Renormalization and dynamics of germs of diffeomorphisms
speaker: Kingshook Biswas (Stony Brook)abstract: We discuss the renormalization and reverse renormalization constructions used in studying the dynamics of germs of holomorphic diffeomorphisms fixing the origin in C with linear part a rotation. Such a germ is said to be linearizable if it is analytically conjugate to its linear part.
Renormalization was first used in this setting by Yoccoz to give a geometric proof sufficiency of the Brjuno condition (an arithmetic condition ensuring that the rotation number does not admit very good rational approximations) for linearizability. It was later used by Perez-Marco to obtain a similar linearization result for germs without periodic orbits accumulating the fixed point.
We explain how renormalization can be adapted to the setting of commuting germs in order to obtain a multi-variable Brjuno type arithmetic condition for linearizability of commuting germs.
Reverse renormalization was also first used by Yoccoz to construct non-linearizable germs demonstrating the optimality of the Brjuno condition for linearizability. It was also used later by Perez-Marco to prove optimality of his arithmetic condition for linearizability without periodic orbits, and to construct examples of germs with uncountable centralizers. A key point in his method was the introduction of Riemann surfaces called tube-log Riemann surfaces in the construction. Cheritat added a new ingredient to this construction, namely the use of Runge's Theorem, which made the construction flexible enough to construct invariant continua with pathological topologies, in particular he constructed Siegel disks with pseudo-circle boundaries.
We explain how this reverse renormalization can be used to construct hedgehogs such that the fixed point is inaccessible from the complement, answering a question of Perez-Marco's.
timetable:
Mon 14 Oct, 11:30 - 12:30, Aula Dini
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