[Paper Review] The centralizer of a C1 generic diffeomorphism is trivial

Bonatti, Christian, Sylvain Crovisier+1 more|ArXiv.org|May 2, 2007

Mathematical Dynamics and Fractals15 references3 citations

TL;DR

This paper establishes that for a generic $C^1$ diffeomorphism on a compact connected manifold, the centralizer—the set of diffeomorphisms commuting with it—is trivial, meaning only integer powers of the diffeomorphism itself commute with it. The proof combines local perturbation techniques and global structural arguments, leveraging residuality in the $C^1$ topology to show that such trivial centralizers are generic, though not open, in the space of $C^1$ diffeomorphisms.

ABSTRACT

In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset.

Motivation & Objective

• To resolve S. Smale's question on whether trivial centralizers are generic in the $C^1$ topology for diffeomorphisms on compact connected manifolds.
• To prove that the set of $C^1$ diffeomorphisms with trivial centralizer is residual in $\mathrm{Diff}^1(M)$, answering affirmatively the second part of Smale's question.
• To show that this set is not open, thus answering negatively the third part of Smale's question, by constructing $C^1$-dense families of $C^\infty$ diffeomorphisms with nontrivial centralizers.
• To establish that for generic $f$, any commuting diffeomorphism must be a power of $f$ on a dense open subset, and that this exponent function is constant globally for generic $f$.

Proposed method

• Use of local perturbation techniques to modify the derivative of $f$ in small neighborhoods where $f$ behaves approximately linearly over many iterates.
• Application of topological towers with large return times to control non-periodic orbits and spread perturbations over long time intervals.
• Employment of upper-semicontinuous maps from $\mathrm{Diff}^1(M)$ to the space of compact subsets of Lipschitz homeomorphisms to analyze continuity of centralizers.
• Leveraging the Baire category theorem via upper-semicontinuity to identify residual sets of continuity points for the centralizer map.
• Combining the $C^1$ Closing Lemma to ensure existence of hyperbolic periodic points in residual sets, enabling contradiction arguments on unbounded return times.
• Using contradiction via growth of derivatives: if centralizer elements are uniformly Lipschitz but correspond to unbounded powers, their derivatives grow exponentially, violating Lipschitz bounds.

Experimental results

Research questions

• RQ1 Is the set of $C^1$ diffeomorphisms with trivial centralizer dense in $\mathrm{Diff}^1(M)$?
• RQ2 Is this set residual in $\mathrm{Diff}^1(M)$, i.e., containing a dense $G_\delta$ subset?
• RQ3 Does the set of $C^1$ diffeomorphisms with trivial centralizer contain an open and dense subset?
• RQ4 Can a generic $C^1$ diffeomorphism be embedded in a continuous one-parameter group (i.e., a flow), or admit a root?
• RQ5 What is the structure of the centralizer of a generic $C^1$ diffeomorphism, and how does it relate to group-theoretic properties like nilpotency or solvability?

Key findings

• The set of $C^1$ diffeomorphisms with trivial centralizer is residual in $\mathrm{Diff}^1(M)$, confirming an affirmative answer to the second part of Smale’s question.
• For generic $f$, any $g$ commuting with $f$ must satisfy $g = f^\alpha$ on an open and dense subset of $M$, where $\alpha$ is locally constant.
• For generic $f$, the exponent $\alpha$ is globally constant, implying $Z^1(f) = \langle f \rangle$, so the centralizer is trivial.
• The set of $C^1$ diffeomorphisms with trivial centralizer is not open, as there exist $C^1$-dense families of $C^\infty$ diffeomorphisms with large centralizers.
• In dimension $\geq 3$, such examples have open sets of periodic points; in lower dimensions, they arise as time-1 maps of flows.
• The centralizer of a $C^1$ generic diffeomorphism cannot contain any nontrivial abelian group beyond $\langle f \rangle$, and the same holds for nilpotent groups.