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Abstract. We prove that two C 3 critical circle maps with the same rotation number in a special set A are C 1+α conjugate for some α > 0 provided their successive renormalizations converge together at an exponential rate in the C 0 sense. The set A has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ∞ critical circle maps with the same rotation number that are not C 1+β conjugate for any β > 0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

Abstract. In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, "hybrid classes". This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.

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