Instabilities for analytic quasi-periodic invariant tori

TL;DR

This paper demonstrates the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori, addressing stability, normal form convergence, and measure-theoretic properties of such tori.

Contribution

It constructs examples of unstable invariant tori with arbitrary frequencies, showing the optimality of exponential stability and exploring measure-theoretic aspects for Liouville vectors.

Findings

Existence of topologically unstable tori of arbitrary frequency.

Convergent Birkhoff Normal Form can be achieved.

Invariant tori with non-measure-zero accumulation.

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.