Let $f, g:S^1\to S^1$ be two $C^3$ critical homeomorphisms of the circle withthe same irrational rotation number and the same (finite) number of criticalpoints, all of which are assumed to be non-flat, of power-law type. In thispaper 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 pointsare integers, this result is a special case of a general theorem recentlyproved by T.~Clark and S.~van Strien \cite{CS}. However, unlike the proof givenin \cite{CS}, which relies on heavy complex-analytic machinery, our proof usespurely real-variable methods, and is valid for non-integer critical exponentsas well. We do not require $h$ to preserve the power-law exponents atcorresponding critical points.
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机译：令$ f，g：S ^ 1 \至S ^ 1 $为圆的两个$ C ^ 3 $临界同胚，它们具有相同的无理旋转数和相同（有限）个临界点，所有这些假设均非-扁平，属于幂律型。在本文中，我们证明如果$ h：S ^ 1 \至S ^ 1 $是$ f $和$ g $之间的拓扑共轭，则$ h $会将$ f $的临界点映射到$ g $的临界点，则$ h $是准对称的。当所有临界点的幂律指数均为整数时，此结果是T.〜Clark和S.〜van Strien \ cite {CS}最近证明的一般定理的特例。但是，不同于\ cite {CS}中给出的证明依赖于复杂的复杂分析机制，我们的证明使用的是纯实变量方法，并且对非整数临界指数也有效。我们不需要$ h $来保持幂律指数在相应的临界点。
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