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

Abstract

AbstractLet 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 $(g_t)$ .