Simon Allais - Periodic points of Hamiltonian diffeomorphisms and generating functions

February 26, 12 pm ET: Simon Allais (ENS Lyon) - Periodic points of Hamiltonian diffeomorphisms and generating functions

Ginzburg and Gürel recently showed that a hamiltonian diffeomorphism of CP^d a hyperbolic periodic point have infinitely many periodic points whereas fixed points of a pseudo-rotation are isolated as an invariant set. In 2019, Shelukhin proved a homology version of the Hofer-Zehnder conjecture in a large class of symplectic manifolds M that includes CP^d: a Hamiltonian diffeomorphism with more homologically visible fixed points than the dimension of the homology of M has infinitely many periodic points. These results rely on the quantum structure of the Floer homology.

In this talk, I will explain how the study of sublevel sets of generating functions can replace the use of J-holomorphic curves and Floer theory in the study of periodic points of CP^d, based on ideas of Givental and Théret in the 90s.