[Math] Best books on ergodic theory

Question

I know nothing about ergodic theory , and one of my professors suggested me to read about it and go deep so I would be grateful if anyone could tell me what the best books on ergodic theory are for introduction and then go further

Answer 1

For me the best would always be Mañé's outstanding introduction Ergodic Theory and Differentiable Dynamics, although you should be careful of the idiosyncratic approach: he avoids the canon of the well established theory when there are much simpler ways. Perhaps it would be a bit too much calling it an introduction, it depends on what you know.

Walter's book An Introduction to Ergodic Theory would be the canon for most people, written to the perfection with everything really in the right place (but sometimes you need some fresh view, and thus why my choice of Mañé's book). It often goes to the extreme, basically emphasizing form instead of content at a few places, which really goes over the board. But it continues being one of the best references in the area (better say than the Katok-Hasselblat book Introduction to the Modern Theory of Dynamical Systems, really impossible to use by the noninitiated, and of course than the Cornfeld-Fomin-Sinai book Ergodic Theory with its somewhat archaic presentation).

In case you want just to get a quick overview of many topics you can have a look at the small book by Pollicott and Yuri Dynamical Systems and Ergodic Theory, although there are really hundreds of mistakes (minor but they are there). This should be a great entry to the other two books (by Mañé and by Walters).