Nonrigidity for circle homeomorphisms with several break points

Abstract

Let $f$ and $g$ be two class $P$-homeomorphisms of the circle $S^{1}$ with break points singularities. Assume that the derivatives $\textrm{Df}$ and $\textrm{Dg}$ are absolutely continuous on every continuity interval of $\textrm{Df}$ and $\textrm{Dg}$ respectively. Denote by $C(f)$ the set of break points of $f$. For $c\in S^{1}$, denote by $\pi_{s, O_{f}(c)}(f)$ the product of $f-$ jumps in break points lying to the $f-$ orbit of $c$ and by $\textrm{SO}(f) = \{O_{f}(c):~c \in C(f)~\textrm{and}~\pi_{s, O_{f}(c)}(f)\neq 1\}$, called the set of singular $f$-orbits. The maps $f$ and $g$ are called break-equivalent if there exists a topological conjugating $h$ such that $h(\textrm{SO}(f))=\textrm{SO}(g)~~ \textrm{and} ~~ \pi_{s, O_{g}(h(c))}(g) = \pi_{s, O_{f}(c)}(f) ~~ \textrm{for all}~~ c\in \textrm{SO}(f)$. Assume that $f$ and $g$ have the same irrational rotation number of bounded type. We prove that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function i.e. it is a continuous on $S^{1}$, but $\textrm{Dh}(x)=0$ a.e. with respect to the Lebesgue measure. As a consequence if for some point $d\in \textrm{SO}(f)$, $\pi_{s, O_{g}(d)}(g)\notin \{\pi_{s, O_{f}(c)}(f): c\in C(f)\}$, then the homeomorphism conjugation $h$ is a singular function. This later result generalizes previous results for one and two break points obtained by Dzhalilov-Akin-Temir and Akhadkulov-Dzhalilov-Noorani. Moreover, if $f$ and $g$ do not have the same number of singular orbits then the homeomorphism conjugating $f$ to $g$ is a singular function.