[Paper Review] Symmetries of vector fields: the diffeomorphism centralizer

Davi Obata|arXiv (Cornell University)|Mar 14, 2019

Advanced Differential Equations and Dynamical Systems16 references4 citations

TL;DR

This paper establishes that for a C¹-generic vector field on a compact manifold with at most finitely many sinks or sources, the diffeomorphism centralizer—the set of C¹ diffeomorphisms commuting with the flow—is quasi-trivial, meaning every such diffeomorphism is a reparametrization of the flow. If the vector field has at most countably many chain-recurrent classes, the centralizer is trivial, i.e., generated solely by the flow itself.

ABSTRACT

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $C^1$-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

Motivation & Objective

• To investigate the structure of the diffeomorphism centralizer for C¹ vector fields, defined as the set of C¹ diffeomorphisms commuting with the flow.
• To determine whether this centralizer is quasi-trivial (each element is a time reparametrization of the flow) or trivial (reparametrization with constant time shift).
• To extend the generic triviality result for diffeomorphism centralizers—previously established for diffeomorphisms—to the setting of vector fields and their flows.
• To provide a criterion for when a diffeomorphism in the centralizer is a reparametrization of the flow.
• To address the analog of Smale’s centralizer problem for flows, asking whether C¹-generic vector fields have trivial or quasi-trivial centralizers.

Proposed method

• Define the C¹-diffeomorphism centralizer of a vector field X as the set of C¹ diffeomorphisms f such that f∘Xₜ = Xₜ∘f for all t∈ℝ.
• Use the notion of quasi-triviality: f is in the centralizer iff f(x) = Xₜ(x) for some continuous time function t(x).
• Prove that for C¹-generic vector fields with finitely many sinks/sources, the centralizer is quasi-trivial via a limit argument on stable manifolds and the linear Poincaré flow.
• Establish uniform boundedness of the time reparametrization function τ via the LND (limiting non-degeneracy) property and C¹-norm control.
• Show that if the number of chain-recurrent classes is at most countable, the continuous time function τ must be constant, implying triviality.
• Use the density of the non-wandering set and orbit dynamics to extend local reparametrization identities globally.

Experimental results

Research questions

• RQ1 Is the C¹-diffeomorphism centralizer of a C¹-generic vector field with finitely many sinks or sources quasi-trivial?
• RQ2 Under what additional conditions does quasi-triviality of the centralizer imply triviality?
• RQ3 Can a diffeomorphism in the centralizer be characterized as a time reparametrization of the flow?
• RQ4 How does the structure of the centralizer differ between vector field centralizers and diffeomorphism centralizers?
• RQ5 Does the centralizer of a C¹-generic vector field in the Axiom A or sectional Axiom A class exhibit triviality?

Key findings

• There exists a C¹-residual subset 𝒪 ⊂ 𝒳¹(M) such that for any X ∈ 𝒪 with at most finitely many sinks or sources, the C¹-diffeomorphism centralizer of X is quasi-trivial.
• If X ∈ 𝒪 has at most countably many chain-recurrent classes, then the C¹-diffeomorphism centralizer of X is trivial.
• For C¹-generic vector fields on a compact surface, the centralizer is trivial, extending Peixoto’s Morse-Smale result.
• A C¹-generic Axiom A vector field has trivial C¹-diffeomorphism centralizer.
• The centralizer of a C¹-generic sectional Axiom A vector field is trivial.
• The result extends to C¹-vector fields far from homoclinic tangencies in dimension three, as per the Palis conjecture in dimension three.