Whiskered invariant tori for bered dynamics. · Whiskered invariant tori for bered dynamics. We...

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Whiskered invariant tori for fibered dynamics. We consider fibered holomorphic dynamics, generated by the skew product over the translation T ω : F: T d ˆ C n Ñ T d ˆ C n Fpθ, zq“pθ ` ω, f pθ, zqq (1) Where ω satisfies a non-resonance condition. -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 "./cubeeps042fullmore.txt" 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) Siegel set for f pθ, z q“ az ` z 2 p1 ` cos θq -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -1.5 -1 -0.5 0 0.5 1 1.5 2 "./planeeps042.txt" (b) A fiber θ Cte Mikel J. de Viana and Rafael de la Llave (Georgia Tech) Whiskered invariant tori for fibered dynamics.

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  • 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` � 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.