[Paper Review] On the Transitivity of Invariant Manifolds of Conservative Flows

Fábio Castro, Fernando A. Oliveira|arXiv (Cornell University)|Feb 28, 2015

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper establishes that for $ C^r $ volume-preserving flows on compact manifolds of dimension at least 3, the closure of invariant manifolds associated with hyperbolic critical elements (periodic orbits or singularities) is chain transitive generically. The proof relies on constructing local perturbations to ensure dense recurrence in invariant manifolds, leveraging a novel flow box theorem for volume-preserving systems and a new local perturbation technique to modify orbits without altering their past trajectories.

ABSTRACT

The main result of this work is the following: for volume preserving flows on compact manifolds with the $C^r$ topology, $1 \leqq r \leqq \infty$ , the closure of every invariant manifold of periodic orbits and singularities is a chain transitive set. We also develop to new local constructions, which surprise by the simplicity of the arguments. One, a local perturbation to change an orbit to a nearby without altering its past. The other is a flow box theorem in the context of volume preserving flows, a result that is well known for Hamiltonians or general flows.

Motivation & Objective

• To establish that the closure of invariant manifolds of hyperbolic critical elements in volume-preserving flows is chain transitive under generic $ C^r $ conditions.
• To address the long-standing open problem of whether periodic orbits are dense in conservative systems, by focusing on the weaker but provable notion of chain transitivity.
• To develop new local perturbation techniques that allow modifying future orbits without altering past trajectories, enabling control over recurrence in invariant manifolds.
• To extend the classical flow box theorem to the setting of volume-preserving flows, a result previously known only for Hamiltonian or general flows.
• To demonstrate that in a residual subset of $ \mathcal{X}^r_\omega(M) $, the closure of invariant manifolds contains a dense set of recurrent orbits, implying chain transitivity.

Proposed method

• Introduces a new local perturbation technique that allows modifying the future evolution of a point's orbit without changing its past trajectory, preserving the flow's volume-preserving property.
• Develops a volume-preserving flow box theorem, providing a local normal form for volume-preserving vector fields near hyperbolic critical elements, analogous to the classical result for general flows.
• Uses the concept of $ (\epsilon,t) $-chains to define chain transitivity, ensuring that any two points in a set can be connected by a finite sequence of flow segments with small jumps and large time steps.
• Applies the Baire category theorem in the $ C^r $ topology to show that the set of vector fields for which invariant manifold closures are chain transitive is residual.
• Leverages the fact that a compact metric space with a dense set of $ \omega $-recurrent points is chain transitive, and constructs such recurrence generically via perturbations.
• Employs inductive perturbation arguments on fundamental domains of stable and unstable manifolds, ensuring that return orbits can be controlled within small balls without intersecting previous return arcs.

Experimental results

Research questions

• RQ1 Is the closure of an invariant manifold of a hyperbolic critical element chain transitive for generic volume-preserving flows on compact manifolds?
• RQ2 Can a local perturbation technique be constructed to alter the future orbit of a point without modifying its past trajectory, while preserving the volume-preserving property?
• RQ3 Does a flow box theorem exist in the context of volume-preserving flows, analogous to the classical result for Hamiltonian or general flows?
• RQ4 Can the closure of an invariant manifold contain a dense set of recurrent orbits in a generic $ C^r $ volume-preserving flow?
• RQ5 Is chain transitivity a generic property of invariant manifold closures in conservative systems, even when attractors or repellers may appear in the closure?

Key findings

• For a residual subset $ \mathcal{R} \subset \mathcal{X}^r_\omega(M) $, the closure of every invariant manifold of a hyperbolic critical element (periodic orbit or singularity) is a chain transitive set.
• The closure of an invariant manifold may contain attractors or repellers in non-generic cases, but such structures do not exist generically.
• A new local perturbation method allows modifying the future evolution of a point's orbit without altering its past, preserving volume and enabling control over recurrence.
• A volume-preserving flow box theorem is established, providing a local normal form for volume-preserving vector fields near hyperbolic critical elements.
• The existence of a dense set of $ \omega $-recurrent points in the closure of an invariant manifold implies chain transitivity, which is shown to be generic via perturbation techniques.
• The result holds for all $ r \in [1, \infty] $, with the $ C^\infty $ case handled via the inductive limit topology of $ C^r $ topologies.