Reference request : Ergodic theory and Number theory

Question

I would like to work on relations between Ergodic theory (or Dynamical systems) and Number theory. I am looking for a good reference book or lecture notes, and also I'd like to get familiar with articles and problems in this area. I checked similar questions that have been asked here but they didn't satisfy me. More precisely I want to know the research problems.

Thanks in advance Maisam

Answer 1

The book "Ergodic theory: with a view towards Number Theory" by Einsiedler & Ward is an excellent intro to some standard results in ergodic theory (e.g. the ergodic theorems, mixing) and provides many number-theoretic applications (e.g. Szemeredi's theorem, the Gauss map, flows on quotients of $\mathbb{H}^2$). It is a graduate-level book, and I recommend a fluency in measure theory before you attempt to tackle it (it does have an appendix covering this material, but I think more conventional texts such as Rudin or Stein & Shakarchi would be more appropriate).

Answer 2

Ergodic Theory of Numbers written by Karma Dajani and Cor Kraaikamp is an easy book to read and doesn't need any preliminaries. Reading that you will be guided through many other books which is mentioned in the part, further reading, specially about Entropy. I saw you had posted a question about history of Entropy so it shows that you have interests in Entropy too. Hope this book comes to use for you.

Answer 3

You mention ergodic theory / dynamical systems and number theory. Certainly, Einsiedler & Ward is worth looking into, as mentioned in @msteve's answer. Dajani & Kraaikamp (see @AmirHoseinSadeghimanesh's answer) and the more recent A First Course in Ergodic Theory by Dajani & Kalle, 2021, also contain examples from number theory.

You may also be interested in the field of arithmetic dynamics, which combines dynamical systems and number theory. See, in particular, the survey article Current Trends and Open Problems in Arithmetic Dynamics by Benedetto, DeMarco, Ingram, Jones, Manes, Silverman & Tucker, 2018, and also Silverman's 2007 book The Arithmetic of Dynamical Systems.