z-logo
open-access-imgOpen Access
The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break
Author(s) -
Akhtam Dzhalilov,
J.J. Karimov
Publication year - 2020
Publication title -
vestnik udmurtskogo universiteta. matematika, mehanika, kompʹûternye nauki
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.354
H-Index - 8
eISSN - 2076-5959
pISSN - 1994-9197
DOI - 10.35634/vm200301
Subject(s) - ergodic theory , singularity , combinatorics , invariant measure , lambda , homeomorphism (graph theory) , mathematics , exponent , invariant (physics) , lyapunov exponent , irrational number , probability measure , physics , mathematical physics , mathematical analysis , geometry , quantum mechanics , linguistics , philosophy , nonlinear system
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 beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots] $, $ m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $ T $ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $ \mu_{G} $ of any circle homeomorphism $ G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $ x_{b} $ and the irrational rotation number $ \rho_{G} $ is singular with respect to the Lebesgue measure $ \lambda $ on the circle, i.e., there is a measurable subset of $ A \subset S^{1} $ such that $ \mu_ {G} (A) = 1 $ and $ \lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $ T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, with one break at the point $ x_{b} $ and rotation number equal to the golden mean, i.e., $ \rho_{T}:= \frac {\sqrt{5} -1}{2} $. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $ \mu_{T} $ of homeomorphism $ T $.

The content you want is available to Zendy users.

Already have an account? Click here to sign in.
Having issues? You can contact us here