Kingshook Biswas
ISI Kolkata
December 14, 2023
Loewner evolution of hedgehogs and 2-conformal measures of circle maps: Given a germ of holomorphic diffeomorphism $f$ with an irrationally indifferent fixed point at the origin in $\mathbb C$, Perez-Marco proved the existence of a unique monotone 1-parameter family of non-trivial connected compacts called Siegel compacts, which contain the fixed point and are invariant under the dynamics of $f$. When the map $f$ is nonlinearizable, these compacts are called hedgehogs; these can be quite complicated topologically, in particular they are never locally connected.
ISI Kolkata
December 14, 2023
Loewner evolution of hedgehogs and 2-conformal measures of circle maps: Given a germ of holomorphic diffeomorphism $f$ with an irrationally indifferent fixed point at the origin in $\mathbb C$, Perez-Marco proved the existence of a unique monotone 1-parameter family of non-trivial connected compacts called Siegel compacts, which contain the fixed point and are invariant under the dynamics of $f$. When the map $f$ is nonlinearizable, these compacts are called hedgehogs; these can be quite complicated topologically, in particular they are never locally connected.
Perez-Marco used the Siegel compacts to give a correspondence between germs $f$ and 1-parameter families $(g_t)$ of analytic circle diffeomorphisms obtained by uniformizing the complement of the Siegel compacts. The family of circle maps $(g_t)$ is in fact the orbit under a locally defined semigroup acting on the space of analytic circle maps. We show that this semigroup has a well defined infinitesimal generator, whose form we determine explicitly using the Loewner equation associated to the family of Siegel compacts. We show that the Loewner measures driving the equation are conformal measures of exponent two for the circle maps $g_t$.