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Dynamics seminar: Genady Levin (HUJI) On renormalizations of rational functions | Einstein Institute of Mathematics

Dynamics seminar: Genady Levin (HUJI) On renormalizations of rational functions

Date: 
Tue, 29/11/202214:00-15:00
Lecturer: 
Genadi Levin
Abstract: We recall Douady-Hubbard's polynomial-like renormalization ("satellite" and "primitive"), Sullivan's "complex bounds", Douady-Hubbard's examples of "no complex bounds" etc., and review some related phenomena and advances. 
Then we discuss two recent results. The first one is a universal drop of complex bounds at a satellite renormalization of high relative period. This implies, e.g., no complex bounds for all "satellite" maps of unbounded combinatorics 
(extending Douady-Hubbard's examples), and degeneration of geometry in this case.
In the second result, we consider the dynamics of a quadratic polynomial which admits infinitely many satellite renormalizations.
The most interesting case is when the "intersection" J_\infty of all renormalizations contains continua. We show that any invariant measure on J_\infty is,
in fact, supported on the postcritical set and has zero Lyapunov exponent. It follows that the exponent at the critical value is equal to zero, too.
Joint works with  Alexander Blokh, Lex Oversteegen, Vladlen Timorin, and Feliks Przytycki.