# Applications of thermodynamic formalism in dynamical systems theory

I'm a senior undergraduate mathematician and physicist. I'm currently engaging in undergraduate research course, and willing to learn what thermodynamic formalism is and how it is applied to mathematical problems.

Since I'm new to dynamical systems theory, I'm curious about what are important achievements in the field which have been resolved using thermodynamic formalism and in what active subdisciplines of dynamics people use the formalism. Applications to mathematical physics(e.g. mathematical formulation of nonequilibrium statistical mechanics) also count. Recommendation of relevant books and lectures will also be greatly appreciated.

• Ruelle's "thermodynamic formalism" used to be the standard, don't know what is preferred now. Apr 5, 2020 at 14:43
• Thank you for the comment. Actually I found Ruelle's book a bit difficult to read and currently working on Parry and Pollicott's 'zeta functions and the periodic orbit structure of hyperbolic dynamics', which was recommended by my advisor. But I'm still not sure what questions are central in hyperbolic dynamics and how thermodynamic formalism is related to them. Apr 5, 2020 at 14:50
• Perhaps springer.com/gp/book/9783764388812 could be what you have in mind, although I think that it doesn't go into physics. Apr 5, 2020 at 22:37

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.

• Accepted your answer. I appreciate it. I've been studying Fluctuation Theorems, which originated in physics but subsequently formulated in terms of Large Deviation Principle and thermodynamic formalism. If you could provide further comments on these topics, it will be greatly appreciated. Your answer is already a precious source of information anyway. Apr 27, 2020 at 10:15
• Applications of thermodynamic formalism to large deviations have been studied as well. For instance, the work of Artur O Lopes goes into that direction. In a paper by Kiefer, Weiss and Peres where they study the dimension gap for Bernoulli measures for the Gauss map, they also study large deviations of certain sets, using TF. Apr 28, 2020 at 0:52