An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks

    Received 15 November 2018; accepted 15 November 2018

    2018, Vol. 14, no. 4, pp.  553-577

    Author(s): Dzhalilov A., Mayer D., Djalilov S., Aliyev A.

    M. Herman showed that the invariant measure $\mu_h$ of a piecewise linear (PL) circle homeomorphism $h$ with two break points and an irrational rotation number $\rho_{h}$ is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise $ C^{2+\varepsilon} $-circle maps $f$ with an irrational rotation number $\rho_f$ and two break points $ a_{0}, c_{0}$, which do not lie on the same orbit and whose total jump ratio is $\sigma_f=1$, as follows: if $\mu_f$ denotes the invariant measure of the $P$-homeomorphism $f$, then for Lebesgue almost all values of $\mu_f([a_0, c_{0}])$ the measure $\mu_f$ is singular with respect to Lebesgue measure.
    Keywords: piecewise-smooth circle homeomorphism, break point, rotation number, invariant measure
    Citation: Dzhalilov A., Mayer D., Djalilov S., Aliyev A., An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  553-577

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