Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3

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Sergio Fenley, Florida State University
Fine Hall 601

These diffeomorphisms exhibit weaker forms of hyperbolicity, which are still extremely common, while at the same time very robusts. We study these in dimension 3 and prove some rigidity or classification results. We assume that the diffeomorphism is homotopic to the identity, and show that certain invariant foliations associated with the diffeomorphism have a structure that is well determined. This has some important consequences when the manifold is either hyperbolic or Seifert: under certain conditions we prove the diffeomorphism is up to iterates and finite covers, leaf conjugate to the time one map of a topological Anosov flow. Some of the main techniques used are the use of (branching) two dimensional foliations in 3-manifolds. This is joint work with Thomas Barthelme, Steven Frankel, and Rafael Potrie.