# Applications of A-posteriori Implicit Function Theorems to the Existence of Quasi-periodic Solutions: Whitney Regularity with Respect to the Frequency

Wednesday, November 4, 2015 - 10:30am - 11:30am

Keller 3-180

Rafael de la Llave (Georgia Institute of Technology)

Many results of persistence of quasi-periodic solutions

(Kolmogorov, Arnold, Moser theory) can be recast

in a-posteriori format. That is, given an approximate solution of

an invariance equation, there is a true solution close to it.

There are many applications of these a-posteriori format. Notably,

one can take as approximate solutions the results of a numerical

calculation. Hence obtain results of the constants in concrete

examples which are more than 99% of other rigorous upper bounds.

One can also take as input of an a-posteriori theorem the results of

formal expansions. One then gets differentiability.

Some more recent developments are that one also obtains results

on Whitney differentiability with respect to the frequency. This

allows to prove the results in near integrable systems with very weak

non-degeneracy assumptions and to give very sharp characterizations of

analyticity domains.

(Kolmogorov, Arnold, Moser theory) can be recast

in a-posteriori format. That is, given an approximate solution of

an invariance equation, there is a true solution close to it.

There are many applications of these a-posteriori format. Notably,

one can take as approximate solutions the results of a numerical

calculation. Hence obtain results of the constants in concrete

examples which are more than 99% of other rigorous upper bounds.

One can also take as input of an a-posteriori theorem the results of

formal expansions. One then gets differentiability.

Some more recent developments are that one also obtains results

on Whitney differentiability with respect to the frequency. This

allows to prove the results in near integrable systems with very weak

non-degeneracy assumptions and to give very sharp characterizations of

analyticity domains.

MSC Code:

70K43