[Paper Review] A survey on partially hyperbolic dynamics

Federico Rodriguez Hertz, M. A. Rodriguez Hertz+1 more|ArXiv.org|Sep 13, 2006

Mathematical Dynamics and FractalsMathematics63 references20 citations

TL;DR

This survey provides a comprehensive overview of partially hyperbolic dynamics, focusing on key problems including examples, invariant foliations, accessibility, ergodicity, Lyapunov exponents, integrability of the central distribution, transitivity, and classification. It synthesizes known results and formulates open problems, particularly emphasizing the role of accessibility in ergodicity and the geometric and topological constraints in dimension three, with results linking integrability, volume growth, and homotopy type via Novikov’s theorem and quasi-isometric embeddings.

ABSTRACT

Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7) Transitivity and (8) Classification. Here we will survey the state of the art on these subjects, and propose related problems.

Motivation & Objective

• To systematize the state of the art in partially hyperbolic dynamics, focusing on eight central problems: examples, invariant foliations, accessibility, ergodicity, Lyapunov exponents, integrability of the central distribution, transitivity, and classification.
• To identify and formulate open problems in the field, particularly emphasizing the role of accessibility in ergodicity and the geometric constraints in low-dimensional systems.
• To explore the interplay between dynamical properties (e.g., ergodicity, growth rates) and topological invariants (e.g., homology, fundamental group, universal cover) in 3-manifolds.
• To examine the implications of integrability or non-integrability of the center bundle on the global dynamics and classification of partially hyperbolic systems.
• To extend known results on volume growth and quasi-isometric properties to partially hyperbolic systems on nilmanifolds and unit tangent bundles of hyperbolic surfaces.

Proposed method

• The paper uses a problem-oriented survey approach, organizing results around eight core problems in partially hyperbolic dynamics.
• It applies geometric and topological tools such as Novikov’s theorem on Reeb components and the theory of quasi-isometric embeddings to analyze the structure of invariant foliations and dynamics on 3-manifolds.
• The authors analyze the action of diffeomorphisms on homology and the fundamental group, using word length in the fundamental group to infer dynamical behavior.
• They employ volume growth estimates and quasi-isometric properties to relate the geometry of unstable leaves to the volume of balls in the universal cover.
• For ergodicity, the paper uses the accessibility approach, showing that accessibility implies ergodicity under certain conditions, particularly when dim(E^c) = 1.
• It applies techniques from Pesin theory and Lyapunov exponent analysis to study the non-uniform hyperbolicity and growth rates of orbits.

Experimental results

Research questions

• RQ1 Does accessibility imply ergodicity for all partially hyperbolic diffeomorphisms, particularly in the case dim(E^c) = 1?
• RQ2 Under what conditions is the central distribution integrable, and how does non-integrability affect the global dynamics?
• RQ3 Can the action of a partially hyperbolic diffeomorphism on the first homology group be partially hyperbolic when the fundamental group is abelian?
• RQ4 Is a partially hyperbolic diffeomorphism on a 3-manifold with integrable E^{s}⊕E^{u}, E^{cs}, or E^{cu} necessarily a K(π,1) manifold with contractible universal cover?
• RQ5 To what extent does the volume growth of balls in the universal cover constrain the geometry of unstable leaves?

Key findings

• If E^{s}⊕E^{u} or E^{c}⊕E^{u} is integrable in a 3-dimensional partially hyperbolic diffeomorphism, then the length of an unstable arc I satisfies length(I) ≤ C·v(diam(I)) + C, where v(r) is the volume of a ball of radius r.
• For nilmanifolds, due to polynomial volume growth, the bound improves to length(I) ≤ C·(diam(I))^4 + C, and for hyperbolic surface unit tangent bundles, exponential volume growth limits the bound to linear growth.
• If E^{s}⊕E^{u}, E^{cs}, or E^{cu} is integrable in a 3-dimensional partially hyperbolic system, then the manifold M is a K(π₁(M),1) space, implying its universal cover is contractible.
• When π₁(M) is abelian and one of the splittings E^{s}⊕E^{u}, E^{cs}, or E^{cu} is integrable, the induced action of f on H₁(M;ℝ) is partially hyperbolic.
• The existence of a codimension-one torus T invariant under f, with f|T preserving an expanding foliation, implies T is not the boundary of a solid torus, restricting the possible 3-manifold types.
• In the case of nilmanifolds, the center foliation supports a transitive expansive continuous flow, and the diffeomorphism preserves a finite number of center leaves under iteration.