Instabilities of invariant quasi-periodic tori
Gerard FarréRoyal Institute of Thechnology (KTH), Stockholm, Sweden
Bassam FayadUniversité Paris Diderot-Université Pierre et Marie Curie, France
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:
- 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.
- Show the optimality of the exponential stability for Diophantine tori.
- 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.
- For sufficiently Liouville vectors, obtain invariant tori that are not accumulated by a positive measure set of quasi-periodic invariant tori.
Cite this article
Gerard Farré, Bassam Fayad, Instabilities of invariant quasi-periodic tori. J. Eur. Math. Soc. 24 (2022), no. 12, pp. 4363–4383DOI 10.4171/JEMS/1206