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.

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.