Hyperbolicity and Rigidity for Fibred Partially Hyperbolic Systems

TL;DR

This paper classifies volume-preserving fibred partially hyperbolic systems with 2D centre, revealing conditions for Lyapunov exponents, invariant structures, and rigidity, with implications for hyperbolicity and topology of centre leaves.

Contribution

It provides a comprehensive classification of such systems, identifying when they exhibit Lyapunov exponents, invariant line fields, or conformal structures, and establishes rigidity results under certain conditions.

Findings

Systems either have distinct centre Lyapunov exponents or invariant structures.

Invariant structures impose topological restrictions on centre leaves.

Symplectic systems with non-zero exponents are non-uniformly hyperbolic.

Abstract

Every volume-preserving centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (1) has two distinct centre Lyapunov exponents, or (2) exhibits an invariant continuous line field (or pair of line fields) tangent to the centre leaves, or (3) admits a continuous conformal structure on the centre leaves invariant under both the dynamics and the stable and unstable holonomies. The last two alternatives carry strong restrictions on the topology of the centre leaves: (2) can only occur on tori, and for (3) the centre leaves must be either tori or spheres. Moreover, under some additional conditions, such maps are rigid, in the sense that they are topologically conjugate to specific algebraic models. When the system is symplectic (1) implies that the centre Lyapunov exponents are non-zero, and thus the system is (non-uniformly) hyperbolic.