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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

edited Feb 16 2010 at 15:01

asked Nov 3 2009 at 21:23

Steve Huntsman
8,139●1●22●56

	too localized: at least provide links, or some background. – Scott Morrison♦ Nov 3 2009 at 21:54
	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. – Steve Huntsman Nov 3 2009 at 22:00

Martin M. W. · Accepted Answer · 2009-11-03 22:05:29Z UTC

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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

Martin M. W.
1,866●9●11

	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. – Matthew Daws Nov 4 2009 at 10:14
	@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. – Steve Huntsman Feb 25 2010 at 20:39

The Arnold cat map

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