THE THERMODYNAMIC FORMALISM FOR CIRCLE MAPS WITH ALGEBRAIC ROTATION NUMBER

Abstract

In present paper we study the orientation preserving circle homeomorphisms with singularity of break type.
Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with
one break point $x_{b}$, at which $ T'(x) $ has a discontinuity of the first kind and both one-sided derivatives
at the point $x_{b} $ are strictly positive.
Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction
has a form $ \rho:=\omega_{k} = [k,\,k,\ldots,k,\ldots] = \frac{-k + \sqrt {{{k} ^ {2}} + 4}} {2}, \, \, k \ge 1 .$
E. Vul and K. Khanin in \cite{VKh} showed that the renormalization transformation on the space of such circle
maps has unique periodic point $( F_{i},G_{i}),\,\,i=1,2$ with period two. Moreover,
$ F_{i}$ and $G_{i})$ are fractional linear maps.
We denote by $T_{i},\,i=1,2$ the circle homeomorphisms associated by pair $(F_{i},G_{i}).$
Let $B(T_{i}),\,i=1,2$ the set of all circle maps which are $C^{1}$ conjugated to $T_{i},\,i=1,2.$
We build a thermodynamic formalism for all maps of $B(T_{i}),\,i=1,2.$