Real bounds and quasisymmetric rigidity of multicritical circle maps

Abstract

Abstract. Let f, g : S 1 → S 1 be two C 3 critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if h : S 1 → S 1 is a topological conjugacy between f and g and h maps the critical points of f to the critical points of g, then h is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien [5]. However, unlike the proof given in [5], which relies on heavy complex-analytic machinery, our proof uses purely realvariable methods, and is valid for non-integer critical exponents as well. We do not require h to preserve the power-law exponents at corresponding critical points.

Beau bounds for multicritical circle maps

“…The fact that the conjugacy h is quasi-symmetric, in the above corollary, is the main theorem proved in [3]. The improvement here is that the quasi-symmetric distortion of h is asymptotically universal .…”

“…Such bounds have been obtained by M. Herman [10] and G.Światek [19]. A detailed proof of such bounds in the case of multicritical circle maps can be found in [3].…”

“…This theorem is, in fact, a special case of Theorem A stated in Section 2.5; see also Section 5. The main consequence of this result is the following quasi-symmetric rigidity statement, which is an improvement over the main theorem in [3]. Given an orientation-preserving homeomorphism h : S 1 → S 1 , we define its local quasisymmetric distortion to be the function σ h :…”

“…Theorem 2.2 was obtained by Herman [10], based on estimates byŚwiatek [19]. Further proofs are to be found in [4] and [17] for the case of a single critical point, and in [3] for the general case.…”

“…Our goal in the present paper is to improve on the bounds presented in [3] by showing that they are beau in the sense of Sullivan [18] (see Theorem A in Section 2.5). This means that such bounds on the scaling ratios of the critical orbits become asymptotically universal, i.e.…”