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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
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- Original Article
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- Copyright
- © The Author(s), 2020. Published by Cambridge University Press
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