Analytic Invariant Submanifolds

• Analytic invariant submanifolds are locally defined analytic sets that remain unchanged under holomorphic maps, vector fields, or group actions.
• They are characterized through resonance, small-divisor conditions, and formal linearization techniques that reveal partial and full linearizable dynamics.
• Their applications span complex dynamics, singularity theory, control systems, and Hilbert module structures, providing key geometric and analytic insights.

An analytic invariant submanifold is a locally defined, real or complex analytic submanifold invariant under the action of a group or family of maps—typically holomorphic diffeomorphisms or vector fields—satisfying additional analytic, algebraic, or geometric constraints. Their existence, classification, and geometric structure are central in complex dynamics, singularity theory, subgeometry, and analytic control theory. Analytic invariant submanifolds serve both as obstructions to global linearization (via resonance and small-divisor phenomena) and as canonical loci carrying geometric or control-theoretic invariants.

1. Analytic Invariant Submanifolds: Basic Concepts

Let $M$ be a real or complex analytic manifold, often $(\mathbb{C}^n,0)$. A germ of an analytic submanifold $S \subset (M,0)$ of (complex) dimension $d$ is defined locally by $n-d$ analytically independent equations:

$$h_1(z) = \cdots = h_{n-d}(z) = 0, \quad h_j(0)=0, \quad \text{with } \{dh_1(0),...,dh_{n-d}(0)\} \text{ linearly independent}.$$

The submanifold $S$ is called invariant under a group $G$ (e.g., abelian group of diffeomorphism germs or a Lie group of symmetries) if for each $g\in G$, $g(S)\subset S$. In the complex analytic context, key cases are:

• Invariance under abelian groups of holomorphic diffeomorphisms
• Invariance under anti-holomorphic involutions
• Invariance under flows of analytic vector fields or affine control systems

Such submanifolds are deeply entwined with notions of resonance, linearizability, ideal-theoretic obstructions, and the structure theory of analytic group actions.

2. Holomorphic and Anti-holomorphic Invariance: Existence and Characterization

Stolovitch's Framework (Stolovitch, 2016)

Consider an abelian group $G = \{F_1,...,F_\ell\}$ of germs of holomorphic diffeomorphisms at $0\in\mathbb{C}^n$, each fixing the origin:

$$F_i(z) = D_i z + f_i(z), \quad D_i=\operatorname{diag}(\mu_{i,1},...,\mu_{i,n}),\ f_i(z) = O(|z|^2).$$

Given a monomial ideal $I \subset \mathcal{O}_n$ generated by monomials $\{z^{R_1}, ..., z^{R_p}\}$ and its zero locus $V(I)$, under suitable hypotheses:

• Diophantine (small-divisor) condition: For each $k\geq 1$,

$$\omega_k(D,I) = \inf\{ \max_{1\leq i\leq \ell} | \mu_i^Q - \mu_{i,j} | \neq 0 : 2\leq |Q| \leq 2^k, 1\leq j \leq n, Q \notin \text{generators of }I \}$$

with Brjuno/Herman-type summability,

• Formal linearizability on $I$, i.e., existence of a formal change of coordinates $\hat\Phi$ tangent to identity reducing all $F_i$ to $D_i z +$ (terms in $I$),

then there exists a holomorphic coordinate change $\Phi$ conjugating $F_i$ to $D_i z + g_i(z)$ where each $g_i(z)\in (I)^n$. Consequently, $V(I)$ is a holomorphic analytic invariant submanifold, and the dynamics on $V(I)$ are linear.

When $I=(0)$ and $D$ jointly satisfy a (full) Brjuno-type condition, full holomorphic linearizability (in Pöschel-Rüssmann sense) is recovered.

Crucially, if the dynamics are resonant only on a proper subset of coordinates (the resonant ideal $I$), then $V(I)$ is, up to holomorphic change of coordinates, a union of invariant linear subspaces—often a union of coordinate planes.

3. Resonant Ideals, Small-divisor Obstructions, and Partial Linearization

The central invariant-theoretic mechanism is the resonant ideal—the ideal generated by monomials attached to resonant multi-indices $Q$ solving $\mu_i^Q=\mu_{i,j}$ for all $i, j$. Resonance obstructs full linearization; small divisors further dictate convergence/divergence of formal series arising in linearization attempts.

Partial linearization is achieved "modulo" this resonant ideal:

• The analytic set $V(I)$ (the analytic invariant submanifold) is characterized as the largest locus where simultaneous resonance occurs.
• On $V(I)$, the group action is holomorphically conjugated to linear form.
• Outside $V(I)$, nonlinear phenomena persist.

This mechanism is essential in the context of families of intersecting totally real submanifolds (e.g., fixed loci of anti-holomorphic involutions), where full simultaneous straightening is obstructed, but a "maximally straightened" germ $S=V(I)$ exists (Stolovitch, 2016).

4. Analytic Invariant Submanifolds in Control Theory and Exterior Differential Systems

In affine control systems of the form

$$\dot{\mathbf{x}} = f(\mathbf{x}) + \sum_j g_j(\mathbf{x}) u_j$$

with analytic vector fields $f, g_j$ spanning a constant-rank distribution $D$, connected analytic submanifolds $N\subset M$ invariant under all admissible controls correspond to level sets of systems of generalized first integrals $p=(p_1, ..., p_d)$ with differentials $dp_\alpha$ satisfying $X\cdot p_\alpha|_N = 0$ for all $X\in D$ (Han et al., 2017).

Algorithmically, invariant analytic submanifolds are constructed as the zero loci of such $p$, derived via reduction of Pfaffian systems and torsion analysis, yielding canonical foliations by invariant leaves.

Analytic regularity is ensured by the analyticity of the vector fields; all such submanifolds are locally real analytic.

5. Invariant Submanifolds in Hilbert Modules and Sheaf Models

For analytic Hilbert modules $\mathcal{H}\subset \mathcal{O}(\Omega)$ (with $K(z,w)$ a reproducing kernel and polynomial density), a submodule $[\mathcal{I}]$ associated to an ideal $\mathcal{I}$ has an associated analytic zero set $V_{[\mathcal{I}]}$ (Biswas et al., 2022). When $V_{[\mathcal{I}]}$ is a smooth submanifold,

• The restriction of the natural coherent sheaf $S^{[\mathcal{I}]}$ yields a holomorphic vector bundle $E\to V_{[\mathcal{I}]}$,
• The reproducing kernel decomposes locally into a holomorphic frame,
• Geometric invariants of $E$ (Chern forms, curvature, second fundamental form) serve as unitary invariants for the original analytic Hilbert module.

This construction ties algebraic (ideal-theoretic), analytic, and topological data of the module to explicit geometric invariants of analytic invariant submanifolds.

6. Analytic 1-Submanifolds and Lie Group Actions

Under analytic, non-contractive or regular actions $\varphi:G\times M\to M$, analytic 1-submanifolds (curves) decompose rigidly (Hanusch, 2016):

• They are either "exponential" (analytic images of 1-parameter subgroups), or
• "Free," admitting a unique decomposition into symmetry-free segments related by discrete group action (z-decomposition or E-decomposition).

Such decompositions reflect the interaction between analytic structure and group symmetry, providing a canonical classification for 1-dimensional analytic invariants. Extensions to higher dimensions and the possible presence of holonomy or more intricate stratification remain open and technically challenging, but the analytic mechanisms (ideal-theoretic invariance, resonance, decomposition) persist as central themes.

7. Applications and Illustrative Examples

• Intersecting Totally Real Submanifolds: In (C$^n$,0), a finite family of totally real n-manifolds $M_i$ intersecting at 0 is associated to anti-holomorphic involutions $\rho_i$. If resonance obstructs full simultaneous linearization, one constructs the analytic invariant set $S=V(I)$ where partial straightening (to totally real linear subspaces) is possible (Stolovitch, 2016).
• Dynamical Systems: Analytic invariant submanifolds serve as canonical loci capturing the linearized dynamics when full linearization is impossible due to resonances or small divisors.
• Control Theory: Level sets of generalized first integrals yield analytic invariant submanifolds (invariant under admissible controls), constructible by torsion analysis in associated Pfaffian systems (Han et al., 2017).
• Hilbert Module Theory: Sheaf-theoretic invariants of analytic submodules correspond to geometric data on the analytic set $V_{[\mathcal{I}]}$, including hermitian vector bundle structures and curvature invariants (Biswas et al., 2022).

Summary Table: Key Features of Analytic Invariant Submanifolds

Setting	Invariance Condition	Invariant Submanifold Construction
Holomorphic dynamics (Stolovitch, 2016)	$G$-invariance, resonance	Zero locus $V(I)$ of resonant ideal
Affine control systems (Han et al., 2017)	Invariance under control orbits	Zero set of generalized first integrals
Analytic Hilbert modules (Biswas et al., 2022)	Module substructure	Zero set $V_{[\mathcal{I}]}$, vector bundle $E$
Lie group actions (Hanusch, 2016)	Symmetry under $G$	Discrete decomposition into free segments or exponential orbits

These developments fundamentally connect local analytic geometry, group/semigroup dynamics, and modern invariant theory, with continued impact on submanifold geometry, linearization theory, and the classification of analytic dynamical phenomena.