[Paper Review] Generic Hamiltonian Dynamics

Mário Bessa, Célia Ferreira+2 more|arXiv (Cornell University)|Mar 17, 2012

Mathematical Dynamics and Fractals19 references4 citations

TL;DR

This paper establishes that for a generic $C^2$ Hamiltonian on a closed symplectic manifold, a full Lebesgue measure set of energy levels are topologically mixing and constitute homoclinic classes. Using a novel connecting lemma for pseudo-orbits in Hamiltonian systems, the authors prove that $C^2$-generically, Hamiltonian flows exhibit robust transitive dynamics across most energy levels, overcoming obstructions from KAM theory.

ABSTRACT

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H in R, there is a full measure subset of energies e in H(M) such that the Hamiltonian level (H,e) is topologically mixing; moreover these level sets are homoclinic classes.

Motivation & Objective

• To establish generic topological mixing in Hamiltonian systems under $C^2$ topology.
• To resolve the conceptual challenge of formulating genericity for Hamiltonians across energy levels.
• To prove that for $C^2$-generic Hamiltonians, most energy levels are homoclinic classes and topologically mixing.
• To develop a connecting lemma for pseudo-orbits in Hamiltonian dynamics as a key technical tool.
• To contrast generic mixing with KAM-theoretic obstructions to mixing in lower regularity settings.

Proposed method

• Constructing a $C^2$-residual set $\mathcal{R}$ of Hamiltonians where generic behavior holds.
• Proving that for $H \in \mathcal{R}$, a full measure subset of energies $e$ yield topologically mixing level sets $\mathcal{E}_{H,e}$.
• Introducing a covering family of cross sections and flowboxes outside a neighborhood $\mathcal{V}$ to control pseudo-orbits.
• Using Kakutani towers and perturbation techniques to construct $C^2$-small Hamiltonian perturbations that connect arbitrary points via pseudo-orbits.
• Applying a connecting lemma for pseudo-orbits in Hamiltonian systems, adapted from symplectomorphism results in [5], to achieve orbit connection.
• Leveraging uniform avoidability of periodic orbits and transversality to ensure perturbations preserve tiling and flowbox structure.

Experimental results

Research questions

• RQ1 Can topological mixing be generic in the $C^2$ topology for Hamiltonian systems on closed symplectic manifolds?
• RQ2 Is there a $C^2$-residual set of Hamiltonians for which most energy levels are topologically mixing?
• RQ3 Do $C^2$-generic Hamiltonians have energy levels that are homoclinic classes?
• RQ4 Can pseudo-orbit connecting arguments be adapted to the continuous-time Hamiltonian setting with $C^2$ regularity?
• RQ5 How does generic mixing in $C^2$ Hamiltonians coexist with KAM-theoretic persistence of invariant tori?

Key findings

• There exists a $C^2$-residual set $\mathcal{R}$ of Hamiltonians such that for each $H \in \mathcal{R}$, a full Lebesgue measure subset of energies $e$ yield topologically mixing level sets $\mathcal{E}_{H,e}$.
• For $H \in \mathcal{R}$, every connected component of the energy level $\mathcal{E}_{H,e}$ is a homoclinic class.
• The energy level sets $\mathcal{E}_{H,e}$ for $H \in \mathcal{R}$ and full measure $e$ are topologically mixing, implying dense orbits and strong recurrence.
• A connecting lemma for pseudo-orbits in Hamiltonian systems is established, enabling orbit connections via $C^2$-small perturbations.
• The result holds despite KAM-theoretic obstructions, as the genericity is in $C^2$ topology, not $C^1$ or lower.
• The proof relies on constructing covering families of cross sections and flowboxes with disjoint supports, ensuring pseudo-orbit control across the energy level.