Dynamical systems for pure mathematicians?

I have taken courses on dynamical systems from an applied perspective, focusing on solving n-order differential equations, chaos, fixed points, etc, and applications to physics.

I however, am interested in having a clearer and deeper theoretical understanding of dynamical systems. Is there a good book that formalizes dynamical systems from a pure math perspective? (Ideally, with not too advanced prerequisites).

Extra: It would be extra nice if it also had a somewhat categorical perspective. In particular, is there such a thing as a "dynamical-system-structure preserving map"? Is this something that is used to analyse dynamical systems? E.g. I can imagine that statistical mechanics is a quotient object of low-level particle mechanics in some kind of category of dynamical systems.

If you've taken an "applied" dynamical systems course, what's missing from what you've seen is mostly the analytic and topological foundations of the subject, as well as introductions to some more advanced topics such as symbolic dynamics or ergodic theory. I would suggest the following textbook:

Barreira, Luis, and Claudia Valls. Dynamical systems: An introduction. Springer Science & Business Media, 2012.

Like most subjects in analysis, dynamical systems does not benefit very much from taking a categorical perspective (unlike, say, algebraic geometry or homotopy theory). The natural notion of isomorphism is topological conjugacy, and there are a few different categories of dynamical systems that one can define, but the heart of the subject lies in analysis and topology, not category theory.

If you're looking for a rigorous exposition from a pure perspective, Katok & Hasselblatt's Introduction to the Modern Theory of Dynamical Systems is probably the gold standard. It's a hefty tome (>800 pages) and would be a beast to read cover to cover, but it touches on pretty much every important subfield.

However, it is a somewhat advanced text. Point-Set Topology and Real Analysis are probably necessary prerequisites for the first quarter or so. Differential Geometry/Calculus on Manifolds, Functional Analysis, and Measure Theory are probably necessary if you were determined to read it cover to cover. That being said, it is quite possible to pick your own adventure through it! Haven't had measure theory? Skip everything on ergodicity and invariant measures.