Speaker: Genadi Levin (Hebrew University of Jerusalem)
Title: Polynomial-like and box dynamics of analytic maps
Abstract. Let g: U to V be a proper holomorphic map of degree d > 1, where U and V
are topological disks in the complex plane. If the closure of U is contained in V, the map is called polynomial-like (PL). Indeed, by Douady-Hubbard’s Straightening Theorem, g is then conjugate to a polynomial of degree d
near its non-escaping set $K_g=\cap_{i\ge 0} g^{-i}(V)$ where K_g is a full (i.e. non-separating) completely invariant compact.
Inspired by McMullen’s theory of robust quadratic polynomials we give a necessary and sufficient condition that
the map g:U to V admits a PL restriction to a neighborhood of its full completely invariant compact set.
A similar statement holds more generally, for locally analytic maps of compact sets, where the PL restriction should be replaced by the “box mapping” restriction.