Singular measures of circle homeomorphisms with two break points

Akhtam Dzhalilov; Isabelle Liousse; Dieter Mayer

paper

Singular measures of circle homeomorphisms with two break points

arXiv:0707.3528

Abstract

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

References in corpus (1)

Singular measures in circle dynamics