I have been unable to find a good reference for a book that study in details ergodic theory on sub shifts of finite type in dimension $D>1.$ The only reference that I got was actually a book by Gerhard Keller, Ergodic States in Ergodic Theory, London mathematical society, Student Text 42. However, it is a bit difficult to read, and sometimes becomes too general (too motivated by physics phenomenons), and not too much symbolic as I would like to see.
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$\begingroup$ Pick a standard ergodic theory book (like Walter's) and try to extend the results to higher dimensions on your own. Once having trouble, consult Keller's book (or survey/research papers). You will find that Keller's book is pretty descent and pedagogically conscious. $\endgroup$– AlgernonCommented Mar 11, 2014 at 8:31
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$\begingroup$ As for being motivated too much by physics phenomena, I am afraid you may have to get used to it. Physics is the main motivation for doing ergodic theory. (I know some people are going to get mad at me here, but I tell the truth nevertheless ;-) ) $\endgroup$– AlgernonCommented Mar 11, 2014 at 8:37
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$\begingroup$ $D>1$ is a serious business...Things are pretty much difficult that just doing this... $\endgroup$– user39115Commented Mar 11, 2014 at 16:49
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$\begingroup$ They are! But I would never know the difficulties if I don't try the one-dimensional arguments first. $\endgroup$– AlgernonCommented Mar 11, 2014 at 19:16
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2 Answers
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I don't think there are many books that deal with this. Is there a particular topic or direction you want to go in? Some areas of papers:
Algebraic SFTs have been treated by Kitchens, Schmidt and others;
Decidability and aperiodicity issues have been treated by Berger, Robinson, Kari, Culik and others;
Entropy issues by Hochman, Meyerovitch, Sahin, Pavlov, Robinson, Desai, Schraudner and others.
answered Mar 10, 2014 at 22:13
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$\begingroup$ Why do you think people in general are not interested? Just because it is too difficult? $\endgroup$ Commented Mar 11, 2014 at 16:45
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$\begingroup$ I think the issue is that too much goes wrong and it's hard to build a general theory of 2D SFTs. This started with the undecidability results, and continued with a whole range of things that fail in 2D, but which are satisfied in 1D. There's a danger that a book would turn into a catalogue of examples. To some extent, though, the situation has been rescued by the relatively recent Hochman & Meyerovitch paper $\endgroup$ Commented Mar 11, 2014 at 17:07
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I learnt quite a bit from lectures by Ronnie Pavlov and Emmanuel Jeandel :http://www.cmm.uchile.cl/~mschraudner/DySyCo/SchoolDySyCo.html
