I'm ending my undergraduate economics course and I'd like to extend my MA research program to dynamical economic systems. Knowing that my mathematical basis is calculus of 1 and 2 variables, linear algebra, probability and statistic, differential equations and real analysis what disciplines should I take to get in in the area?
[Math] dynamical systems applied to economics
dynamical systemseconomics
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Smale also wrote a little book Mathematics of Time which, as I recall, is also a pretty good survey.
Another favorite of mine is Foundations of Mechanics by Abraham and Marsden.
Thermodynamic formalism (TF) is a very exciting area which still to these days remains quite active. One of the most prominent applications of TF is fractal geometry. On of the most amazing results in my opinion, is Bowen's formula, which he derived to study the dimension of quasi-circles but it was then generalized to a more general setting. It essentially says that the dimension of certain fractals can be characterized as the unique solution to an equation involving a dynamically defined quantity (the pressure function, a central object of TF), which is also related to the spectrum of a certain class of dynamically defined operators. This allowed Ruelle to use some spectral theory to conclude that if you perturbate your fractals in a certain way, then their dimension varies very nicely (analytically in many cases). These notes are a good starting point to give you a more precise idea of what I am talking about:
http://www.mat.uc.cl/~giommi/notas3_escuela.pdf
If you want to go further into the details of these ideas, I highly recommend the book by Falconer, Fractal Geometry, where he goes into the details of the theory. Many of his works are also very important in establishing connections between TF and fractal geometry. His books are very nice to read also.
As you mentioned in one of the comments, the work of Pollicott is also very important in TF. He has a long history of trying to establish connections between TF and number theory, and lately he has been trying to find formulas for a circle packing problem:
https://arxiv.org/pdf/1704.06896.pdf
His works tend to be a bit harder to read though.
There are other uses of TF within the area of dynamical systems and ergodic theory, which concern the existence of invariant measures and equilibrium, which is a very big deal in general. The theory of Ruelle here is quite important, and you can read some nice notes by Walkden here:
https://personalpages.manchester.ac.uk/staff/charles.walkden/magic/lecture09.pdf
I think this should give you a good taste of different applications of TF.
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So for starters, I'd say to anyone who wants to do dynamical systems well (from a math perspective) you'll want to know measure theory (not sure if your real analysis was at that level or not) and topology. A little differential geometry/topology can't hurt either though some of that can be acquired on the fly (speaking from personal experience).
As for context on dynamical systems in economics (to see what it may entail), formal dynamical systems in economics is not yet really a thing, but it is a direction I think a number of people wish to take the field. Speaking as a micro theorist, much of the solution concepts from game theory are static, in the sense that while they are stable under best-response dynamics, generically you can't say much else (for example, are they attracting or repelling, etc).
A classic example is in general equilibrium theory; there isn't a real mechanism that satisfies normative conditions by which an out-of-equilibrium price vector converges to an equilibrium price vector.
A (very straightforward to a dynamical systems buff) example of some dynamics in game theory was proved originally by Milgrom and Roberts (Link). For a class of games with what amounts to complementary strategies for players, the basin of attraction of the set of equilibria under best-response dynamics is the entire strategy space. Think of this like classical Bertrand competition with homogeneous goods: if you cut prices by epsilon, I wish to just undercut you, and so on and so forth and in limit this iterated logic converges to the Nash equilibrium of zero.
Steven Smale wrote a wonderful paper discussing applications of dynamical systems theory to general equilibrium in AER (Link) which you might find interesting. He has a couple of other great papers too on game theory/economics, if you're interested I'd look them up.
Hope this was helpful and a good jumping-off point!