K. Khanin, S. Kocic
Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks
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ABSTRACT. We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e. that the renormalizations of any two $C^{2+lpha}$-smooth circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, approach each other exponentially fast in the $C^2$-topology. As a corollary, we obtain the following rigidity result: for almost all irrational rotation numbers, any two circle diffeomorphisms with a break, with the same rotation number and the same size of the break, are $C^1$-smoothly conjugate to each other.