Conformally Symplectic Diffeomorphisms

• Conformally symplectic diffeomorphisms are smooth maps on symplectic manifolds that scale the symplectic form by a constant or bounded function, generalizing classical symplectic maps.
• They obey strict topological and cohomological constraints on the scaling factor η, ensuring the preservation or systematic scaling of volume forms and invariant geometric structures like NHIMs.
• These maps are pivotal in modeling dissipative systems, such as mechanical damped oscillators and quasi-integrable resonant systems, with applications in constructing variational principles and scattering maps.

A conformally symplectic diffeomorphism is a smooth map $f: M \rightarrow M$ on a symplectic manifold $(M, \omega)$ that transforms the symplectic form by a scalar factor, i.e., $f^*\omega = \eta \omega$, where $\eta$ is typically a positive constant or, more generally, a function bounded away from zero and infinity. This property generalizes symplectic maps (which correspond to $\eta = 1$) and arises naturally in the paper of dissipative dynamical systems. The interplay between the conformal scaling, manifold topology, dynamical behavior, and the geometry of invariant objects—including normally hyperbolic invariant manifolds and scattering maps—governs the global structure and rigidity of these systems (Gidea et al., 20 Aug 2025).

1. Conformally Symplectic Maps: Definition and Topological Constraints

A diffeomorphism $f: M \rightarrow M$ is conformally symplectic if

$f^*\omega = \eta \omega,$

where $\omega$ is a symplectic form and $\eta > 0$ is a constant or (in more general discrete-time contexts) a function with $\sup \eta < \infty$, $\inf \eta > 0$. The volume form $\omega^n$ scales as $\eta^n$ under $f$, where $2n$ is the dimension of $M$.

The topological properties of $M$ restrict possible conformal factors $\eta$. If $\omega$ is non-exact ($[\omega] \neq 0$ in de Rham cohomology), the induced map $f^{\sharp}$ on cohomology satisfies

$f^{\sharp}[\omega] = \eta [\omega],$

so $\eta$ must be an eigenvalue of integer-structured cohomology-action matrices. In some cases (e.g., on certain compact symplectic manifolds), the possible $\eta$ are algebraic numbers determined by topological invariants. If $M$ is compact and $\omega$ is non-exact, often the only possibility is $\eta=1$ (i.e., classical symplectic maps). For exact symplectic forms ($\omega = d\alpha$), the transformation satisfies

$f^*\alpha = \eta \alpha + dP^f,$

where $P^f$ is a globally defined smooth function; this is essential in generating function approaches.

2. Dynamical Geometry: Normally Hyperbolic Invariant Manifolds

A central feature in conformally symplectic dynamics is the organization provided by invariant geometric objects, notably normally hyperbolic invariant manifolds (NHIMs). For a submanifold $\Lambda$ invariant under $f$, the tangent bundle splits as

$T_x M = E^s_x \oplus T_x \Lambda \oplus E^u_x,$

with $E^s$ exhibiting exponential contraction, $E^u$ exponential expansion, and $T\Lambda$ weaker rates. The Lyapunov rates (for expansion/contraction) in these bundles, together with the conformal factor $\eta$, are tightly constrained:

• The NHIM $\Lambda$ inherits a symplectic structure from $\omega$ if and only if certain "pairing rules" are satisfied:

$$(\mu_+^* / \mu_-^*) = \eta, \qquad (\lambda_+^* / \lambda_-^*) = \eta,$$

where $\mu_+$, $\mu_-$ refer to tangential rates and $\lambda_+$, $\lambda_-$ to stable/unstable rates.

• The rates and the conformal factor must also obey

$$\mu_+ \lambda_+ \eta^{-1} < 1, \qquad \mu_- \lambda_- \eta < 1.$$

These inequalities are necessary for the persistence of NHIMs with symplectic structure under conformally symplectic dynamics (Gidea et al., 20 Aug 2025).

3. Structure and Exactness of Scattering Maps

When the stable and unstable manifolds of a NHIM intersect transversely, one can define wave maps $\Omega_+, \Omega_-$ from neighborhoods in the manifolds to $\Lambda$, yielding the scattering map

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

where $H_-, H_+ \subset \Lambda$ are open sets. Even if $f$ is dissipative ($\eta \ne 1$), the scattering map $S$ is symplectic with respect to the induced structure on $\Lambda$. When $\omega = d\alpha$ and $f^*\alpha = \eta \alpha + dP^f$, $S$ is exact symplectic: $S^*\alpha = \alpha + dP_\alpha^S,$ with an explicit primitive $P_\alpha^S$ constructed from infinite sums involving the action contribution along orbits. Iteratively,

$$\alpha = \eta^{-N}(f^*)^N \alpha - d\left(\sum_{j=0}^{N-1} \eta^{-j-1} P^f \circ f^j \right),$$

and the generating function for the scattering map is

$P_\alpha^S = (P_+ - P_-) \circ (\Omega_-)^{-1},$

where $P_\pm$ are limits along stable/unstable directions, built from the action sum (Gidea et al., 20 Aug 2025).

4. Topological and Cohomological Implications for $\eta$

Since $f^*\omega = \eta \omega$, the possible values of $\eta$ are determined by the spectral properties of $f^*$ acting on $H^2(M;\mathbb{R})$. For instance, if $\omega$ is exact, no such restriction arises, and $\eta$ may be arbitrary (subject to boundedness). However, when $\omega$ is non-exact, possible values of $\eta$ are among the eigenvalues of the induced action, and the conformal symplectic diffeomorphisms' existence is thus connected to the topological type of $M$.

In some cases, e.g., tori or nilmanifold models, only a discrete set of algebraic values for $\eta$ can arise, which has implications for the classification of dynamical and geometric structures compatible with conformal symplecticity.

5. Variational Principles and Extensions to Presymplectic Settings

For exact symplectic forms, the explicit variational formulation for scattering maps involves infinite sums of action contributions, yielding generating functions for the reduced maps. This approach generalizes previous variational treatments (e.g., in twist maps and weak KAM theory) to the dissipative, conformally symplectic context.

The analysis extends naturally to presymplectic settings (where $\omega$ is nondegenerate only on a subbundle of $TM$), which are relevant in systems with multiple time scales or many expansion/contraction rates, such as quasi-integrable systems near multiple resonances.

6. Examples and Broader Applications

Conformally symplectic diffeomorphisms are foundational in the geometric description of dissipative systems, notably:

• Mechanical systems with velocity-proportional friction, including spin–orbit models and damped oscillators.
• Quasi-integrable dynamical systems near multiple resonances, where multiple Lyapunov rates and presymplectic structures frequently emerge.
• Models in nonequilibrium statistical mechanics and partial thermostats.
• Systems analyzed using Aubry–Mather theory, where dissipation leads to global attractors characterized by weak KAM phenomena but now equipped with symplectic (or conformally symplectic) reductions and invariant structures.

7. Summary Table: Relations among Structures

Structure	Symplectic Map	Conformally Symplectic Map ($\eta \ne 1$)	Dissipation
NHIMs	Symplectic	Symplectic if rate/η pairing is satisfied	Possible
Scattering Maps	Symplectic	Symplectic; exact if ω exact	Possible
Primitive (Action Form)	$f^*\alpha = \alpha + dP^f$	$f^*\alpha = \eta \alpha + dP^f$	Yes
Volume	Preserved	Scaled by η^n	Yes/No

Conclusion

Conformally symplectic diffeomorphisms interpolate between symplectic and dissipative dynamics, with the scaling factor $\eta$ regulating the extent of dissipation and governed by the topology and cohomology of the underlying manifold. These maps facilitate the persistence and symplectic (or exact symplectic) structure of critical invariant objects—NHIMs, their stable/unstable foliations, and scattering maps—so long as critical inequalities and pairing rules relating dynamical rates and the conformal factor are satisfied. Their explicit variational characterization, robustness in the dissipative context, and compatibility with presymplectic generalizations position them as a central mathematical object in the geometric approach to dissipative systems and multi-scale dynamics (Gidea et al., 20 Aug 2025).