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Article contents
Loewner evolution of hedgehogs and 2-conformal measures of circle maps
Published online by Cambridge University Press: 28 September 2020
Abstract
Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in
${\mathbb C}$
(i.e.
$f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$
). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families
$(g_t)$
of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps
$(g_t)$
is the orbit of a locally defined semigroup
$(\Phi _t)$
on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls
$(K_t)$
. We show that the Loewner measures
$(\mu _t)$
driving the equation are 2-conformal measures on the circle for the circle maps
$(g_t)$
.
MSC classification
Secondary:
37E10: Maps of the circle
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2020. Published by Cambridge University Press
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