Real bounds and quasisymmetric rigidity of multicritical circle maps

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Let f, g:S¹→ S¹ be two C³ 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:S¹→ S¹ 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 CS. However, unlike the proof given in CS, 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 to preserve the power-law exponents at corresponding critical points.