On Herman’s Theorem for Piecewise Smooth Circle Maps with Two Breaks

Abstract

In this paper we consider general orientation preserving circle homeomorphisms \(f\in C^{2+\varepsilon } (S ^{1}\setminus \{ a^{(0)}, c^{(0)} \} )\;, \varepsilon >0\), with an irrational rotation number \(\rho _{f}\) and two break points \(a^{(0)}, c^{(0)} \). Denote by \(\sigma _f(x_b):=\frac{Df_{-}(x_b)}{Df_{+}(x_b)},\, x_b=a^{(0)},c^{(0)}\), the jump ratios of f at the two break points and by \(\sigma _f:= \sigma _f(a^{(0)})\cdot \sigma _f(c^{(0)}) \) its total jump ratio. Let h be a piecewise-linear (PL) circle homeomorphism with two break points \(a_{0}\), \(c_{0}\), irrational rotation number \(\rho _{h}\) and total jump ratio \(\sigma _h=1\). Denote by \(\mathbf {B}_{n}(h)\) the partition determined by the break points of \(h^{q_{n}}\) and by \(\mu _{h}\) the unique h-invariant probability measure. It is shown that the derivative \(Dh^{q_{n}}\) is constant on every element of \(\mathbf {B}_{n}(h)\) and takes either two or three values. Furthermore we prove, that \( \log Dh^{q_{n}}\) can be expressed in terms of the \(\mu _{h}-\) measures of some intervals of the partition \(\mathbf {B}_{n}(h)\) multiplied by the logarithm of the jump ratio \( \sigma _{h}(a_{0})\) of h at the break point \(a_{0}\). M. Herman showed, that the invariant measure \(\mu _h\) is absolutely continuous iff the two break points belong to the same orbit. We complement Herman’s result for the above class of piecewise \( C^{2+\varepsilon } \)-circle maps f with irrational rotation number \(\rho _f\) and two break points \( a^{(0)}, c^{(0)}\) not lying on the same orbit with total jump ratio \(\sigma _f=1\) as follows: if \(\mu _f\) denotes the invariant measure of the P-homeomorphism f, then for almost all values of \(\mu _f([a^{(0)}, c^{(0)}])\) the measure \(\mu _f\) is singular with respect to Lebesgue measure.