Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms. (English) Zbl 1497.37036

Damjanović, Danijela; Wilkinson, Amie; Xu, Disheng

Duke Math. J. 170, No. 17, 3815-3890 (2021).

One of the classical questions in perturbation theory is: what kind of perturbation can be added to a diffeomorphism in such a way that both the diffeomorphism and the perturbed one belong to smooth flows? This paper gives an answer for algebraic geodesic flows in negative curvature in the conservative setting.
Consider as examples the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle. The authors consider a large class of algebraic systems and smooth ergodic perturbations. They show that the smooth centralizer is either virtually \(\mathbb{Z}^l\) or contains a smooth flow, where the centralizer of a diffeomorphism \(f: M \to M\) is the set of diffeomorphisms \(g\) that commute with \(f\) under composition: \(f\circ g = g\circ f\).
The authors use a combination of many powerful techniques and some of them are new. Among them: a novel geometric approach to build new partially hyperbolic elements in hyperbolic Weyl chambers via Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livšic theory, nonstationary normal forms.

Reviewer: Xu Zhang (Weihai)

MSC:

37D30	Partially hyperbolic systems and dominated splittings
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D35	Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
37C20	Generic properties, structural stability of dynamical systems

Keywords:

centralizer; geodesic flows; higher-rank actions; partially hyperbolic; rigidity

References:

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