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

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 $$(g_t)$$ .