Regular and Exotic behavior in Hamiltonian and low-dimensional dynamical systems

REXHALODYN aims at understanding some of the regular and exotic features of Hamiltonian systems, circle maps, and generalized interval exchange transformations, or GIETs. Hamiltonians provide a powerful description of classical mechanics phenomena.

Circle maps and GIETs are archetypal examples that illustrate fundamental concepts and phenomena within dynamical systems.

Moreover, the latter two offer simplified models that capture essential features of more complex systems; e.g., they appear naturally as Poincar maps of (locally) Hamiltonian flows on compact surfaces, and thus are closely related to Hamiltonians in this context.

Within this framework, the words regular and exotic do not have a precise mathematical meaning; however, we use them here to stand for features in each system that are either well-behaved (regular), such as the existence of stable quasi-periodic motions in Hamiltonian, and rigidity phenomena in circle maps and GIETs; and for features that are ill-behaved (exotic), such as sensitive dependence to initial conditions in Hamiltonians or existence of singular invariant measures in circle maps and GIETs.

Understanding these complementary features deepens our understanding of the systems, their properties, and the underlying mathematical structures that give rise to a rich spectrum of behaviors.The main objectives of this project are:-Prove the existence of lower-dimensional invariant tori, associated with a resonant torus with any number of resonances, for general classes (e.g., convex) of near-integrable Hamiltonians.-Obtain precise estimates for the Hausdorff dimension of the unique invariant probability measure of multicritical circle maps with irrational rotation number of bounded type.-Obtain rigidity results for multicritical circle maps with irrational rotation number.The research is planned for two years and takes as a basis many of the results and methods developed by the applicant in his previous works.