C^1-rigidity of circle maps with breaks for almost all rotation numbers

Abstract

We prove that, for almost all irrational ρ ∈ (0, 1), every two C 2+α-smooth, α ∈ (0, 1), circle diffeomorphisms with a break point, i.e., a singular point where the derivative has a jump discontinuity, with the same rotation number ρ and the same size of the break c ∈ R + \{1}, are C 1-smoothly conjugate to each other. Résumé Nous démontrons que pour presque tous les irrationnels ρ ∈ (0, 1), deux difféomorphismes du cercle C 2+α lisses, α ∈ (0, 1), avec un point de singularité de type rupture où la dérivée a une discontinuité de saut, avec le même nombre de rotation ρ et la même taille de rupture c ∈ R + \{1}, sont C 1-conjugués l'un à l'autre.

Renormalization of circle diffeomorphisms with a break-type singularity

“…In addition, Khanin and Vul's result plays an important role to show 'K-regularity' of renormalizations which is developed in [14] and [17]. 'K-regularity' and the convergence of renormalizations together with a certain condition on rotation number ensure C 1 -rigidity of circle maps with singularities.…”

“…The rigidity problem for circle diffeomorphisms with breaks with quadratic rotation numbers was solved by Khanin and Khmelov [12] and for circle diffeomorphisms with breaks with half bounded rotation numbers by Khanin and Teplinsky [17]. Recently, this problem was completely solved by Khanin et al [14] for almost all irrational rotation numbers.…”

Circle homeomorphisms with breaks with no $\boldsymbol{C^{2-\nu}}$ conjugacy

“…

The rigidity theory for circle homeomorphisms with breaks has been studied intensively in the last 20 years. It was proved [15,21,17,19] that under mild conditions of the Diophantine type on the rotation number any two C 2+α smooth circle homeomorphisms with a break point are C 1 smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be C 2−ν even if the maps are analytic outside of the break points.

…”