Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar
Hyperbolicity of Periodic Points of Hamiltonian Maps
The basic invariant of a fixed point of a Hamiltonian diffeomorphism, besides its existence (which is implied by the proven Arnol'd Conjecture), is the number of eigenvalues of unit norm of the linearization of the map at the fixed point. When there are no such eigenvalues, the fixed point is said to be purely hyperbolic, and has characteristically different local dynamics from the contrasting partially elliptic case. In this talk, I will discuss how period doubling bifurcations can be used to make periodic points purely hyperbolic without appreciably changing Floer-theoretic invariants. Via a limiting process one can approximate Hamiltonian diffeomorphisms by hameomorphisms which behave as if they have only hyperbolic periodic points. We will review the dynamical background for such constructions, and if time permits, discuss upper and lower bounds on the growth rate of periodic points of these hameomorphisms.