Geometric, topological and dynamical properties of conformally symplectic systems, normally hyperbolic invariant manifolds, and scattering maps

TL;DR

This paper extends symplectic scattering theory to conformally symplectic dynamics on possibly unbounded manifolds by developing a coherent framework relating topology, geometry, and dynamics. It establishes that NHIMs admit a symplectic (and, under exactness, exact) structure when rate data and conformal factors satisfy natural pairing inequalities, while their stable/unstable foliations become presymplectic in this setting. A central result is that scattering maps—describing homoclinic excursions to NHIMs—are symplectic, with exactness when the ambient form is exact; multiple proofs (vanishing lemmas, coordinates, Cartan, and Stokes arguments) illuminate the cancellations responsible for this phenomenon. The work also characterizes topological obstructions to exactness and conformal factors, showing, in concrete manifolds, that admissible $\eta$ often form algebraic or restricted sets, and provides variational interpretations of the scattering map. Together, these results generalize key aspects of symplectic instability theory to dissipative, conformally symplectic contexts and offer practical tools for analyzing quasi-integrable systems with multiple rates and presymplectic geometries.

Abstract

Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $ω$ on a manifold M into a multiple of itself, $f^* ω= ηω$. We assume $ω$ is bounded, as some of the results may fail otherwise. We show that there are deep interactions between the topological properties of the manifold, the dynamical properties of the map, and the geometry of invariant manifolds. We show that, when the symplectic form is not exact, the possible conformal factors $η$ are related to topological properties of the manifold. For some manifolds the conformal factors are restricted to be algebraic numbers. We also find relations between dynamical properties (relations between growth rate of vectors and $η$) and symplectic properties. Normally hyperbolic invariant manifolds (NHIM) and their (un)stable manifolds are important landmarks that organize long-term dynamical behaviour. We prove that a NHIM is symplectic if and only if the rates satisfy certain pairing rules and if and only if the rates and the conformal factor satisfy certain (natural) inequalities. Homoclinic excursions to NHIMs are quantitatively described by scattering maps. These maps give the trajectory asymptotic in the future as a function of the trajectory asymptotic in the past. We prove that the scattering maps are symplectic even if the dynamics is dissipative. We also show that if the symplectic form is exact, then the scattering maps are exact, even if the dynamics is not exact. We give a variational interpretation of scattering maps in the conformally symplectic setting. We also show that similar properties of NHIMs and scattering maps hold in the case when $ω$ is presymplectic. In dynamical systems with many rates (e.g., quasi-integrable systems near multiple resonances), pre-symplectic geometries appear naturally.

Figures (5)

• Figure 1: Hyperbolic rates
• Figure 2: A NHIM without a uniform tubular neighborhood. After Eldering12
• Figure 3: The scattering map.
• Figure 4:
• Figure 5: A 1-cell $\sigma$ and its completion to a 2D-cell $\tilde{\sigma}$.

Theorems & Definitions (149)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Remark 2.9
• Definition 2.10