THE THERMODYNAMIC FORMALISM FOR CIRCLE MAPS WITH ALGEBRAIC ROTATION NUMBER

Authors

  • Javlon Karimov Turin Polytechnic University in Tashkent

Keywords:

circle homeomorphism, break point, rotation number, invariant measure, symbolic dynamics, shift map, thermodynamic formalism, renormalization transformation

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 $B(T_{i}),\,i=1,2.$

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Published

2021-09-01

How to Cite

Karimov, J. (2021). THE THERMODYNAMIC FORMALISM FOR CIRCLE MAPS WITH ALGEBRAIC ROTATION NUMBER. Acta of Turin Polytechnic University in Tashkent, 11(3). Retrieved from http://acta.polito.uz/index.php/journal/article/view/89