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Loewner evolution of hedgehogs and 2-conformal measures of circle maps

Part of: Complex dynamical systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  28 September 2020

KINGSHOOK BISWAS
KINGSHOOK BISWAS*
Affiliation:
Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India (e-mail: [email protected])
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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 $(g_t)$ .

Keywords

Loewner evolutionconformal measuresindifferent fixed pointscircle maps

MSC classification

Primary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Secondary: 37E10: Maps of the circle
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2734 - 2753
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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