Rigour and Formal Reference for Ergodic Theory

Question

I am not even a beginner to Ergodic Theory, but I want to start to read about it. I am coming from a math background and for me its quite important that the definitions to be stated and the formalism to be kept as concise as possible. Obviously then good intuitions on the subject is a plus. Can any body recommend me something? By browsing this the followings have been suggested:

1. Invitation to Ergodic Theory by César Ernesto Silva
2. Basic Ergodic Theory by Mahendra Ganpatrao Nadkarni
3. Ergodic Theory by Einsiedler, Manfred, Ward, Thomas
4. Ergodic Theory by Karl Petersen
5. The Ergodic Theory of Discrete Simple Paths by Paul C. Sheilds
6. Introduction to Ergodic Theory by Paul Helmos
7. Harry Furstenberg - Recurrence in ergodic theory and combinatorial number theory
8. Dynamical systems and ergodic theory – Mark Pollicott, Michiko Yuri.
9. Lecture Notes by Ben Green
10. "Randomness and Recurrence in Dynamical Systems" By Rodney Nillsen

As far as I could see there were no comments from the viewpoint that I am interested in a textbook; I really have hard time following physicist's approach to problem solving which usually looks very sketchy to me. Therefore I prefer a textbook far away from such approaches.

Answer 1

Manfred Einsiedler and Thomas Ward's book is also a nice choice if you are interested in the aspects of ergodic which are related to number theory. The book contains many detailed proofs of theorems with clear explanation, and gives many examples. This volume does not contain entropy theory, which will be discussed in the second volume, and you can find the draft version in the homepage of Thomas Ward.

Answer 2

Halmos's text was the one that ironed out my misconceptions. I would strongly recommend it. (I had a strong statistical mechanics background from which to launch, so this may not be exactly congruent to your starting place. Regardless, I believe Halmos will deliver on your request for a rigorous treatment with intuition development.)

Answer 3

1. Silva's text would be a good start -- it's what his REU students at Williams read in preparation to do research, even if they have no prior knowledge of measure theory. It's rigorous, and it's not that expensive, either.

2. I used Petersen's text for a directed reading in ergodic theory, and it was incredibly difficult. I do not recommend it for those without a solid background in abstract measure theory. It's definitely a graduate textbook.

Answer 4

Basic Ergodic Theory by M.G.Nadkarni is a very good text for beginners (it's cheap also :) ). The same is true for Invitation to Ergodic Theory by Silva...have a look at both of them and then decide. The book by Furstenberg is written by the master himself but I don't know whether it is suited for a first course or not.