Conjugacies between P-homeomorphisms with several breaks

TL;DR

This paper investigates conjugacies between circle homeomorphisms with break points, showing that differences in total jump ratios lead to conjugacies being singular functions, under certain smoothness conditions.

Contribution

It establishes that conjugacies between such homeomorphisms are singular functions when their total jump ratios differ, extending understanding of their structural differences.

Findings

Conjugacies are singular functions if total jump ratios differ.

Smoothness condition: $C^{2+ ext{epsilon}}$ on continuity intervals.

Total jump ratios determine the nature of conjugacy.

Abstract

Let $f_{i},i=1,2$ be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives $Df_{i}$ have jumps, and with identical irrational rotation number $\rho=\rho_{f_{1}}=\rho_{f_{2}}.$ The jump ratio of $f_{i}$ at the break point $b$ is denoted by $\sigma_{f_{i}}(b)$, i.e. $\sigma_{f_{i}}(b):=\frac{Df_{i}(b-0)}{Df_{i}(b+0)}$. Denote by $\sigma_{f_{i}}, i=1,2,$ the total jump ratio given by the product over all break points $b$ of the jump ratios $\sigma_{f_{i}}(b)$ of $f_{i}$. We prove, that for circle homeomorphisms $f_{i}, i=1,2$, which are $C^{2+\varepsilon}, \varepsilon>0$, on each interval of continuity of $Df_{i}$ and whose total jump ratios $\sigma_{f_{1}}$ and $\sigma_{f_{2}}$ do not coincide, the congugacy between $f_{1}$ and $f_{2}$ is a singular function.