How can I compute the SRB measure for the cat map? Also any pointers to references for obtaining Markov partitions and recurrence times would be lovely. Thanks
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asked Nov 3, 2009 at 21:23
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$\begingroup$ too localized: at least provide links, or some background. $\endgroup$– Kim MorrisonCommented Nov 3, 2009 at 21:54
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$\begingroup$ The cat map's description is available on Wikipedia, as an Anosov diffeomorphism of the torus, it is guaranteed to have an SRB measure. The Sinai-Ruelle-Bowen measure uniquely "describes the time averages of observables on motions with initial data randomly sampled with respect to the Lebesgue measure". The quote is from Tasaki, Gilbert, and Dorfman, "An analytical construction of the SRB measures for Baker-type maps", Chaos 8, 424 (1998), which is the closest thing to a reference that I could find. $\endgroup$– Steve HuntsmanCommented Nov 3, 2009 at 22:00
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Unless I'm misinterpreting the question, the SRB measure is just Lebesgue measure... the cat map is hyperbolic, preserves area, and is topologically transitive. See Theorem 3.10 and the following remarks here: http://books.google.com/books?id=uu-qeVBvQNEC&pg=PA141#v=onepage&q=&f=false
answered Nov 3, 2009 at 22:05
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$\begingroup$ Yes, I would agree: the CAT map lifts to a map on R^2 and as it's linear, given by a matrix with determinant one, it preserves Lebesgue measure. Hence the same is true on the Torus. $\endgroup$ Commented Nov 4, 2009 at 10:14
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$\begingroup$ @Martin: I had asked this question under an account that I'd forgotten the login for, and was only just able to accept it as I got it merged into my active account. Sorry for the delay in acceptance. $\endgroup$ Commented Feb 25, 2010 at 20:39