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
$$(g_t)$$
.
Koenigs Function and Fractional Iteration of Functions Analytic in the Unit Disk with Real Coefficients and Fixed Points
