July 18, 2010

The Livsic theorem on the cohomological equation

Posted by Akhil Mathew under differential geometry, topology | Tags: , , , , |
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We continue the discussion of the cohomological equation started yesterday and explain a situation where it can be solved.

Given a compact space {X}, a continuous map {T: X \rightarrow X}, and a continuous {g: X \rightarrow \mathbb{R}} with vanishing periodic data as in yesterday’s post, there are not many ways to construct a solution {f} of the cohomological equation

\displaystyle g = f \circ T - f.

The basic thing to note that if {f(x)} is known, then recursively we can determine {f} on the entire orbit of {x} in terms of {g}. In case the map {T} is topologically transitive, say with a dense orbit generated by {x_0}, then by continuity the entire map {f} is determined by its value on {x_0}.

This also provides the method for obtaining {f} in the topologically transitive case. Namely, one picks {f(x_0)} aribtrarily, defines {f(T^ix_0)} in the only way possible by the cohomological equation. In this way one has {f} defined on the entire orbit {T^{\mathbb{Z}}(x_0)} such that on this orbit, the equation is satisfied. If one can show that {f} is uinformly continuous on {T^{\mathbb{Z}}(x_0)}, then it extends to the whole space and must by continuity satisfy the cohomological equation there too.

This is the strategy behind the proof of the theorem of Livsic from the seventies, whose proof we shall sketch:

Theorem 1 (Livsic) Let {M} be a compact Riemannian manifold, {T: M \rightarrow M} a topologically transitive Anosov diffeomorphism. If {g: M \rightarrow \mathbb{R}} is an {\alpha}-Holder function such that {T^n p =p} implies {\sum_{i=0}^{n-1} g(T^i p) = 0}, then there exists an {\alpha}-Holder {f: M \rightarrow \mathbb{R}} such that

\displaystyle g = f \circ T -f .

(more…)

 

July 11, 2010

A few more facts about Anosov diffeomorphisms

Posted by Akhil Mathew under differential geometry | Tags: , , , |
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Today, we will apply the technical lemma proved yesterday to proving a few special properties of Anosov diffeomorphisms. The first one states that if you have an approximate orbit, then you can approximate it by a real orbit. This may not sound like much, but it is false for isometries—and in fact, it gives another way of proving the structural stability result.

As usual, start with a compact manifold {M} and an Anosov diffeomorphism {f: M \rightarrow M}. We can put a metric {d} on {M} (e.g. by imbedding {M} in euclidean space, or using a Riemannian metric, etc.). To formalize this, fix {\delta>0}. We introduce the notion of a {\delta}-orbit. This is a two sided sequence {\{x_n\}_{n \in \mathbb{Z}}} such that {d(x_{n+1}, f(x_n)) \leq \delta}.

Theorem 1 (Anosov shadowing lemma) Fix {\epsilon>0} sufficiently small. There is {\delta>0} such that any {\delta}-orbit {\{x_n\}} can be shadowed by a unique real orbit {\{y_n\}}, i.e. {y_{n+1} = f(y_n)} and {d(x_n, y_n) < \epsilon} for all {n \in \mathbb{Z}}. (more…)

 

July 7, 2010

The technical approximation lemma for Anosov diffeomorphisms

Posted by Akhil Mathew under differential geometry | Tags: , , |
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We now state and prove the ugly technical theorem invoked yesterday, that you can refine certain “approximate” solutions of conjugacy-like equations involving Anosov diffeomorphisms (and maps close to them—though actually one can prove that Anosov diffeomorphisms are open in the C^1 topology). The proof is rather complicated, but it will basically rely on familiar techniques: hyperbolic linearization (in Banach spaces!), the contraction principle, and simple algebraic manipulation.

Theorem 1 Let {f} be an Anosov diffeomorphism of the compact manifold {M}. Then if {\delta>0} is sufficiently small, there is {\epsilon>0} satisfying the following condition. Suppose {d_{C^1}(f,g)<\epsilon}, and one has an “approximately commutative diagram” for a map \phi: X \to M:


with {X} a topological space and {h: X \rightarrow X} a homeomorphism: i.e. {d(g \circ \phi, \phi \circ h)< \epsilon}. Then there is a unique continuous {\psi: X \rightarrow U} close to {\phi} (namely {d(\psi, \phi)<\delta}) such that the modified diagram

commutes exactly.

So, how are we going to prove this? First, we want some sort of linearity, but we can’t add two elements of a manifold. Thus, we use the Whitney embedding theorem to assume without loss of generality that {M} is a closed submanifold of {\mathbb{R}^N}. (more…)

 

July 6, 2010

Hyperbolicity, Anosov diffeomorphisms, and structural stability

Posted by Akhil Mathew under differential geometry | Tags: , , , |
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I now know what I’m working on for my REU project; I’ll be studying (with two other undergraduates) a type of cohomology for dynamical systems. Misha Guysinsky, our mentor, has not explained the specific problem yet—perhaps that’ll come when we meet with him on Thursday. So I’ve spent the last weekend trying to learn a few basic facts about (especially hyperbolic) dynamical systems, which I will try to explain here.

1. Why do we care about hyperbolicity?

So, first a definition: let {f: M \rightarrow M} be a {C^1}-morphism of a smooth manifold {M}. Suppose {p \in M} is a fixed point. Then {p} is called hyperbolic if the derivative {Df_p: T_p(M) \rightarrow T_p(M)} has no eigenvalues on the unit circle. This comes from linear algebra: an endomorphism of a vector space is called hyperbolic if its eigenvalues are off the unit circle. Hyperbolicity is an important condition in dynamics, and I want to illustrate this with a few examples. (more…)