One of the main goals of global analysis is to understand the structure of meaningful subsets of the group of diffeomorphisms of a manifold M. In this lecture we will consider the set of partially hyperbolic diffeomorphisms on M, where M is a three dimensional compact orientable manifold. Up to algebraic and geometric construction there are some classic different examples:
• Hyperbolic linear automorphisms in T3,
• Circle extensions of Anosov surface maps,
• time-one maps of Anosov flows that are either suspensions of hyperbolic surface maps or mixing flows.
In this lecture we add some hypotheses to smooth partially hyperbolic maps to show that the above examples are all possible types of partially hyperbolic under these hypotheses (up to isotopy classes).

REFERENCES:
[1] J. Franks; Anosov diffeomorphisms. In Amer. Math. Soc., editor, Global Analysis. Proc. Sympos. Pure Math 14 (1968), pages 61–93.
[2] A. Hammerlindl and R. Potrie; Pointwise partial hyperbolicity in three dimensional nilmanifolds. Journal of the London Mathematical Society 89 (2014), no. 3, 853–875.
[3] P. Carrasco, E. PUJALS, and F. Hertz; Classification of partially hyperbolic diffeomorphisms under some rigid conditions. Ergodic Theory and Dynamical Systems (2020), 1–12.
[4] R. Sagin and J. Yang; Lyapunov exponents and rigidity of Anosov automorphisms and skew products. Advances in Mathematics 355 (2019).
[5] P. D. Carrasco, F. Rodriguez-Hertz, J. Rodriguez-Hertz, and R. Ures; Partially hyperbolic dynamics in dimension three. Ergodic Theory and Dynamical Systems 38 (2017) no. 3, 2801–2837.
[6] A. Gogolev; Surgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifolds. Geometry and Topology 22 (2018), no. 4, 2219–2252.

[IPM Youth Seminars on Topology and Dynamics]