Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

Abstract

We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e. that the renormalizations of any two C 2+α -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.

The rigidity conjecture

“…Rigidity is known to hold for many classes of systems with bounded combinatorics and sufficient smoothness. For example: Kleinian groups [27], circle diffeomorphisms [13,33], critical circle homeomorphisms [6,10,11,31,32], unimodal maps [7, 17-19, 24, 25, 29], circle maps with breakpoints [14,15].…”

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

“…We will refer to (A) as the Denjoy estimate. As we showed in [10], the constant λ ∈ (0, 1) can be chosen uniformly for all T with the same size of the break c and Hölder exponent α.…”

“…q n−1 +iqn )| < 1, the bound (3.123) follows by summing up the inequalities In the proof of Theorem 1.2, we will use the following properties of renormalizations of circle diffeomorphisms with a break that were proven in [10]. Let T be a C 2+α -smooth circle diffeomorphisms with a break of size c ∈ R + \{1} at x br = x 0 and an irrational rotation number ρ ∈ (0, 1).…”

“…Theorem 1.1 ( [10]) Let α ∈ (0, 1) and let c ∈ R + \{1}. There exists λ ∈ (0, 1) such that, for every two C 2+α -smooth circle diffeomorphisms with a break T and T , with the same irrational rotation number ρ ∈ (0, 1), and the same size of the break c, there exists C > 0, such that the renormalizations f n and f n of T and T , respectively, satisfy f n − f n C 2 ≤ Cλ n , for all n ∈ N.…”

A Sufficient Condition for Absolute Continuity of Conjugations between Interval Exchange Maps

“…with the same bounded-type combinatorics and zero mean nonlinearities C 1 -smoothly conjugate to each other. In the case of circle maps it was shown that for almost all rotation numbers every two C 2+ε -smooth circle homeomorphisms with a break point, with the same irrational rotation number and the same size of the break are C 1 -smoothly conjugate to each other [7][8][9]. Let us note that statement on the regularity of conjugating map can be obtained by using the convergence of renormalizations of given maps.…”