# What are the current research directions in the geometric theory of dynamical systems?

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.

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.

• Thank you! The book by Bonatti, Diaz and Viana looks great and I had not heard of it before. A question on MO etiquette - if I find an answer very satisfactory, but more replies would still be welcome, should I flair the answer as accepted anyway? Commented May 3, 2021 at 1:01

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.

• Would this be in general a multi-valued map? Or is it a.e. single valued with respect to some suitable measure? Commented May 3, 2021 at 1:02
• Schwarz restricts it to those points that do have a unique furthest point. He says, "One focus [in the literature] has been on Steinhaus's conjecture concerning the ubiquity of points $p$ such that $\mathcal{F}_p$ is a single point." Commented May 3, 2021 at 1:10