Geometric, topological and dynamical properties of conformally symplectic systems, normally hyperbolic invariant manifolds, and scattering maps (2508.14794v1)

Abstract: Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $\omega$ on a manifold $M$ into a multiple of itself, $f^* \omega = \eta \omega$. We assume $\omega$ 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 $\eta$ 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 $\eta$) 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 $\omega$ is presymplectic. In dynamical systems with many rates (e.g., quasi-integrable systems near multiple resonances), pre-symplectic geometries appear naturally.

• The paper establishes a geometric-dynamical framework for conformally symplectic maps including explicit criteria for the symplecticity of NHIMs and rate-pairing rules.
• The study employs vanishing lemmas and topological analyses to derive algebraic restrictions on conformal factors and identify obstructions to exactness.
• It demonstrates that scattering maps retain (exact) symplectic properties, offering practical insights for applications in celestial mechanics and optimal control.

Geometric, Topological, and Dynamical Structure in Conformally Symplectic Systems and NHIMs

Introduction and Scope

This paper develops a comprehensive geometric and dynamical theory for diffeomorphisms on manifolds equipped with a symplectic form that is preserved up to a constant (but possibly not unit) conformal factor—termed conformally symplectic maps. Such maps arise naturally in a wide range of applications, for instance, mechanics with velocity-proportional friction, discounted variational problems, or physical systems under thermostat constraints. The primary focus is on precise interactions between the topology of the underlying manifold, the conformal/dynamical properties of the map, and the geometry of invariant manifolds. Special emphasis is given to normally hyperbolic invariant manifolds (NHIMs) and the associated scattering maps arising from their (un)stable manifold intersections.

Conformally Symplectic Maps: Definitions and Topological Constraints

A diffeomorphism $f: M \to M$ is conformally symplectic with respect to a symplectic form $\omega$ if $f^* \omega = \eta \omega$ for a positive constant $\eta$ (the conformal factor). This generalizes the symplectic ($\eta=1$) and volume-preserving cases. For $\eta \neq 1$, the volume defined by $\omega$ is multiplied by $\eta^{d/2}$ per iteration, constraining the possible compactness of invariant objects and precluding compact, positive-volume invariant sets except for trivial or pathological cases.

The topology of $M$ imposes algebraic restrictions: for non-exact forms, the conformal factor $\eta$ must be an eigenvalue of the map induced by $f$ on $H^2(M; \mathbb{R})$. In explicit models (e.g., $M = \mathbb{T}^d \times \mathbb{T}^d$), this can severely restrict $\eta$ to algebraic numbers (e.g., products of eigenvalues of an automorphism on the torus), thereby making the construction of conformally symplectic maps a subtle problem intertwined with algebraic topology.

For exact symplectic forms ($\omega=d\alpha$), the notion of an exact conformally symplectic map requires $f^* \alpha = \eta \alpha + dP^f_\alpha$ for some primitive $P^f_\alpha$—the existence of which is itself topologically obstructed when $\eta$ is not an eigenvalue of the induced map on $H^1(M;\mathbb{R})$.

Dynamical–Geometric Interplay: NHIMs and Rate-Pairing Rules

NHIM Formation and Symplecticity

A NHIM is a manifold $\Lambda \subset M$ invariant under $f$, with tangent bundle splitting $T_x M = T_x \Lambda \oplus E^s_x \oplus E^u_x$, and exhibiting exponential contraction and expansion rates in the stable/unstable directions separated by a spectral gap from the rates tangent to $\Lambda$. These structures concentrate the persistent, nontrivial dynamical behaviors in high-dimensional non-conservative systems.

A central theorem establishes precise inequalities (involving the rates of contraction/expansion and $\eta$) under which $\omega|_\Lambda$ is guaranteed non-degenerate, making $\Lambda$ symplectic and $f|_\Lambda$ conformally symplectic. Furthermore, the non-degeneracy of $\omega|_\Lambda$ enforces explicit pairing rules between forward and backward optimal rates; for optimal rates $\mu_\pm^*$ tangent to $\Lambda$ and corresponding stable/unstable rates $\lambda_\pm^*$,

$$\frac{\mu_+^*}{\mu_-^*} = \eta, \quad \frac{\lambda_+^*}{\lambda_-^*} = \eta$$

must hold. This aligns and generalizes the Lyapunov exponent pairing in symplectic/dissipative systems [dettmann1996proof] [wojtkowski1998conformally].

Vanishing Lemmas and Presymplectic Reductions

The geometry of the stable and unstable manifolds of NHIMs is further clarified through vanishing lemmas, showing that for subbundles whose rates force $\|Df^n(u)\|\|Df^n(v)\|\eta^{-n}\rightarrow 0$, necessarily $\omega(u,v)=0$. This ensures, for the allowed range of rates, that the (un)stable manifolds are always presymplectic (coisotropic), and the kernel of $\omega$ integrates to the strong (un)stable foliation. These results extend to cases where $\omega$ is presymplectic, leading to a fully consistent geometric picture in degenerate contexts.

Scattering Maps and Symplecticity in Homoclinic Dynamics

When the stable and unstable manifolds of a NHIM intersect transversally along homoclinic channels, the asymptotics of orbits induces a canonical scattering map $S: H_{-}^{\Gamma} \to H_{+}^{\Gamma}$ on subsets of $\Lambda$. The paper proves that:

• Scattering maps are always symplectic with respect to $\omega|_\Lambda$, regardless of whether $f$ is dissipative ($\eta \neq 1$). This is a nontrivial result because the scattering map mixes (un)stable manifold dynamics, and the dissipative part cancels due to geometric constraints.
• If $\omega$ is exact, then the scattering map is also exact symplectic, even if $f$ itself is only exact with respect to specific gauge choices for the action form.
• Primitive functions for the wave and scattering maps are constructed, with explicit series (often convergent only with a suitable gauge), and for generic escaping orbits, gauge choices affect convergence or divergence.

These properties rely on multiple—mathematically distinct—proofs: vanishing lemma arguments, Cartan’s magic formula, coordinate-based calculations, and graph transforms, highlighting the robustness of the conclusion.

Topological Obstructions and Exactness

A detailed analysis is provided of topological restrictions on the possible conformal factors. Specifically, for non-exact forms, the permissible $\eta$’s must be eigenvalues of the second cohomology action—always algebraic, and much more restricted than for symplectic ($\eta=1$) maps. The possible set $\mathcal{R}$ of conformal factors generally forms a discrete, algebraic subgroup of $\mathbb{R}_{+}^{*}$. This solves questions posed in prior work, such as whether $\mathcal{R}$ can be strictly intermediate between $\{1\}$ and $\mathbb{R}_{+}^{*}$ [ArnaudF24].

Generalization and Further Models

The paper generalizes its geometric and dynamical mechanisms to presymplectic systems and to product-type systems with partially conformal structures—relevant to dissipative chains, mechanical thermostats, or partially damped PDEs. The vanishing lemma strategy adapts directly to settings where the conformal factor is not constant but only bounded above and below, further establishing the broad applicability of the results.

Applications and Implications

Mathematical Implications

• Dynamical systems theory: The pairing rules and geometric structure provided offer not only a classification of admissible NHIMs for conformally symplectic systems, but also place stringent rate-based obstructions for possible behavior, applicable to dissipative Hamiltonian perturbations and beyond.
• Algorithmic computation: The explicit, stepwise constructions in the case of exact symplectic forms confirm that invariant objects and their connecting orbits (e.g., using the parameterization method) can be computed with guaranteed symplecticity properties, even in the infinite-volume, noncompact context.
• Topology and dynamics: The connection between algebraic topology (de Rham or simplicial cohomology) and conformally symplectic factors adds fundamental constraints in the construction and perturbation of models, with implications for homological stability and structural rigidity.

Applied Directions and Prospects

• Hamiltonian and celestial mechanics: The results underpin rigorous mechanisms for energy diffusion and instability (Arnold diffusion) in high-dimensional systems, including both conservative and weakly dissipative cases, applicable to long-term solar system stability and more general control/optimization scenarios.
• Control theory and economics: The discounted variational principles emerging from exact conformally symplectic structure connect directly to models in optimal control and mathematical finance, providing a geometric foundation for computational algorithms.

Computational and AI Implications

While this work is foundational and geometric rather than algorithmic per se, its implications for AI lie in structure-preserving algorithms for high-dimensional dynamics, hybrid/constrained simulation (e.g., for robotic locomotion or meta-stability analysis), and topologically constrained learning (e.g., in symplectic or volume-preserving neural ODEs). For reinforcement learning in continuous spaces, the variational analogs may provide a mathematically robust framework for modeling dissipative environments with conserved geometric quantities.

Conclusion

This paper delivers an in-depth, rigorous account of how geometry, topology, and dynamics interact for conformally symplectic maps and their associated invariant manifolds—particularly NHIMs—and in the construction and properties of scattering maps. The characterization of symplecticity via rate inequalities and pairing rules, the explicit quantification of obstructions from topology, and the extension to presymplectic and partially conformal systems yields a unified geometric framework with repercussions across dynamical systems, computational modeling, and applied mathematics. The established results set a strong foundation for future investigation into large-scale diffusion in weakly non-conservative systems, the algorithmic computation of invariant objects, and the design of geometric integrators in high-dimensional physical and artificial environments.