Egor Shelukhin

Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

We find robust obstructions to representing a Hamiltonian diffeomorphism as a full k-th power,

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

k \ge 2

g

k≥2, and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer’s metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms.

-areas

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on

On the

Ham(T^2)

L^p

and some of them are Calabi.

-geometry of autonomous Hamiltonian diffeomorphisms of surfaces

We show that for each

On the large-scale geometry of the L^p-metric on the symplectomorphism group of the two-sphere

p \geq 1,