Conformally Symplectic Maps

• Conformally symplectic maps are diffeomorphisms that rescale the symplectic form by a positive factor, enabling analysis of both dissipative and expanding dynamics.
• They impose topological constraints by linking the conformal factor to cohomological properties, ensuring invariant manifolds satisfy strict pairing rules.
• Applications include dissipative KAM theory, Aubry–Mather dynamics, scattering analysis, and extensions to presymplectic systems in varied physical contexts.

Conformally symplectic maps are diffeomorphisms of a symplectic manifold that scale the symplectic form by a constant or bounded factor. Specifically, a diffeomorphism $f: M \to M$ on a manifold $(M, \omega)$ is conformally symplectic if $f^*\omega = \eta \omega$, where $\eta > 0$ is the conformal factor—often constant, though bounded functions $\eta(x)$ arise in presymplectic contexts. This mapping class includes dissipative or expanding dynamics and is strictly broader than symplectic maps (which correspond to $\eta=1$). The analysis of conformally symplectic maps interacts fundamentally with the topology of $M$, the structure and rates of invariant manifolds, the properties of normally hyperbolic invariant manifolds (NHIMs), scattering theory, and presymplectic generalizations (Gidea et al., 20 Aug 2025).

1. Definition and Fundamental Properties

Conformally symplectic maps satisfy the pull-back relation: $f^*\omega = \eta\,\omega$ for a nondegenerate two-form $\omega$ on $M$ and a conformal factor $\eta > 0$ (often constant, but potentially a bounded function). Unlike symplectic diffeomorphisms, which preserve $\omega$ pointwise, conformally symplectic maps rescale the symplectic volume by iterates of $\eta$. If $\eta < 1$, the system is dissipative; if $\eta > 1$, it is expanding. In dynamics and geometry, $\eta$ is often forced to be constant by the nondegeneracy conditions. When $\omega$ is presymplectic, $\eta$ may vary smoothly across $M$.

The symplectic volume transforms as $(f^n)^*\omega = \eta^n \omega$, implying a geometric distinction from conservative systems. These maps appear in the paper of thermostatted systems, Aubry–Mather theory for dissipative flows (Marò et al., 2016), and regularization techniques for integrable systems such as the Kepler problem (Marle, 2010).

2. Topological and Dynamical Constraints on the Conformal Factor

The interaction between topology and dynamics emerges from the cohomological restrictions of $\omega$ under $f$. When $\omega$ is not exact, the induced map on de Rham cohomology satisfies $[f^*\omega] = \eta [\omega]$ in $H^2(M)$, forcing $\eta$ to be an eigenvalue of the induced map on cohomology. For closed manifolds supporting nonzero $H^2(M)$, $\eta$ is constrained to be an algebraic number. For example, if the cohomology class $[\omega]$ is fixed by $f$, then $\eta=1$; otherwise, $\eta$ must be an eigenvalue of $f^*$.

Dynamically, vanishing lemmas connect the growth rates of vectors under the derivative $Df$ to the conformal factor. If contraction along a subbundle exceeds the rate dictated by $\eta$, the symplectic form must vanish on that subbundle. More precisely, for expansion/contraction rates $\lambda_\pm$ (stable/unstable directions) and $\mu_\pm$ (tangent directions on NHIMs), pairing rules constrain these by: $$\frac{\mu_+}{\mu_-} = \eta, \quad \frac{\lambda_+}{\lambda_-} = \eta$$ These relations link the topological invariants of $M$ to dynamical features of $f$, governing possible rates compatible with conformally symplectic geometry (Gidea et al., 20 Aug 2025).

3. Normally Hyperbolic Invariant Manifolds (NHIMs) and Symplecticity

NHIMs $\Lambda \subset M$ are invariant submanifolds for which normal directions have stronger expansion/contraction than tangential directions. In the conformally symplectic setting, the existence of a symplectic structure on $\Lambda$ depends on the rates. The restriction $\omega|_\Lambda$ is nondegenerate if: $$\mu_+ \lambda_+ \eta^{-1} < 1 \qquad \text{and} \qquad \mu_- \lambda_- \eta < 1$$ Under these inequalities, the ambient symplectic geometry descends to $\Lambda$, making the NHIM symplectic. Pairing rules in the rates between tangential and normal directions play a critical role in ensuring $\omega|_\Lambda$ remains nondegenerate.

In systems with multiple invariant bundles (e.g., quasi-integrable systems with resonances), presymplectic geometry often arises. The main results—pairing rules and criteria for nondegeneracy—extend naturally to presymplectic contexts, given sufficient codimension of the kernel of $\omega$ (Gidea et al., 20 Aug 2025).

4. Scattering Maps and Symplecticity

Scattering maps are constructed from homoclinic channels—transverse intersections of stable and unstable manifolds of a NHIM. For $y \in \Gamma$ (the channel), there are unique "shadow" points $x_+$ and $x_-$ in $\Lambda$ reached via stable and unstable wave maps, $\Omega_+$ and $\Omega_-$. The scattering map is defined as

$$S = \Omega_+ \circ (\Omega_-)^{-1} : \Lambda \to \Lambda$$

Despite the dissipative nature of $f$, the scattering map $S$ inherits symplecticity from the ambient structure. Specifically,

$$(\Omega_+)^*(\omega|_\Lambda) = \omega|_{W^s_\Lambda}$$

and similarly for $\Omega_-$, implying $S$ is symplectic on $\Lambda$ (exact if $\omega$ is exact). This result is robust under dissipation: the global dynamics may contract or expand symplectic volume, yet the local transition map between asymptotic states on $\Lambda$ preserves or exactly preserves the symplectic form (Gidea et al., 20 Aug 2025, Marle, 2010).

For presymplectic forms, the induced scattering map is presymplectic, and when restricted to transversal symplectic submanifolds, it remains symplectic.

5. Variational Interpretation of Scattering in Conformally Symplectic Systems

If $f$ is exact conformally symplectic, meaning there exists an action $1$-form $\alpha$ such that $f^*\alpha = \eta \alpha + dP^f_\alpha$, iterative application yields

$$\alpha = \eta^{-n} (f^*)^n \alpha - d\!\!\sum_{j=0}^{n-1} \eta^{-j-1} P^f_\alpha \circ f^j$$

Integrating along connecting trajectories between $x_-$ and $x_+$ via the wave maps defines primitive functions $P^+$ and $P^-$. The difference $P^+ - P^-$ generates the scattering map. This formalism embodies a discounted action principle analogous to Aubry–Mather theory for dissipative systems (Marò et al., 2016), and underpins the discrete variational nature of scattering in conformally symplectic settings.

6. Extensions to Presymplectic Geometry

In many applications, dynamical systems exhibit multiple time scales, giving rise to presymplectic (degenerate) two-forms. Provided the symplectic quotient preserves sufficient nondegeneracy, the principal results—rate conditions for invariant manifolds, and symplecticity of scattering maps—extend to presymplectic cases. The geometric criteria for nondegeneracy tolerate partial degeneracy (codimension of the kernel), and the induced quotient dynamics maintains symplecticity on the appropriate subspaces (Gidea et al., 20 Aug 2025).

7. Applications and Broader Implications

Conformally symplectic maps arise in a range of mathematical and physical contexts:

• Dissipative KAM theory for whiskered tori and quasi-periodic attractors (Calleja et al., 2019, Calleja et al., 2020).
• Hamiltonian regularization procedures such as in the Kepler problem (Marle, 2010).
• Analysis of Aubry–Mather theory and global attractors in dissipative flows (Marò et al., 2016).
• The paper of global dynamics of Lorentzian manifolds and contact/cobordism categories in general relativity (Morava, 2022).

Topological constraints on $\eta$ provide rigidity in the realization of conformally symplectic dynamics on given manifolds. The symplecticity of scattering and wave maps reveals that even dissipative or non-exact systems retain essential structural invariants, facilitating analysis and computation of long-term behaviors.

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This overview synthesizes the definition, topological and dynamical constraints, invariant manifold theory, scattering map construction and symplecticity, variational interpretation, presymplectic extensions, and broader applications of conformally symplectic maps in rigorous, technical, and comprehensive detail, as required for current arXiv-level research.