The conformal measures of a normal subgroup of a cocompact Fuchsian group

In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

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

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 $$(g_t)$$ .