer D., Djalilov S., Aliyev A.,  An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  553-577

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