July 18, 2010
We continue the discussion of the cohomological equation started yesterday and explain a situation where it can be solved.
Given a compact space , a continuous map , and a continuous with vanishing periodic data as in yesterday’s post, there are not many ways to construct a solution of the cohomological equation
The basic thing to note that if is known, then recursively we can determine on the entire orbit of in terms of . In case the map is topologically transitive, say with a dense orbit generated by , then by continuity the entire map is determined by its value on .
This also provides the method for obtaining in the topologically transitive case. Namely, one picks aribtrarily, defines in the only way possible by the cohomological equation. In this way one has defined on the entire orbit such that on this orbit, the equation is satisfied. If one can show that is uinformly continuous on , 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 be a compact Riemannian manifold, a topologically transitive Anosov diffeomorphism. If is an -Holder function such that implies , then there exists an -Holder such that
July 17, 2010
We start by considering a very simple problem. Let be a set, be a bijection function, and a function. We want to know when the cohomological equation
can be solved for some .
It turns out that this very simple question has an equally simple answer. The answer is that the equation can be solved if and only if for every finite (i.e., periodic) orbit , we have . The necessity of this is evident, because if we have such a solution, then
because induces a bijection of with itself. This condition is called the vanishing of the periodic obstruction.
July 15, 2010
We continue the discussion of topological entropy started yesterday.
-1. Basic properties-
So, recall that we attached an invariant to a transformation of a compact metric space . We showed that it was purely topological, i.e. invariant under semiconjugacies. However, we have yet to establish its basic properties and compute some examples.
In general, we can’t expect topological entropy to be additive, i.e. , even if and commute. The reason is that the identity—or any isometry—has zero entropy, while there are homeomorphisms with nonzero entropy.
However, we do have:
Theorem 1 If , then .
(Here if , this makes sense even for noninvertible.)
We handle the two cases and (which together imply the result). In each, we will use the second definition of entropy that we gave in terms of coverings (which historically actually came first). Namely, the definition was
July 14, 2010
In the theory of dynamical systems, it is of interest to have invariants to tell us when two dynamical systems are qualitatively “different.” Today, I want to talk about one particularly important one: topological entropy.
We will be in the setting of discrete dynamical systems: here a discrete dynamical system is just a pair for a compact metric space and a continuous map.
Recall that two such pairs are called topologically conjugate if there is a homeomorphism such that . This is a natural enough definition, and it is clearly an equivalence relation. For instance, it follows that there is a one-to-one correspondence between the orbits of and those of . In particular, if has a fixed point, so does . Admittedly this necessary criterion for determining whether are topologically conjugate is rather trivial.
Note incidentally that topological conjugacy needs to be considered even when one is studying smooth dynamical systems—in many cases, one can construct a homeomorphism as above but not a diffeomorphism. This is the case in the Hartman-Grobman theorem, which states that if is a smooth map with a fixed point where the derivative is a hyperbolic endomorphism of the tangent space, then it is locally conjugate to the derivative (that is, the corresponding linear map). (more…)
July 7, 2010
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 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 be an Anosov diffeomorphism of the compact manifold . Then if is sufficiently small, there is satisfying the following condition. Suppose , and one has an “approximately commutative diagram” for a map :
with a topological space and a homeomorphism: i.e. . Then there is a unique continuous close to (namely ) such that the modified diagram
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 is a closed submanifold of . (more…)