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

TL;DR

This paper explores the Loewner evolution of hedgehogs in complex dynamics, establishing a connection between Loewner measures and 2-conformal measures of circle maps, and providing explicit formulas for the infinitesimal generator.

Contribution

It introduces a novel link between Loewner evolution, hedgehogs, and 2-conformal measures, with explicit characterization of the infinitesimal generator for the associated semigroup.

Findings

The Loewner measures are 2-conformal measures on the circle.

An explicit form of the infinitesimal generator is derived.

A correspondence between germs and circle map families is established.

Abstract

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha}, \alpha \in \mathbb{R} - \mathbb{Q}$). Perez-Marco showed the existence of a unique continuous monotone one-parameter family of nontrivial 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 $(z \mapsto \overline{g_t(\overline{z})})$.