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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

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  • $\begingroup$ i would be interested in answers to this qstn too have you looked at this book by Einsiedler and Ward: springer.com/mathematics/dynamical+systems/book/… $\endgroup$
    – DBS
    Mar 15, 2015 at 13:32
  • $\begingroup$ I read Ergodic theory With Topological aspect,I looked at it but i didn't read it completely $\endgroup$
    – M.H
    Mar 15, 2015 at 13:36

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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).

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    $\begingroup$ The only downside of Einsiedler-Ward is the lack of entropy theory. (They are preparing a separate book on this topic, but it is not yet published.) If you are interested in this important topic, measure theoretic and topological entropy, the variational principle, thermodynamics formalism etc. take a look at the older book by Walters, Introduction to ergodic theory. $\endgroup$
    – MHS
    Mar 18, 2015 at 13:49
  • $\begingroup$ Hi - I got this book. It is very good and very challenging. Do you know if there is a solution manual for his questions? It would be good to verify many of the problems in the book. $\endgroup$
    – Novice
    Feb 25, 2018 at 13:15
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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.

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    $\begingroup$ There is also the more recent A First Course in Ergodic Theory by Dajani & Kalle, 2021. Like Dajani & Kraaikamp, it contains examples from number theory. $\endgroup$
    – J W
    Jan 15, 2022 at 18:46
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    $\begingroup$ Re Dajani & Kraaikamps's Ergodic Theory of Numbers, an MAA review can be found here. $\endgroup$
    – J W
    Jan 16, 2022 at 12:02
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    $\begingroup$ @JW Thanks for the update ^_^ $\endgroup$
    – AmirHosein Sadeghimanesh
    Jan 18, 2022 at 14:57
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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.

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