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I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks). I am asking here for an advice of a book (or maybe a set of papers) to look at during our reading meetings. We would like to discuss some topic of current research (say not older than 5 years), but we do not want to be too specific, because is an activity extra to our own research. At the same time, we would like to build a useful basis for further research.

For me the the classical references are:

  • An Introduction to Ergodic Theory (Graduate Texts in Mathematics) by Peter Walters.
  • Ergodic Theory (Cambridge Studies in Advanced Mathematics) by Karl E. Petersen.
  • Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications) by Anatole Katok and Boris Hasselblatt.
  • Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics) by Robert Edward Bowen, Jean-René Chazottes and David Ruelle.
  • Ergodic Theory: With a View Towards Number Theory (Graduate Texts in Mathematics) by Thomas Ward, Manfred Einsiedler.

Many thanks.

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  • $\begingroup$ It is not clear what your 5-year limit means. 3 of the 5 books you listed are more than 20 years old. $\endgroup$ Feb 20, 2014 at 1:13
  • $\begingroup$ Exactly. Them are the classic books I know. I would appreciate also to know what are the new books. if you could suggest some, I will be happy. $\endgroup$
    – user39115
    Feb 20, 2014 at 9:59
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    $\begingroup$ See also mathoverflow.net/questions/82661/… $\endgroup$ Jul 29, 2014 at 14:35

2 Answers 2

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For a beautiful overview, focusing on entropy and the variational principle, you can't beat Walters' book. I've given reading courses from it, and it is very well written and excellent for self-study.

The Einsiedler-Ward book (the first of a projected three volumes, parts of the second and third books are viewable at http://maths.dur.ac.uk/~tpcc68/welcome.html) also covers the basics extremely well and at a leisurely pace, with more emphasis on topics crucial to deep applications of dynamics to number theory via homogeneous spaces, which is a very active research area.

Given your short time frame, I'd suggest using Walters as the basic text, but also browsing Einsiedler-Ward for topics of more current research interest. Should any students get interested enough to delve more deeply, then a more systemic reading of Einsiedler-Ward would provide them with a very solid foundation.

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  • $\begingroup$ The link seems to be broken, but maybe this is the right one now: tbward0.wixsite.com/books $\endgroup$
    – J W
    Oct 1, 2021 at 14:26
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Personally, I like Mañe's book Teoría Ergódica. I do think it's a classical book full of exercises. Also, it's orientated to start studying smooth ergodic theory. On the other hand the book has loads of mistakes (which makes it interesting to read, you realise that you are understanding everything when you spot the mistakes) and, as far as I know, both versions (English and Portuguese) are sold out. Also, Dynamics Beyond Uniform Hyperbolicity by Bonatti, Diaz and Viana can provide a good set of open problems to work in, but it is a bit hard to read.

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