[Paper Review] Persistence of laminations

Pierre Berger|ArXiv.org|Jul 30, 2008

Mathematical Dynamics and Fractals3 references4 citations

TL;DR

This paper presents a modern, generalized proof of the persistence of normally hyperbolic laminations under $C^r$-perturbations, extending the classical Hirsch-Pugh-Shub theorem to endomorphisms, non-compact laminations, and the complex analytic setting. Using deformation theory of complex structures and cone field constructions, it establishes the existence of $C^r$-persistent laminations and holomorphic families of complex laminations under holomorphic perturbations, with applications to Hénon maps and fibered holomorphic dynamics.

ABSTRACT

We present a modern proof of some extensions of the celebrated Hirsch-Pugh-Shub theorem on persistence of normally hyperbolic compact laminations. Our extensions consist of allowing the dynamics to be an endomorphism, of considering the complex analytic case and of allowing the laminations to be non compact. To study the analytic case, we use the formalism of deformations of complex structures. We present various persistent complex laminations which appear in dynamics of several complex variables: Henon maps, fibered holomorphic maps... In order to proof the persistence theorems, we construct a laminar structure on the stable and unstable of the normally hyperbolic laminations.

Motivation & Objective

• To extend the classical Hirsch-Pugh-Shub $C^r$-persistence theorem for normally hyperbolic laminations to the case of endomorphisms, which may be non-bijective and singular.
• To generalize the persistence result to non-compact laminations by requiring the pullback of the dynamics to preserve the closure of the lamination.
• To establish the persistence of complex laminations under holomorphic one-parameter families of endomorphisms using deformation theory of complex structures.
• To construct invariant laminar structures on stable and unstable sets of normally hyperbolic laminations via cone fields and holomorphic lifting.
• To prove that the persistent laminations inherit complex analytic structures, even when the total space is not a trivial product bundle.

Proposed method

• Use of cone fields on the pullback of the tangent bundle to define stable and unstable directions, ensuring uniform contraction/expansion relative to the leaf tangent spaces.
• Construction of a $C^r$-lamination structure on the pullback bundle of normal directions via local charts and $C^r$-transitions in the Grassmannian bundle.
• Application of deformation theory of complex structures to construct holomorphic families of complex laminations over a complex parameter space.
• Proof of $J$-invariance of tangent spaces to the laminations by analyzing intersections of iterated tangent images under the dynamics.
• Use of the pullback map $f^*$ to define a dynamics on the lamination itself, ensuring plaque-expansiveness and compatibility with perturbations.
• Leveraging the uniqueness of complex structures (via Lemma 5.2) to ensure that the lifted laminations are holomorphic immersions.

Experimental results

Research questions

• RQ1 Can the $C^r$-persistence of normally hyperbolic laminations be extended to endomorphisms that are not diffeomorphisms?
• RQ2 Does the persistence result hold for non-compact laminations when the pullback map preserves the closure of the lamination?
• RQ3 Can the HPS theorem be generalized to the holomorphic setting, ensuring the existence of holomorphic families of persistent complex laminations?
• RQ4 Under what conditions is the persistent complex lamination structure on the total space isomorphic to a product structure $\mathcal{L} \times B_0$?
• RQ5 How can one construct a $C^r$-smooth lamination structure on the normal bundle of a lamination to ensure $C^r$-persistence?

Key findings

• Theorem 0.1 establishes $C^r$-persistence of $r$-normally expanding endomorphisms on compact laminations, even when the map is non-bijective.
• Theorem 0.2 extends persistence to $r$-normally hyperbolic endomorphisms with bijective pullbacks, preserving the $C^r$-structure under perturbations.
• Theorem 0.3 proves the existence of a holomorphic family of complex laminations $(\mathcal{L}_t)$ over a complex parameter space $B_0$, invariant under holomorphic perturbations of the endomorphism.
• The persistent complex laminations need not arise from a product structure $\mathcal{L} \times B_0$, as shown in Example 5.5, but such a product structure exists under sufficient conditions (Proposition 5.3).
• The tangent spaces to the persistent laminations are shown to be $J$-invariant by analyzing the intersection of iterated tangent images under the dynamics.
• The construction of the laminar structure on the normal bundle via cone fields and $C^r$-lifting into the Grassmannian ensures the existence of a well-defined $C^r$-lamination on the total space.