A translation of "The Geometry of the Group of Symplectic Diffeomorphisms"

TL;DR

The work develops Hofer’s metric on the group $\mathrm{Ham}(M,\Omega)$, linking dynamical questions to intrinsic symplectic geometry. It leverages Gromov–Floer–Sikorav techniques (pseudoholomorphic curves, Lagrangian intersections) to derive nontrivial lower bounds on displacement energy, establish nondegeneracy of the Hofer metric in key settings (notably $\mathbb{R}^{2n}$), and analyze the spectrum of lengths via fibred and coupling–form constructions. It further connects to ergodic theory, growth of one-parameter subgroups, and variational geodesic theory, yielding diameter results on surfaces, constraints from the fundamental group, and a rich interplay between algebraic and geometric structures in symplectic topology. The results illuminate how energy, geometry, and topology constrain Hamiltonian dynamics and provide foundational tools for understanding long-time behavior, rigidity phenomena, and spectral invariants in symplectic manifolds.

Abstract

This book offers an introduction to Hofer's metric on the group of Hamiltonian diffeomorphisms. It presents results on the diameter, geodesics, and the growth of one-parameter subgroups, along with applications to dynamics and ergodic theory.

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