Analytic AbC constructions in Hamiltonian dynamics and ergodic theory
Time: Fri 2021-10-08 14.00
Doctoral student: Gerard Farré Puiggalí , Matematik (Avd.)
Opponent: Professor Krikorian Raphaël, CY Cergy Paris Université (University of Cergy-Pontoise), Frankrike
Supervisor: Universitetslektor Maria Saprykina,
This thesis contains several results on the theory of perturbationof integrable dynamical systems. Most of the results are obtainedthrough the use of constructive methods developed in the seventies byD. Anosov and A. Katok in the context of ergodic theory. Using different variants of their method we give examples of dynamical systemswhich are close to being integrable but exhibit surprising instability orergodic properties. In Paper A we provide different examples of realanalytic Hamiltonian systems with an invariant quasi-periodic toruswith different types of unstable behaviour. In particular this includesexamples of topologically unstable tori of arbitrary frequency, extending a well known result of R. Douady for the smooth case. In paperB, analytic and weakly mixing volume preserving diffeomorphisms areconstructed on odd dimensional spheres. In Paper C we prove thatthe motion of the solutions for most of the initial conditions around aDiophantine quasi-periodic torus T0 are close to being quasi-periodicfor a double exponentially long time with respect to the inverse of thedistance to T0. This result relates to a conjecture by Herman statingthat there should be a positive measure set of initial conditions withquasi-periodic motion around the original Diophantine quasi-periodictorus T0.