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I'm trying to read a paper of Boris Kalinin on the cohomology of dynamical systems for a project. The material is geared towards topologically transitive Anosov diffeomorphisms (which is how the initial (abelian) results were proved by Livsic). However, he axiomatizes things for homeomorphisms of metric spaces. It is necessary in the proof to use a version of the Anosov closing lemma, but it looks stronger than the one I've seen. I'd like to know whether elementary techniques will suffice to prove it.

Background: The statement of the closing lemma that I learned initially uis as follows. Let $M$ be a compact manifold, $f$ an Anosov diffeomorphism. Put a metric $d$ on $M$, and fix $\epsilon>0$. Then there is $\delta$ such that if $n \in \mathbb{N}$ and $d(f^n(x), x)<\delta$, then there is $p \in X$ with $f^n(p)=p$ and $d(p,x)<\epsilon$. In other words, "approximately periodic points" can be approximately closely by actual ones.

The property Kalinin stipulates in section 1 of his paper is that there is a type of exponential closeness. In other words, Kalinin wants that $c,\delta, \gamma>0$ exist such that any $x \in X$ with $d(f^n(x),x)<\delta$ can have the whole orbit be exponentially approximated by a periodic orbit (of $p$ with $f^n(p)=p$). More precisely, one has $d(f^i(p), f^i(x))< c d(f^i(x),x) e^{-\gamma \min(i,n-i)}$ for each $i=0, \dots, n-1$. This means that the orbits of $p$ and $x$ get even closer in the middle, and this is a strenghtening of the usual closing condition.

One can motivate this fact for Anosov diffeomorphisms geometrically by considering hyperbolic linear maps and drawing a picture of the stable and unstable subspaces, and I am told that it is a straightforward (and "effective") generalization of the usual statement of the closing lemma. This seems more like an intuitive aid rather than a rigorous proof for general Anosov diffeomorphisms, though.

However, Kalinin goes on to say more. He assumes that there exists $y \in X$ such that $d(f^i(x), f^i(y)) \leq \delta e^{-\gamma i}$, $d(f^i(y)), f^i(p)) \leq \delta e^{-\gamma(n-i)}$. Immediately thereafter, he states that this is true for Anosov diffeomorphisms in view of the closing lemma.

Questions:

1) Can one prove the (stronger) version of the closing lemma in Kalinin's paper using the usual statement itself standard techniques (i.e. successive approximation, basic linear algebra for hyperbolic maps, or lemmas like this one)? The books I have seen do not mention it, and certainly say nothing about a point $y$ as in the statement.

2) Does anyone know a good reference for this material (or for the general theory of Anosov diffeomorphisms, for that matter)?

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2 Answers 2

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The closing lemma as stated by Kalinin can be found in many textbooks e.g. Katok-Hasselblatt "Introduction to the modern theory of dynamical systems", corollary 6.4.17.

The closing lemma really gives a periodic point close to x, with iterates also close to the iterates of x until the orbit of x returns. That's not just the fact that periodic points are close to non-wandering points.

The point y is obtained by taking the intersection of the local stable set of x with the nth pull-back of the local unstable set of $f^n(p)$. Draw a picture to understand what's going on. There are "geometric" proofs of the closing lemma that build y before p. And of course the original article of Livsic contains such a proof.

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Thanks for your answer. It's ironic that I'm currently at PSU and the library doesn't have an available copy of Katok-Hasselblatt, so I hadn't checked that. Fortunately, I should be able to borrow one soon... –  Akhil Mathew Jul 7 '10 at 20:58

Besides the Hasselblatt and Katok bible, these are the references on Anosov diffeomorphisms that I found worth buying in the past six months: HK's Handbook of Dynamical Systems vol 1A, Gallavotti's books (available online and giving some great physical background and intuition, but probably not of any interest for you otherwise) and last but certainly not least Bowen's Equilibrium states and the ergodic theory of Anosov diffeomorphisms, which covers the closing lemma (3.8). There is a PDF of this online that should be easy to find.

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I should elaborate: Gallavotti's "short treatise" and "aspects" books are probably the most relevant here. –  Steve Huntsman Jul 7 '10 at 21:02
    
Thanks! The last one in particular looks helpful. –  Akhil Mathew Jul 7 '10 at 23:34

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