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arXiv · 1912.01575

Instabilities for analytic quasi-periodic invariant tori

Gerard Farré·Bassam Fayad
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Abstract

We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic motion: $\quad i)$ Show the existence of topologically unstable tori of arbitrary frequency. Moreover, the Birkhoff Normal Form at the invariant torus can be chosen to be convergent, equal to a planar or non-planar polynomial. $\quad ii)$ Show the optimality of the exponential stability for Diophantine tori. ' $\quad iii)$ Show the existence of real analytic Hamiltonians that are integrable on half of the phase space, and such that all orbits on the other half accumulate at infinity. $\quad iv)$ For sufficiently Liouville vectors, obtain invariant tori that are not accumulated by a positive measure set of quasi-periodic invariant tori.

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Gerard Farré, Bassam Fayad. 2021-05-04. Instabilities for analytic quasi-periodic invariant tori. https://arxiv.org/abs/1912.01575

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