On conjugations of circle homeomorphisms with two break points

Habibulla Akhadkulov; Akhtam Dzhalilov; Dieter Mayer

doi:10.1017/etds.2012.159

paper

On conjugations of circle homeomorphisms with two break points

arXiv:1110.6125 · doi:10.1017/etds.2012.159

Abstract

Let $f_{i} \in C^{2 + α} (S^{1} ∖ {a_{i}, b_{i}}), α > 0, i = 1, 2$ be circle homeomorphisms with two break points $a_{i}, b_{i}$ , i.e. discontinuities in the derivative $f_{i}$ , with identical irrational rotation number $r h o$ and $μ_{1} ([a_{1}, b_{1}]) = μ_{2} ([a_{2}, b_{2}])$ , where $μ_{i}$ are invariant measures of $f_{i}$ . Suppose the products of the jump ratios of $D f_{1}$ and $D f_{2}$ do not coincide, i.e. $\frac{D f _{1} ( a _{1} - 0 )}{D f _{1} ( a _{1} + 0 )} \times \frac{D f _{1} ( b _{1} - 0 )}{D f _{1} ( b _{1} + 0 )} \neq = \frac{D f _{2} ( a _{2} - 0 )}{D f _{2} ( a _{2} + 0 )} \times \frac{D f _{2} ( b _{2} - 0 )}{D f _{2} ( b _{2} + 0 )}$ . Then the map $ψ$ conjugating $f_{1}$ and $f_{2}$ is a singular function, i.e. it is continuous on $S^{1}$ , but $D ψ = 0$ a.e. with respect to Lebesgue measure

16 pages, 2 figures, to appear in Ergodic Theory and Dynamical Systems

On conjugations of circle homeomorphisms with two break points

Abstract

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