@article{Karimov_2021, title={THE THERMODYNAMIC FORMALISM FOR CIRCLE MAPS WITH ALGEBRAIC ROTATION NUMBER}, volume={11}, url={https://acta.polito.uz/index.php/journal/article/view/89}, abstractNote={<p>In present paper we study the orientation preserving circle homeomorphisms with singularity of break type.<br>Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with<br>one break point $x_{b}$, at which $ T’(x) $ has a discontinuity of the first kind and both one-sided derivatives<br>at the point $x_{b} $ are strictly positive.<br>Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction<br>has a form $ \rho:=\omega_{k} = [k,\,k,\ldots,k,\ldots] = \frac{-k + \sqrt {k} ^ {2 } + 4 } {2}, \, \, k \ge 1 .$<br>E. Vul and K. Khanin in \cite{VKh} showed that the renormalization transformation on the space of such circle<br>maps has unique periodic point $( F_{i},G_{i}),\,\,i=1,2$ with period two. Moreover,<br>$ F_{i}$ and $G_{i})$ are fractional linear maps.<br>We denote by $T_{i},\,i=1,2$ the circle homeomorphisms associated by pair $(F_{i},G_{i}).$<br>Let $B(T_{i}),\,i=1,2$ the set of all circle maps which are $C^{1}$ conjugated to $T_{i},\,i=1,2.$<br>We build a thermodynamic formalism for all maps of $B(T_{i}),\,i=1,2.$</p>}, number={3}, journal={Acta of Turin Polytechnic University in Tashkent}, author={Karimov, Javlon}, year={2021}, month={Sep.} }