Analytic spaces

I’ve been trying to understand some complex analytic geometry as of late; here is an overview of Oka’s theorem.

Consider the space and the sheaf of holomorphic functions on it. One should think of this as the analog of complex affine space , with the Zariski topology, and with the sheaf of regular functions.

In algebraic geometry, if is an ideal, or if is a coherent sheaf of ideals, then we can define a closed subset of corresponding to the roots of the polynomials in . This construction gives the notion of an affine variety, and by gluing these one gets general varieties.

More precisely, here is what an affine variety is. If is a coherent sheaf of ideals, then we define a ringed space ; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in . In this way an affine variety is not just a subset of , but a locally ringed space.

Now we want to repeat this construction in the holomorphic category. If is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on , then we define the space cut out by to be . We can think of these as “affine analytic spaces.”

Definition 1 An analytic space is a locally ringed space which is locally isomorphic to an “affine analytic space.”

 

Applications of ergodic theory to equidistribution

We will now apply the machinery already developed to a few concrete problems.

Proposition 1 Let be a compact abelian group and the rotation by . Then is uniquely ergodic (with the Haar measure invariant) if is dense in .

 

The proof is straightforward. Suppose is invariant with respect to rotations by . Then for , we have

and hence

for any , which means that must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle is uniquely ergodic.

 

Application: Equidistribution

 

Theorem 3 Let be irrational and let be continuous and -periodic. Then

 

Unique ergodicity

So, let’s fix a compact metric space and a transformation which is continuous. We defined the space of probability Borel measures which are -invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which is ergodic). We are interested in knowing what looks like, based solely on the topological properties of . Here are some techniques we can use:

1) If has no fixed points, then cannot have any atoms (i.e. ). Otherwise would have infinite measure.

2) The set of recurrent points in (i.e. such that there exists a sequence with ) has -measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that is wandering if there is a neighborhood of such that . In other words, the family of sets is disjoint. If not, say that is non-wandering. Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example.

 

The space of invariant measures

Up until now, we have concentrated on a transformation of a fixed measure space. We now take a different approach: is fixed, and we look for appropriate measures (on a fixed -algebra).  First, we will show that this space is nonempty.  Then we will characterize ergodicity in terms of extreme points.

This is the first theorem we seek to prove:

Theorem 1 Let be a continuous transformation of the compact metric space . Then there exists a probability Borel measure on with respect to which is measure-preserving.

 

Consider the Banach space of continuous and the dual , which, by the Riesz representation theorem, is identified with the space of (complex) Borel measures on . The positive measures of total mass one form a compact convex subset of in the weak* topology by Alaoglu’s theorem. Now, induces a transformation of : . The adjoint transformation of is given by , where for a measure , . We want to show that has a fixed point on ; then we will have proved the theorem.

There are fancier methods in functional analysis one could use, but to finish the proof we will appeal to the simple

Lemma 2 Let be a compact convex subset of a locally convex space , and let be the restriction of a continuous linear map on . Then has a fixed point in .

 

Topological transitivity and Poincaré recurrence

So, now it’s time to connect the topological notions of dynamical systems with ergodic theory (which makes use of measures).  Our first example will use the notion of topological transitivity, which we now introduce.  The next example will return to the story about recurrent points, which I talked a bit about yesterday.

Say that a homeomorphism of a compact metric space is topologically transitive if there exists with dense in .  (For instance, a minimal homeomorphism is obviously topologically transitive.)  Let be a countable basis for the topology of . Then the set of all such (with dense) is given by

In particular, if it is nonempty, then each is dense—being -invariant and containing —and this set is a dense by Baire’s theorem.

Proposition 1 Let have a Borel probability measure positive on every nonempty open set, and let be measure-preserving and ergodic. Then the set of with is of measure 1, so is topologically transitive.

 

Indeed, each must have measure zero or one by ergodicity, so measure 1 by hypothesis. Then take the countable intersection.

Poincaré recurrence

We now move to the abstract measure-theoretic framework, not topological.

Theorem 2 (Poincaré recurrence) Let be a measure-preserving transformation on a probability space . If is measurable, then there exists with such that for each , there is a sequence with .

In other words, points of are -frequently in .

 

Ergodicity and mixing

Ergodicity

Let be a probability space and a measure-preserving transformation. In many cases, it turns out that the averages of a function given by

actually converge a.e. to a constant.

This is the case if is ergodic, which we define as follows: is ergodic if for all with , or . This is a form of irreducibility; the system has no smaller subsystem (disregarding measure zero sets). It is easy to see that this is equivalent to the statement: measurable (one could assume measurable and bounded if one prefers) and -invariant implies constant a.e. (One first shows that if is ergodic, then implies , by constructing something close to that is -invariant.)

In this case, therefore, the ergodic theorem takes the following form. Let be integrable. Then almost everywhere,

This is a very useful fact, and it has many applications.

 

The Birkhoff (pointwise) ergodic theorem

Let be a measure space with measure ; let be a measure-preserving transformation. Last time we looked at how the averages

behave in . But, now we want pointwise convergence.

The pointwise ergodic theorem

We consider the pointwise ergodic theorem of Garrett George Birkhoff: 

Theorem 1 (Birkhoff) Let . Then the averages converge almost everywhere to a function with a.e.

 

The von Neumann ergodic theorem

So, what’s ergodic theory all about?

The idea is that we are given a system together with some operation on it. For instance, could be a homeomorphism of a topological space, i.e. a discrete dynamical system. We are interested in studying the iterates of this process. In many case, averaging over the iterates of this process yields in the limit something that is actually invariant under this process.

For instance, suppose is a measure-preserving transformation of a measure space (which means if is measurable, then so is and ). How might one arise? Well, suppose is a compact symplectic manifold, a Hamiltonian vector field, and the volume form. Then the flows of leave invariant the volume form , so any such diffeomorphism is a measure-preserving transformation of the measure induced by the volume form. Anyway, back to the general story. Then the action of on a function is given by

The Birkhoff ergodic theorem states that the averages

converge a.e. to a function invariant under , provided . In many interesting cases, the invariant limit will actually be constant a.e. For instance, this is guaranteed if the transformation is ergodic, i.e. has no nontrivial invariant subsets. There are many spectacular applications to number theory of this result, e.g. the existence of Khintchine’s constant. Cf. also this post of Harrison Brown.

 

A theorem of Mazur-Ulam on isometries of vector spaces

Today’s will be a relatively short post, and primarily algebraic. I had mentioned a couple of days back the following result:

Theorem 1 (Mazur-Ulam) An isometry of a normed linear space onto another normed linear space with is linear.

We needed the theorem to show that a distance-preserving map between Riemannian manifolds is an isometry.   Recall that ‘s being an isometry means that it preserves distances between points.

Apparently the theorem has no relation to Barry Mazur, but to another Mazur.

I will follow Lax’s Functional Analysis in the proof.

 

Reduction

It is enough to prove that

 

for all .

Indeed, if this is the case, then by , we get by induction. So

 for all . By density of the dyadic fractions and continuity, we find for all . Also (1) and what’s already proved imply , so , which proves linearity.

 

Strategy of the proof

The idea is to use a purely norm-theoretic way of describing the midpoint of . This must be reflected by , so it will prove (1). In particular, given , we will define sets with , with the sets defined solely in terms of the norm structure on . It will thus follow that maps these sets onto their analogs on —which will prove (1) and the theorem.