Rational rotation numbers for maps of the circle

Abstract

We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps likewith c = l. We prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0. In other words, the intervals on which frequency-locking occurs fill up the set of full measure.

The Attractor of Renormalization¶and Rigidity of Towers of Critical Circle Maps

“…For every m ≥ 1 such that the continued fraction expansion of ρ(f ) has at least m terms set I m ≡ [0, f q m (0)]. An important consequence ofŚwiatek-Herman real a priori bounds ( [Sw1,He]), is that when ρ(f ) is irrational, there exists M = M(f ) such that for all m ≥ M the intervals I m and I m+1 are K− commensurable, with a universal bound K > 1. For a more detailed statement of this and some proofs see [dFdM1].…”

Rigidity of critical circle mappings I

“…All estimates performed in this paper rely heavily on the real a-priori bounds of M. Herman [12] and G.Światek [18]. These bounds are revisited in §3.…”

Rigidity of critical circle mappings II

“…The proof of this theorem uses both real and complex-analytic tools. We use the real a-priori bounds ofŚwiatek and Herman (see [14], [6] and [3]) in the form stated in §2. The complex-analytic part we develop here combines some of the powerful new ideas on conformal rigidity and renormalization recently developed by McMullen in [9], [10], [11] with the basic theory of holomorphic commuting pairs introduced in [2] and a generalization of the Lyubich-Yampolsky approach to the complex bounds given in [8], [15].…”