The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break

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 $T$.