Geodesic
An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks
Russian journal of nonlinear dynamics, Tome 14 (2018) no. 4, pp. 553-577

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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. 
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     author = {A. Dzhalilov and D. Mayer and S. Djalilov and A. Aliyev},
     title = {An {Extention} of {Herman{\textquoteright}s} {Theorem} for {Nonlinear} {Circle} {Maps} with {Two} {Breaks}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {553--577},
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     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2018_14_4_a8/}
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A. Dzhalilov; D. Mayer; S. Djalilov; A. Aliyev. An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks. Russian journal of nonlinear dynamics, Tome 14 (2018) no. 4, pp. 553-577. http://geodesic.mathdoc.fr/item/ND_2018_14_4_a8/