Nonrigidity for circle homeomorphisms with several break points

TL;DR

This paper investigates the rigidity of circle homeomorphisms with multiple break points, showing that non break-equivalence leads to conjugacies that are singular functions with zero derivatives almost everywhere.

Contribution

It generalizes previous results by characterizing the conjugacy's regularity for circle homeomorphisms with several break points and different singular orbit structures.

Findings

Non break-equivalent maps have conjugacies that are singular functions.

Different numbers of singular orbits imply the conjugacy is a singular function.

Results extend prior work from one or two break points to multiple 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.