Real bounds and quasisymmetric rigidity of multicritical circle maps

Let $f, g:S^1\to S^1$ 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 $h:S^1\to S^1$ 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 \cite{CS}. However, unlike the proof given in \cite{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.