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Generic Rigidity for Circle Diffeomorphisms with Breaks

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Abstract

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).

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  1. Department of Mathematics, The University of Mississippi, Hume Hall 305, University, MS, 38677-1848, USA

    Saša Kocić

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  1. Saša Kocić
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Correspondence to Saša Kocić.

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Communicated by M. Lyubich

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Kocić, S. Generic Rigidity for Circle Diffeomorphisms with Breaks. Commun. Math. Phys. 344, 427–445 (2016). https://doi.org/10.1007/s00220-016-2615-x

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