Real bounds and quasisymmetric rigidity of multicritical circle maps

Abstract

Let f , g : S 1 → S 1 f, g:S^1\to S^1 be two C 3 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 h:S^1\to S^1 is a topological conjugacy between f f and g g and h h maps the critical points of f f to the critical points of g g , then h 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 preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods and is valid for non-integer critical exponents as well. We do not require h h to preserve the power-law exponents at corresponding critical points.