Singular measures of circle homeomorphisms with two break points

TL;DR

This paper proves that for certain circle homeomorphisms with two break points and specific derivative conditions, the invariant measure is singular relative to Lebesgue measure.

Contribution

It establishes the singularity of the invariant measure for circle homeomorphisms with two break points under specified derivative and jump ratio conditions.

Findings

Invariant measure is singular w.r.t. Lebesgue measure.

Conditions on derivative and jump ratios are crucial.

Results extend understanding of measure singularity in dynamical systems.

Abstract

Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.