Smooth Models of Fibered Partially Hyperbolic Systems
Jonathan DeWitt, Meg Doucette, Oliver Wang
TL;DR
The paper develops a smoothing theory for fibered partially hyperbolic diffeomorphisms with center tangential to a topological fibering, proving that under vanishing obstructions and a base-dynamics domination condition, one can deform to a smoothly fibered system on an adjusted smooth total space. It advances a global smoothing program by constructing smooth models over nilmanifold bases and employing Kirby–Siebenmann smoothing, spectral sequences, and controlled isotopies to preserve center dynamics while achieving a smooth center fibering. A principal contribution is the Main Theorem, which guarantees a smooth, partially hyperbolic lift g that fibers over an Anosov automorphism with bounded distortion, and that in high dimensions the total space becomes diffeomorphic after a finite cover. The work also identifies obstructions to smooth lifts of Anosov diffeomorphisms, providing a concrete counterexample using a high-dimensional sphere bundle over a torus, illustrating that leaf conjugacy can preserve coarse dynamics while losing smooth solvability on the base. Overall, the results connect dynamical rigidity questions with high-dimensional topology and smoothing theory, revealing when exotic smooth structures are necessary to realize smooth models of fibered partially hyperbolic systems.
Abstract
We study fibered partially hyperbolic diffeomorphisms. We show that as long as certain topological obstructions vanish and as long as homological minimum expansion dominates the distortion on the fibers that a fibered partially hyperbolic system can be homotoped to a fibered partially hyperbolic system with a $C^{\infty}$-center fibering. In addition, we study obstructions to the existence of smooth lifts of Anosov diffeomorphisms to bundles. In particular, we give an example of smooth topologically trivial bundle over a torus, where an Anosov diffeomorphism can lift continuously but not smoothly to the bundle.