TITLE: What are the current research directions in the geometric theory of dynamical systems?
QUESTION [6 upvotes]: By geometric theory of dynamical systems, I mean the kind found in the book by Palis, or papers like this one. In other words, dynamics on manifolds, but not specifically hyperbolic dynamical systems.
What are some recommended papers, survey articles, lecture notes, or books to read to explore this topic further? I really like this flavour of dynamics and would like to know what the modern research directions/questions are.
Thanks in advance!

REPLY [4 votes]: I think the book by Bonatti, Diaz and Viana: "Dynamics beyond uniform hyperbolicity' can give you a nice overview of one possible point of view. https://link.springer.com/book/10.1007/b138174
The book by Katok-Hasselblatt and their Handbook contains a lot of other points of view.

REPLY [3 votes]: Perhaps not the type of dynamical system in which you are interested, but
an interesting example:

Schwartz, Richard Evan. "The Farthest Point Map on the Regular Octahedron." Experimental Mathematics (2021): 1-12.
Preliminary version: arXiv abs.

The farthest point map associates to each point $p$ on the surface
the set of points $\mathcal{F}_p$ that are furthest from $p$, with distance
measured by shortest paths (geodesics segments).
Of special interest are the points $p$ for which $\mathcal{F}_p$
is a single point.
Even on the regular octahedron, the dynamics are quite intricate, but
calculable.