Dynamical systems for pure mathematicians?

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.