[Paper Review] The group of Hamiltonian homeomorphisms and topological Hamiltonian flows

Yong‐Geun Oh|arXiv (Cornell University)|Jan 10, 2006

Geometric and Algebraic Topology5 citations

TL;DR

This paper extends core concepts from symplectic topology—such as the Hofer norm, spectral invariants, and the Calabi quasi-morphism—to the group of Hamiltonian homeomorphisms, Hameo(M, ω), by introducing autonomous topological Hamiltonian flows. It establishes conservation of energy for these flows and generalizes intrinsic norms and quasi-morphisms to the topological setting, extending Entov-Polterovich's Calabi quasi-morphism from S² to topological Hamiltonian paths.

ABSTRACT

In this paper, we study the dynamical aspects of the group Hameo(M, ω) of Hamiltonian homeomorphisms which was recently introduced by the author. We define the notion of autonomous topological Hamiltonian flows and extend the well-known conservation of energy to such flows. We extend the definitions of the Hofer length and of the spectral invariants ρa to topological Hamiltonian paths, and generalize the Hofer norm and the spectral norm γ: Ham(M, ω) → R+ to the corresponding intrinsic norms on Hameo(M, ω) respectively. Using this extension, we also extend Entov-Polterovich’s Calabi quasi-morphism on S² to the space of topological Hamiltonian paths.

Motivation & Objective

• To define and study autonomous topological Hamiltonian flows as a natural extension of smooth Hamiltonian dynamics.
• To generalize the Hofer length and spectral invariants ρa to topological Hamiltonian paths.
• To extend the Hofer and spectral norms from the group of Hamiltonian diffeomorphisms to Hameo(M, ω).
• To extend the Calabi quasi-morphism from S² to the space of topological Hamiltonian paths.
• To establish conservation of energy for topological Hamiltonian flows, generalizing a classical result in symplectic geometry.

Proposed method

• Introduce the notion of autonomous topological Hamiltonian flows as limits of smooth Hamiltonian flows under the C⁰ topology.
• Define the Hofer length and spectral invariants ρa for topological Hamiltonian paths via approximation by smooth paths.
• Extend the Hofer norm and spectral norm γ to Hameo(M, ω) using the extended invariants.
• Prove that the extended norms are intrinsic and well-defined on Hameo(M, ω), independent of approximating sequences.
• Use the extended spectral invariants to construct a Calabi quasi-morphism on the space of topological Hamiltonian paths.
• Apply the extension to S², generalizing Entov-Polterovich’s Calabi quasi-morphism to the topological setting.

Experimental results

Research questions

• RQ1 Can the concept of Hamiltonian flow be meaningfully extended to the C⁰-closure of the group of Hamiltonian diffeomorphisms?
• RQ2 How can the Hofer length and spectral invariants be generalized to topological Hamiltonian paths?
• RQ3 Is the Calabi quasi-morphism on S² extendable to the space of topological Hamiltonian paths?
• RQ4 Does energy conservation hold for autonomous topological Hamiltonian flows?
• RQ5 Are the extended norms on Hameo(M, ω) intrinsic and independent of the choice of approximating smooth paths?

Key findings

• The paper defines autonomous topological Hamiltonian flows and proves that energy is conserved along such flows, extending a classical result to the topological setting.
• The Hofer length and spectral invariants ρa are extended to topological Hamiltonian paths via approximation, and the resulting invariants are well-defined and intrinsic.
• The Hofer and spectral norms on Ham(M, ω) are generalized to intrinsic norms on the group Hameo(M, ω), preserving key properties.
• The Calabi quasi-morphism on S² is extended to the space of topological Hamiltonian paths, establishing a new quasi-morphism in the topological category.
• The extended spectral invariants and norms are stable under C⁰-limits, ensuring consistency with the smooth case.