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Summary: ON RIGIDITY OF CRITICAL CIRCLE MAPS
ARTUR AVILA
Abstract. We give examples of analytic critical circle maps which are not C1+ rigid.
1. Introduction
A critical circle map is a C1
homeomorphism f : R/Z R/Z with a unique critical point at 0.
Fix d 3 odd and let be the space of analytic critical circle maps such that the critical point has
order d.
Yoccoz showed that if f has irrational rotation number then f is topologically conjugate to a
irrational rotation [Y]. Thus if f, g have the same irrational rotation number then there exists a
unique homeomorphism h = hf,g : R/Z R/Z such that h(0) = 0 and h f = g h. The following
result was announced by Teplinsky and Khanin [TK].
Theorem 1.1. Let f, g have the same irrational rotation number. Then h is C1
.
This theorem immediately provokes the question of whether one could promote h to C1+
for
some > 0. This was known not to be possible if one assumes that f and g are merely C
, unless
further hypothesis on the rotation number are made [FM1], [FM2]. The following result of Khmelev
and Yampolski [KY] seemed to indicate that the analytic case could be different (the question of
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