Invariant Foliations in Dynamical Systems

• Invariant foliations are geometric partitions of manifolds where dynamics preserve the leaf structure, enabling local decoupling and stability analysis.
• They appear in diverse fields such as bi-Hamiltonian geometry, algebraic dynamics, and stochastic systems, providing a unified framework for studying integrability and singular behavior.
• Invariant foliations support data-driven model reduction by straightening dynamics and isolating key transverse modes through invariant coordinate transformations.

Invariant foliations are geometric structures central to the study of dynamical systems, differential geometry, and algebraic geometry. Defined on smooth, complex, or symplectic manifolds, an invariant foliation partitions phase space into disjoint leaves that are preserved under the dynamics, symmetries, or flows of interest. Each leaf is an immersed submanifold or analytic variety, and invariance means that the evolution map (flow, diffeomorphism, etc.) maps leaves into leaves. Invariant foliations encode robust decoupling in the local and global behavior of systems—decoupling transverse directions, reducing local stability analyses, organizing conditional measures, and enabling data-driven reduced order modeling. Their intrinsic structure is apparent across pure and applied mathematics: in bi-Hamiltonian geometry, algebraic dynamics, singular foliation theory, stochastic systems, and algorithmic model reduction.

1. Definitions and Abstract Framework

Let $M$ be a manifold (smooth, complex, or symplectic). A foliation $\mathcal{F}$ of codimension $p$ is a coherent subsheaf $T_{\mathcal{F}} \subset TM$ of generic rank $n-p$, involutive under the Lie bracket $$[T_{\mathcal{F}}, T_{\mathcal{F}}] \subset T_{\mathcal{F}}$$; the leaves integrate $T_{\mathcal{F}}$ to immersed submanifolds. For a vector field, flow, or automorphism $f$, $\mathcal{F}$ is invariant if $f_*(T_{\mathcal{F}})=T_{\mathcal{F}}$ locally, so $f$ maps leaves into leaves. In complex or algebraic settings, the notion dualizes to Pfaff systems defined via twisted $p$-forms $\omega \in H^0(X, \Omega_X^p \otimes L)$ and the Frobenius integrability condition $\omega \wedge d\omega = 0$ (Corrêa, 2021). In bi-Hamiltonian geometry, invariant distributions $D \subset TM$ are those whose fibers $D_x$ are invariant under the recursion operator $N=P_1P_0^{-1}$ associated with compatible Poisson structures and vanish under the Nijenhuis torsion (Kozlov, 2022).

2. Classification, Existence, and Integrability

Local classification of invariant foliations exploits normal forms, splitting the tangent space by spectral or block-decomposition. In bi-Hamiltonian settings, Turiel’s splitting applies the Jordan–Kronecker decomposition to the recursion operator $N$, reducing the study to irreducible factors with characteristic polynomial $\chi_N(t)=\prod_i\chi_i(t)$ (Kozlov, 2022). Linear invariant distributions on each block are classified as sums of intersections of kernels and images:

$$\bigl(\ker(P-\lambda)^k \cap \mathrm{Im}(P-\lambda)^\ell\bigr)$$

subject to interlacing inequalities on exponents (Kozlov, 2022). Integrability is governed by the Nijenhuis torsion—image-type distributions are always involutive, kernel-type are so only when eigenvalues are locally constant. In parabolic PDEs, invariant foliations near normally hyperbolic equilibrium manifolds are constructed via Lyapunov–Perron methods, with each leaf a $C^1$-graph tangent to the stable directions about each equilibrium (Pruess et al., 2012).

3. Foliations in Algebraic and Holomorphic Dynamics

For holomorphic endomorphisms, invariant foliations can appear as pencils of curves: taping the Green current $T$ and equilibrium measure $\mu = T \wedge T$ in a pluripotential-theoretic product structure enforces the existence of a local invariant foliation in a neighborhood of supp$(T)\setminus\mathcal{E}$, and, when extended through the exceptional set, as a global invariant pencil of lines (Tapiero, 20 Mar 2024). In algebraic geometry, the parameter space of foliations on $\mathbb{P}^2$ is a projective space acted on by $PGL(3)$; stability conditions and the dual discriminant map classify foliations in terms of singular loci and invariants, with instability arising from one-dimensional singularities or isolated singularities of high multiplicity (Esteves et al., 2011). For families of foliations on surfaces, invariance of plurigenera stabilizes for $m \gg 0$, relying on Zariski decompositions and vanishing theorems (Cascini et al., 2015).

4. Invariant Foliations in Stochastic and Forced Systems

Random dynamical systems—SDEs and SPDEs—admit invariant foliations defined as families of random Lipschitz graphs over spectral bundles, with trajectories exhibiting exponential dichotomy and stochastic perturbations. The Lyapunov–Perron method generates stable and unstable random foliations, with error estimates quantifying deviation between deterministic and stochastic leaves to $\mathcal{O}(\varepsilon^2)$ in small noise (Sun et al., 2011); in Lévy-driven systems, the Marcus integral and Ornstein–Uhlenbeck shifts are required (Chao et al., 2018). For forced quasi-periodic systems, existence and uniqueness of invariant foliations about analytic invariant tori is established via the parametrization method, under Diophantine non-resonance conditions and small-divisor estimates. Such foliations encode reduced order transversal dynamics, enabling explicit calculation of amplitude-dependent frequencies and damping (Szalai, 21 Mar 2024).

5. Data-Driven Model Reduction using Invariant Foliations

Invariant foliations underpin model order reduction in high-dimensional physical and engineered systems. Key is the construction of a coordinate system that straightens out invariant manifolds or tori, with leaves encoding reduced order dynamics. The defining invariance equation in continuous or discrete (map) settings reads, e.g.,

$$W(\Phi_t(x), e^{\Lambda t}s) = \Phi_t(W(x,s)), \quad \text{(continuous)}$$

$$\boldsymbol{R}(\boldsymbol{U}(\boldsymbol{x}, \theta), \theta) = \boldsymbol{U}(\boldsymbol{F}(\boldsymbol{x}, \theta), \theta+\omega) \quad \text{(discrete)}$$

Simultaneous regression of encoders/decoders and reduced maps is performed using multivariate polynomials with hierarchical tensor compression to maintain linear complexity with problem dimension (Szalai, 2022, Szalai, 21 Mar 2024). Invariant manifolds are extracted as fixed leaves of the foliation. The approach corrects for nonlinear coordinate-induced frequency and damping distortions and outperforms autoencoders or Koopman eigenfunction approaches especially in regime of near-resonance and limited regularity.

6. Singular Foliations, Cohomological Invariants, and Stability

The theory of singular foliations incorporates local finite generation and involutive modules, with leaves constructed via the Stefan–Sussmann theorem. Linearizability along a leaf is a Morita-invariant property, preserved under Hausdorff Morita equivalence. Key in the proof is the characterization of tubular neighborhood embeddings via Euler-like vector fields and compatible projection-lift descent along source-connected groupoids (Zambon, 24 Jun 2025). Cohomological invariants of foliations—basic and antibasic cohomology—yield analytic and topological constraints, with the antibasic Betti numbers independent of metric and diffeomorphism, and a Hodge decomposition theorem in the Riemannian case (Habib et al., 2019). Mapping-torus and log-tangent foliations, as well as elliptic-tangent bundles, serve as canonical examples where linearizability and cohomological computation are explicit.

7. Open Problems and Future Directions

Several challenges remain open. Full topological and global classification of foliations—especially about degenerate bi-Hamiltonian points, in higher codimension, or in the presence of singularities—is incomplete (Kozlov, 2022, Corrêa, 2021). Data-driven methods lack a global uniqueness theory for invariant foliations, especially when internal resonances or limited smoothness are present (Szalai, 21 Mar 2024). The relationship between cohomological obstructions and geometric or transverse structures needs further clarification. Stability of plurigenera in low $m$ and the behaviour of non-taut Riemannian flows and infinite-dimensional basic cohomology represent ongoing areas of research (Habib et al., 2019, Cascini et al., 2015).

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Table: Core Aspects of Invariant Foliations across Domains

Domain	Defining Structure	Key Invariance/Integrability Criteria
Bi-Hamiltonian Geometry	Distributions $D \subset TM$, recursion operator $N$	Nijenhuis torsion vanishing; block classification (Kozlov, 2022)
Algebraic Geometry	Subsheaves/Pfaff systems, rational/meromorphic invariants	Frobenius and Zariski decompositions, stability/GIT (Corrêa, 2021, Esteves et al., 2011, Cascini et al., 2015)
Stochastic Dynamics	Random Lipschitz graphs over spectral bundles in Hilbert space	Lyapunov–Perron fixed point; small noise asymptotics (Sun et al., 2011, Chao et al., 2018)
Data-Driven Model Reduction	Encoder/decoder maps, polynomial approximation, hierarchical tensor compression	Invariance equation; polynomial and Fourier parameterization (Szalai, 2022, Szalai, 21 Mar 2024, Szalai, 21 Mar 2024)
Singular Foliations/Theory	Locally finitely generated $C^\infty$ modules, Euler-like fields	Morita-invariance, tubular neighborhood linearization (Zambon, 24 Jun 2025)

The interplay of invariant foliations with integrability, model reduction, algebraic and topological invariants, and dynamical rigidity establishes them as a pivotal notion informing both the theory and data-driven analysis of complex systems.