I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read after Brin and Stuck’s very good general introduction? I like pretty much all the fields presented so far, especially topological dynamics and differentiable dynamics.

$\begingroup$ I have added some tags to your question. For example, (referencerequest) should not stand on its own  it should be accompanied by an areaspecific tag. Also your questions was missing a top level tag. Plese do edit the tags further if needed. (In particular, I was not sure whether or not to include the (topologicaldynamics) tag  but since you've mention it in your post I have added it.) $\endgroup$ – Martin Sleziak Feb 19 at 6:43

$\begingroup$ Thank you very much! $\endgroup$ – James Baxter Feb 19 at 6:48

$\begingroup$ It probably goes without saying, but it might be worth to check past questions about book recommendations in this area both on MathOverflow and on Mathematics Stack Exchange. $\endgroup$ – Martin Sleziak Feb 19 at 6:51

2$\begingroup$ maybe consider KatokHasselblatt, a pretty lengthy detailed book $\endgroup$ – rori Feb 19 at 6:59

$\begingroup$ @JamesBaxter it depends on your personality I guess. For most people, it is hard to read long texts without having a particular application in mind. A recommendation could be to pick some particular result in the field of dynamical systems, which you like but whose proof you do not completely understand, and then read the chapters about techniques/notions used in the proof. $\endgroup$ – rori Feb 19 at 11:29
There is an encyclopedic book of A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995 which covers most of the subject, and requires minimal background.
Among the more specialized, smaller books, I like Z. Nitecki, Differentiable dynamics.
But dynamics is a vast subject, with many branches, and there are very many good books. On the very popular subject of Holomorphic dynamics, the best introductory book on my opinion is J. Milnor, Dynamics in one complex variable (available of the arxiv).
There is also the classic Palis, de Melo: Geometric Theory of Dynamical Systems.
If you want to follow up on chapter 3 of Brin & Stuck, I recommend An Introduction to Symbolic Dynamics by Doug Lind and Brian Marcus (Cambridge, 1995).