Download - Whiskered invariant tori for bered dynamics. · Whiskered invariant tori for bered dynamics. We consider bered holomorphic dynamics, generated by the skew product over the translation

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Page 1: Whiskered invariant tori for bered dynamics. · Whiskered invariant tori for bered dynamics. We consider bered holomorphic dynamics, generated by the skew product over the translation

Whiskered invariant tori for fibered dynamics.

We consider fibered holomorphic dynamics, generated by the skewproduct over the translation Tω:

F : Td ˆ Cn Ñ Td ˆ Cn

Fpθ, zq “ pθ ` ω, fpθ, zqq(1)

Where ω satisfies a non-resonance condition.

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(a) Siegel set forf pθ, zq “ az ` z2p1` ε cos θq

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(b) A fiber θ “ Cte

Mikel J. de Viana and Rafael de la Llave (Georgia Tech) Whiskered invariant tori for fibered dynamics.

Page 2: Whiskered invariant tori for bered dynamics. · Whiskered invariant tori for bered dynamics. We consider bered holomorphic dynamics, generated by the skew product over the translation

Objective: Given an approximately F -invariant torus T0, computean F -invariant torus T, near T0.

Methodology:

A-posteriori format: Set up functional equations.

Nash-Moser iteration: Asymptotics for the derivative cocycle.

Melnikov Conditions.

Lower dimensional tori (d ă n): Add parameters (after[Moser, 1967] ) which act as ”counterterms”.

Use a-posteriori format to reduce the number of parameters(after [Herman, Sevryuk]).

Mikel J. de Viana and Rafael de la Llave (Georgia Tech) Whiskered invariant tori for fibered dynamics.