Posts Tagged ‘dynamical systems’

A report of my Princeton generals exam

April 26, 2011

Well, some people might be wondering why I haven’t updated my blog since two weeks ago…Here’s the answer: I have been preparing for this generals exam — perhaps the last exam in my life.

For those who don’t know the game rules: The exam consists of 3 professors (proposed by the kid taking the exam, approved by the department), 5 topics (real, complex, algebra + 2 specialized topics chosen by the student). One of the committee member acts as the chair of the exam.

The exam consists of the three committee members sitting in the chair’s office, the student stands in front of the board. The professors ask questions ranging in those 5 topics for however long they want, the kid is in charge of explaining them on the board.

I was tortured for 4.5 hours (I guess it sets a new record?)
I have perhaps missed some questions in my recollection (it’s hard to remember all 4.5 hours of questions).

Conan Wu’s generals

Commitee: David Gabai (Chair), Larry Guth, John Mather

Topics: Metric Geometry, Dynamical Systems

Real analysis:

Mather: Construct a first category but full measure set.

(I gave the intersection of decreasing balls around the rationals)

Guth: F:S^1 \rightarrow \mathbb{R} 1-Lipschitz, what can one say about its Fourier coefficients.

(Decreasing faster than c*1/n via integration by parts)

Mather: Does integration by parts work for Lipschitz functions?

(Lip imply absolutely continuous then Lebesgue’s differentiation theorem)

Mather: If one only has bounded variation, what can we say?

(f(x) \geq f(0) + \int_0^x f'(t) dt)

Mather: If f:S^1 \rightarrow \mathbb{R} is smooth, what can you say about it’s Fourier coefficients?

(Prove it’s rapidly decreasing)

Mather: Given a smooth g: S^1 \rightarrow \mathbb{R}, given a \alpha \in S^1, when can you find a f: S^1 \rightarrow \mathbb{R} such that
g(\theta) = f(\theta+\alpha)-f(\theta) ?

(A necessary condition is the integral of g needs to vanish,
I had to expand everything in Fourier coefficients, show that if \hat{g}(n) is rapidly decreasing, compute the Diophantine set \alpha should be in to guarantee \hat{f}(n) being rapidly decreasing.

Gabai: Write down a smooth function from f:\mathbb{R}^2 \rightarrow \mathbb{R} with no critical points.

(I wrote f(x,y) = x+y) Draw its level curves (straight lines parallel to x=-y)

Gabai: Can you find a such function with the level curves form a different foliation from this one?

(I think he meant that two foliations are different if there is no homeo on \mathbb{R}^2 carrying one to the other,
After playing around with it for a while, I came up with an example where the level sets form a Reeb foliation, and that’s not same as the lines!)

We moved on to complex.

Complex analysis:

Guth: Given a holomorphic f:\mathbb{D} \rightarrow \mathbb{D}, if f has 50 0s inside the ball B_{1/3}(\bar{0}), what can you say about f(0)?

(with a bunch of hints/suggestions, I finally got f(0) \leq (1/2)^{50} — construct polynomial vanishing at those roots, quotient and maximal modulus)

Guth: State maximal modulus principal.

Gabai: Define the Mobius group and how does it act on \mathbb{H}.

Gabai: What do the Mobius group preserve?

(Poincare metric)

Mather: Write down the Poincare metric, what’s the distance from \bar{0} to 1? (infinity)

(I don’t remember the exact distance form, so I tried to guess the denominator being \sqrt{1-|z|}, but then integrating from 0 to 1 does not “barely diverge”. Turns out it should be (1-|z|^2)^2.)

Gabai: Suppose I have a finite subgroup with the group of Mobius transformations acting on \mathbb{D}, show it has a global fixed point.

(I sketched an argument based on each element having finite order must have a unique fixed point in the interior of \mathbb{D}, if two element has different fixed points, then one can construct a sequence of elements where the fixed point tends to the boundary, so the group can’t be finite.)

I think that’s pretty much all for the complex.

Algebra:

Gabai: State Eisenstein’s criteria

(I stated it with rings and prime ideals, which leaded to a small discussion about for which rings it work)

Gabai: State Sylow’s theorem

(It’s strange that after stating Sylow, he didn’t ask me do anything such as classify finite groups of order xx)

Gabai: What’s a Galois extension? State the fundamental theorem of Galois theory.

(Again, no computing Galois group…)

Gabai: Given a finite abelian group, if it has at most n elements of order divisible by n, prove it’s cyclic.

(classification of abelian groups, induction, each Sylow is cyclic)

Gabai: Prove multiplicative group of a finite field is cyclic.

(It’s embarrassing that I was actually stuck on this for a little bit before being reminded of using the previous question)

Gabai: What’s SL_2(\mathbb{Z})? What are all possible orders of elements of it?

(I said linear automorphisms on the torus. I thought it can only be 1,2,4,\infty, but turns out there is elements of order 6. Then I had to draw the torus as a hexagon and so on…)

Gabai: What’s \pi_3(S^2)?

(\mathbb{Z}, via Hopf fibration)

Gabai: For any closed orientable n-manifold M, why is H_{n-1}(M) torsion free?

(Poincare duality + universal coefficient)

We then moved on to special topics:

Metric Geometry:

Guth: What’s the systolic inequality?

(the term ‘aspherical’ comes up)

Gabai: What’s aspherical? What if the manifold is unbounded?

(I guessed it still works if the manifold is unbounded, Guth ‘seem to’ agree)

Guth: Sketch a proof of the systolic inequality for the n-torus.

(I sketched Gromov’s proof via filling radius)

Guth: Give an isoperimetric inequality for filling loops in the 3-manifold S^2 \times \mathbb{R} where S^2 has the round unit sphere metric.

(My guess is for any 2-chain we should have

\mbox{vol}_1(\partial c) \geq C \mbox{vol}_2(c)

then I tried to prove that using some kind of random cone and grid-pushing argument, but soon realized the method only prove

\mbox{vol}_1(\partial c) \geq C \sqrt{\mbox{vol}_2(c)}.)

Guth: Given two loops of length L_1, L_2, the distance between the closest points on two loops is \geq 1, what’s the maximum linking number?

(it can be as large as c L_1 L_2)

Dynamical Systems:

Mather: Define Anosov diffeomorphisms.

Mather: Prove the definition is independent of the metric.

(Then he asked what properties does Anosov have, I should have said stable/unstable manifolds, and ergodic if it’s more than C^{1+\varepsilon}…or anything I’m familiar with, for some reason the first word I pulled out was structurally stable…well then it leaded to and immediate question)

Mather: Prove structural stability of Anosov diffeomorphisms.

(This is quite long, so I proposed to prove Anosov that’s Lipschitz close to the linear one in \mathbb{R}^n is structurally stable. i.e. the Hartman-Grobman Theorem, using Moser’s method, some details still missing)

Mather: Define Anosov flow, what can you say about geodesic flow for negatively curved manifold?

(They are Anosov, I tried to draw a picture to showing the stable and unstable and finished with some help)

Mather: Define rotation number, what can you say if rotation numbers are irrational?

(They are semi-conjugate to a rotation with a map that perhaps collapse some intervals to points.)

Mather: When are they actually conjugate to the irrational rotation?

(I said when f is C^2, C^1 is not enough. Actually C^1 with derivative having bounded variation suffice)

I do not know why, but at this point he wanted me to talk about the fixed point problem of non-separating plane continua (which I once mentioned in his class).

After that they decided to set me free~ So I wandered in the hallway for a few minutes and the three of them came out to shake my hand.

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A survey on ergodicity of Anosov diffeomorphisms

March 7, 2011

This is in part a preparation for my 25-minutes talk in a workshop here at Princeton next week. (Never given a short talk before…I’m super nervous about this >.<) In this little survey post I wish to list some background and historical results which might appear in the talk.

Let me post the (tentative) abstract first:

——————————————————

Title: Volume preserving extensions and ergodicity of Anosov diffeomorphisms

Abstract: Given a C^1 self-diffeomorphism of a compact subset in \mathbb{R}^n, from Whitney’s extension theorem we know exactly when does it C^1 extend to \mathbb{R}^n. How about volume preserving extensions?

It is a classical result that any volume preserving Anosov di ffeomorphism of regularity C^{1+\varepsilon} is ergodic. The question is open for C^1. In 1975 Rufus Bowen constructed an (non-volume-preserving) Anosov map on the 2-torus with an invariant positive measured Cantor set. Various attempts have been made to make the construction volume preserving.

By studying the above extension problem we conclude, in particular the Bowen-type mapping on positive measured Cantor sets can never be volume preservingly extended to the torus. This is joint work with Charles Pugh and Amie Wilkinson.

——————————————————

A diffeomorphism f: M \rightarrow M is said to be Anosov if there is a splitting of the tangent space TM = E^u \oplus E^s that’s invariant under Df, vectors in E^u are uniformly expanding and vectors in E^s are uniformly contracting.

In his thesis, Anosov gave an argument that proves:

Theorem: (Anosov ’67) Any volume preserving Anosov diffeomorphism on compact manifolds with regularity C^2 or higher on is ergodic.

This result is later generalized to Anosov diffeo with regularity C^{1+\varepsilon}. i.e. C^1 with an \varepsilon-holder condition on the derivative.

It is a curious open question whether this is true for maps that’s strictly C^1.

The methods for proving ergodicity for maps with higher regularity, which relies on the stable and unstable foliation being absolutely continuous, certainly does not carry through to the C^1 case:

In 1975, Rufus Bowen gave the first example of an Anosov map that’s only C^1, with non-absolutely continuous stable and unstable foliations. In fact his example is a modification of the classical Smale’s horseshoe on the two-torus, non-volume-preserving but has an invariant Cantor set of positive Lebesgue measure.

A simple observation is that the Bowen map is in fact volume preserving on the Cantor set. Ever since then, it’s been of interest to extend Bowen’s example to the complement of the Cantor set in order to obtain an volume preserving Anosov diffeo that’s not ergodic.

In 1980, Robinson and Young extended the Bowen example to a C^1 Anosov diffeomorphism that preserves a measure that’s absolutely continuous with respect to the Lebesgue measure.

In a recent paper, Artur Avila showed:

Theorem: (Avila ’10) C^\infty volume preserving diffeomorphisms are C^1 dense in C^1 volume preserving diffeomorphisms.

Together with other fact about Anosov diffeomorphisms, this implies the generic C^1 volume preserving diffeomorphism is ergodic. Making the question of whether such example exists even more curious.

In light of this problem, we study the much more elementary question:

Question: Given a compact set K \subseteq \mathbb{R}^2 and a self-map f: K \rightarrow K, when can the map f be extended to an area-preserving C^1 diffeomorphism F: \mathbb{R}^2 \rightarrow \mathbb{R}^2?

Of course, a necessary condition for such extension to exist is that f extends to a C^1 diffeomorphism F (perhaps not volume preserving) and that DF has determent 1 on K. Whitney’s extension theorem gives a necessary and sufficient criteria for this.

Hence the unknown part of our question is just:

Question: Given K \subseteq \mathbb{R}^2, F \in \mbox{Diff}^1(\mathbb{R}^2) s.t. \det(DF_p) = 1 for all p \in K. When is there a G \in \mbox{Diff}^1_\omega(\mathbb{R}^2) with G|_K = F|_K?

There are trivial restrictions on K i.e. if K separates \mathbb{R}^2 and F switches complementary components with different volume, then F|_K can never have volume preserving extension.

A positive result along the line would be the following slight modification of Moser’s theorem:

Theorem: Any C^{r+1} diffeomorphism on S^1 can be extended to a C^r area-preserving diffeomorphism on the unit disc D.

For more details see this pervious post.

Applying methods of generating functions and Whitney’s extension theorem, as in this paper, in fact we can get rid of the loss of one derivative. i.e.

Theorem: (Bonatti, Crovisier, Wilkinson ’08) Any C^1 diffeo on the circle can be extended to a volume-preserving C^1 diffeo on the disc.

With the above theorem, shall we expect the condition of switching complementary components of same volume to be also sufficient?

No. As seen in the pervious post, restricting to the case that F only permute complementary components with the same volume is not enough. In the example, K does not separate the plane, f: K \rightarrow K can be C^1 extended, the extension preserves volume on K, and yet it’s impossible to find an extension preserving the volume on the complement of K.

The problem here is that there are ‘almost enclosed regions’ with different volume that are being switched. One might hope this is true at least for Cantor sets (such as in the Bowen case), however this is still not the case.

Theorem: For any positively measured product Cantor set C = C_1 \times C_2, the Horseshoe map h: C \rightarrow C does not extend to a Holder continuous map preserving area on the torus.

Hence in particular we get that no volume preserving extension of the Bowen map can be possible. (not even Holder continuous)

Recurrence and genericity – a translation from French

January 31, 2011

To commemorate passing the French exam earlier this week (without knowing any French) and also to test this program ‘latex to wordpress‘, I decided to post my French-translation assignment here.

Last year, I went to Paris and heard a French talk by Crovisier. Strangely enough, although I can’t understand a single word he says, just by looking at the slides and pictures, I liked the talk. That’s why when being asked the question ‘so are there any French papers you wanted to look at?’, I immediately came up with this one which the talk was based on.

Here is a translation of selected parts (selected according to my interest) in section 1.2 taken from the paper `Récurrence et Généricité‘ ( Inventiones Mathematicae 158 (2004), 33-104 ) by C. Bonatti and S. Crovisier. In which they proved a connecting lemma for pseudo-orbits.

Interestingly, just in this short section they referred to two results I have discussed in earlier posts of this blog: Conley’s fundamental theorem of dynamical systems and the closing lemma. In any case, I think it’s a cool piece of work to look at! Enjoy~ (Unfortunately, if one wants to see the rest of the paper, one has to read French >.<)

Precise statements of results

1. Statement of the connecting lemma for pseudo-orbits

In all the following work we consider compact manifold {M} equipped with an arbitrary Riemannian metric and sometimes also with a volume form {\omega} (unrelated to the metric). We write {\mbox{Diff}^1(M)} for the set of diffeomorphisms of class {C^1} on {M} with the {C^1} topology and {\mbox{Diff}^1_\omega(M) \subset \mbox{Diff}^1(M)} the subset preserving volume form {\omega}.

Recall that, in any complete metric space, a set is said to be residual if it contains a countable intersection of open and dense sets. A property is said to be generic if it is satisfied on a residual set. By slight abuse of language, we use the term generic diffeomorphisms: the phase ‘generic diffeomorphisms satisfy property P‘ means that property P is generic.

Let f \in \mbox{Diff}^1(M) be a diffeomorphism of M. For all \varepsilon>0, an \varepsilon-pseudo-orbit of f is a sequence (finite or infinite) of points(x_i) such that for all i, d(x_{i+1},f(x_i)) < \varepsilon. We define the following binary relations for pairs of points (x,y) on M:

– For all \varepsilon > 0, we write x \dashv_\varepsilon y if there exists an \varepsilon-pseudo-orbit (x_0, x_1, \cdots, x_k) where x_0 = x and x_k = y for some k \geq 1.

– We write {x \dashv y} if {x \dashv_\varepsilon y} for all {\varepsilon>0}. We sometimes write {x \dashv_f y} to specify the dynamical system in consideration.

– We write {x \prec y} (or {x \prec_f y}) if for all neighborhoods {U, V} of {x} and {y}, respectively, there exists {n \geq 1} such that {f^n(U)} intersects {V}.

Here are a few elementary properties of these relations.

1. The relations {\dashv} and {\dashv_\varepsilon} are, by construction, transitive. The chain recurrent set {\mathcal{R}(f)} is the set of points {x} in {M} such that {x \dashv x}.

2. The relation {x \prec y} is not a-priori transitive. The non-wandering set {\Omega(f)} is the set of points {x} in {M} such that {x \prec x}.

Marie-Claude Arnaud has shown in [Ar] that the relation {\prec} is transitive for generic diffeomorphisms. By using similar methods we show:

Theorem 1: There exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} (or in {\mbox{Diff}^1_\omega(M)}) such that for all diffeomorphisms {f} in {\mathcal{G}} and all pair of points {(x, y)} in {M} we have:

\displaystyle x \dashv_f y \Longleftrightarrow x \prec_f y.

This theorem is a consequence of the following general perturbation result:

Theorem 2: Let {f} be a diffeomorphism on compact manifold {M}, satisfying one of the following two hypotheses:

1. all periodic orbits of {f} are hyperbolic,

2. {M} is a compact surface and all periodic orbits are either hyperbolic or elliptic with irrational rotation number (its derivative has complex eigenvalues, all of modulus {1}, but are not powers of roots of unity).

Let {\mathcal{U}} be a {C^1}-neighborhood of {f} in {\mbox{Diff}^1(M)} (or in {\mbox{Diff}^1_\omega(M)}, if {f} preserves volume form {\omega}). Then for all pairs of points {(x,y)} in {M} such that {x \dashv y}, there exists a diffeomorphism {g} in {\mathcal{U}} and an integer {n>0} such that {g^n(x) = y}.

Remark: In Theorem 2 above, if the diffeomorphism {f} if of class {C^r} with {r \in (\mathbb{N} \backslash \{0\})\cup \{ \infty \}}, then the {C^1}-perturbation {g} can also be chosen in class {C^r}. Indeed the diffeomorphism {g} is obtained thanks to a finite number of {C^1}-perturbations given by the connecting lemma (Theorem 2.1), each of these perturbations is itself of class {C^r}.

Here are a few consequences of these results:

Corollary: There exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all diffeomorphism {f} in {\mathcal{G}}, the chain recurrent set {\mathcal{R}(f)} coincides with the non-wandering set {\Omega(f)}.

Corollary: Suppose {M} is connected, then there exists a residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that if {f \in \mathcal{G}} satisfies {\Omega(f) = M} then it is transitive. Furthermore, {M} is the unique homoclinic class for {f}.

For volume preserving diffeomorphism {f}, the set {\Omega(f)} always coincide with the whole manifold {M}. We therefore find the analogue of this corollary in the conservative case (see section 1.2.4).

2. Dynamical decomposition of generic diffeomorphisms into elementary pieces

Consider the symmetrized relation {\vdash\dashv } of {\dashv} defined by {x \vdash\dashv y} if {x \dashv y} and {y\dashv x}. This relation then induces an equivalence relation on {\mathcal{R}(f)}, where the equivalence classes are called chain recurrence classes.

We say a compact {f}-invariant set {\Lambda} is weakly transitive if for all {x, y \in \Lambda}, we have {x \prec y}. A set {\Lambda} is maximally weakly transitive if it is maximal under the partial order {\subseteq} among the collection of weakly transitive sets.

Since the closure of increasing union of weakly transitive sets is weakly transitive, Zorn’s lemma implies any weakly transitive set is contained in a maximally weakly transitive set. In the case where the relation {\prec_f} is transitive (which is a generic property), the maximally weakly transitive sets are the equivalence classes of the symmetrized relation induced by {\prec} on the set {\Omega(f)}. Hence we obtain, for generic diffeomorphisms:

Corollary: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}} the chain recurrence classesare exactly the maximally weakly transitive sets of {f}.

The result of Conley (see posts on fundamental theorem of dynamical systems) on the decomposition of {\mathcal{R}(f)} into chain recurrence classes will therefore apply (for generic diffeomorphisms) to the decomposition of {\Omega(f)} into maximally weakly transitive sets.

{\cdots}

3. Chain recurrence classes and periodic orbits

Recall that after the establishment of closing lemma by C. Pugh (see the closing lemma post), it is known that periodic points are dense in {\Omega(f)} for generic diffeomorphisms, we would like to use these periodic orbits to better understand the dynamics of chain recurrence classes.

Recall the homoclinic class {H(p, f)} of a hyperbolic periodic point {p} is the closure of all transversal crossing points of its stable and unstable manifolds. This set is by construction transitive, as we have seen in section 1.2, the results of [CMP] imply that, for generic diffeomorphisms any homoclinic class is maximally weakly transitive. By applying corollary 1.4, we see that:

Remark: For generic diffeomorphisms homoclinic classes are also chain recurrence classes.

However, for generic diffeomorphisms, there are chain recurrence classes which are not homoclinic classes, therefore contains no periodic orbit, we call such chain recurrence class with no periodic points aperiodic class.

Corollary: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}}, any connected component with empty interior of {\Omega(f) = \mathcal{R}(f)} is periodic and its orbit is a homoclinic class.

The closing lemma of Pugh and Remark 5 show:

Remark: For generic {f}, any isolated chain recurrence class in {R(f)} is a homoclinic class. In particular this applies to classes that are topological attractors or repellers.
{\cdots}

For non-isolated classes, a recent work (see [Cr]) specifies how a chain recurrence class is approximated by periodic orbits:

Theorem: There exists residual set {\mathcal{G}} in {\mbox{Diff}^1(M)} such that for all {f \in \mathcal{G}}, all maximally weakly transitive sets of {f} are Hausdorff limits of sequences of periodic orbits.

More general chain recurrence classes satisfy the upper semi-continuity property: if {(x_i) \subseteq \mathcal{R}(f)} is a sequence of points converging to a point {x} then for large enough {n}, the class of {x_n} is contained in an arbitrary small neighborhood of the class of {x}.

Coding Fractals by trees

November 15, 2010

Recently I’ve been editing a set of notes taken by professor Kra during Furstenberg’s course last year. (Well…I should have done this last year >.<) One of the main ideas in the course was to build a correspondence between trees and Fractal sets – and hence enables one to prove statements about dimensions and projections of Fractals by looking at the corresponding trees. I want to sketch some highlights of the basic idea here.

Let Q=Q^{(n)} denote the unit cube in \mathbb{R}^{n}.

Let A\subset\mathbb{R}^{n} and x\in A. For t\in\mathbb{R}, consider the family
t(A-x)\cap Q.

Question:Does the limit exist as t\to\infty? (here we consider the Hausdorff limit on compact subsets of Q)

i.e. we ‘zoom in’ the set around the point x, always crop the set to the cube Q and consider what does the set ‘look like’ when the strength of the magnifying glass approaches infinity.

Example:
If A is smooth (curve of surface), then this limit exists and is a subspace intersected with Q.

Generally one should not expect the above limit to exist for fractal sets, however if we weaken the question and ask for the limit when we take a sequence (n_k)\rightarrow\infty, then it’s not hard to see self-similar sets would have non-trivial limits. Hence in some sense fractal behavior is characterized by having non-trivial limit sets.

Definition: Let A\subset Q^{(n)},
A' is a mini-set of A if for some \lambda\geq 1 and u\in\mathbb{R}^{n}, A' =(\lambda A+u)\cap Q

A'' is a micro-set of A if there is sequence of minisets A'_{n} of A and A'' = \lim_{n\to\infty}A'_{n}

A is homogeneous if all of its microsets are minisets. As we should see below, homogeneous fractals are ‘well-behaved’ in terms of dimensions.

Here comes a version of our familiar definition:

Definition:A set A\subset X has Hausdorff \alpha-measure 0 if for every \varepsilon > 0, there exists a covering of A\subset \bigcup_{n=1}^{\infty}B_{\rho_{n}} with \sum_{n}\rho_{n}^{\alpha}  \alpha.

Thus it makes sense to define:

Definition: The Hausdorff dimension of A is

\dim(A) = \inf\{\alpha>0\colon \text{Hausdorff } \alpha \text{ measure is } 0 \}.

Now comes trees~ For A\subset[0,1] closed, consider expansion in base 3 of numbers in A. In the expansion of each number in A, there will be certain digits which appear. Following this digit, there may be certain digits that can appear. This defines a tree, which is a tree of shrinking triadic intervals that intersects the set A.

Definition: Let \Lambda be a finite set of symbols (which we will refer to as the alphabet and elements of \Lambda are letters of the alphabet).

A word w is any concatenation of finitely many symbols in \Lambda, the number of symbols in w is called the length of w, denoted by \ell(w).

A tree \tau in the alphabet \Lambda is a set of words satisfying
1. If uv\in\tau then u\in\tau.
2. If u\in\tau, then there exists a letter a\in\Lambda such that ua\in\tau.

Notation: if w\in\tau, denote \tau^{w} = \{v\colon wv\in\tau\}.

Definition: A section \Sigma of \tau is a finite set of words for which there exists N such that if s\in\tau and \ell(s) \geq N, then there exists r\in\Sigma and a word w such that s = rw.

Definition: The Hausdorff dimension of a tree is defined by \dim(\tau)=\inf\{ \beta \ | \ \inf_{\Sigma}\{\sum_{r\in\Sigma} e^{-\beta\ell(r)}\} = 0 \} where \Sigma is any section of \tau.

Theorem: The Hausdorff dimension of a tree equals the Hausdorff dimension of set it corresponds to.

The proof is merely going through the definition, hence we won’t present here. It suffice to check the dimension is unchanged if we only consider open ball covers with balls of radius $p^{-n}$ for any given $p \in \N$. Which is indeed true.

Now let’s give a simple example to illustrate how to make use of this correspondence:

It is easy to prove \dim(X \times Y) \geq \dim(X) \times \dim(Y), here we produce an example were the inequality is strict.

Let A \subseteq [0,1] be the set of numbers whose decimal expansion vanishes from the k_{2n} to k_{2n+1}-th digits to the right of 0.

To compute \dim(A), look at levels such that a small number of intervals will cover the set A. The number of intervals of size 10^{k_{2n+1}} is less than 10^{k_{2n}}. Then if \frac{10^{k_{2n}}}{k_{2n+1}} goes to 0, we’ll have that the Hausdorff dimension is 0. So we just
need k_{2n}/k_{2n+1}\to 0.

Let B be the set of numbers whose decimals vanish from k_{2n-1} to k_{2n}-1-th digits. By the same computation, \dim(B) = 0.

What about A\times B? The tree that corresponds to this is a regular tree which every point has 10 branches. So the Hausdorff dimension is 1.

Note: Here the structure of the tree gives you easy information on the set, but it is hard to look directly at the set and see this information.

Many theorems regarding dimension of projections and intersections of fractal sets turns out to be simple to prove after we coded the sets by trees.

Remarks from the Oxtoby Centennial Conference

November 2, 2010

A few weeks ago, I received this mysterious e-mail invitation to the ‘Oxtoby Centennial Conference’ in Philadelphia. I had no idea about how did they find me since I don’t seem to know any of the organizers, as someone who loves conference-going, of course I went. (Later I figured out it was due to Mike Hockman, thanks Mike~ ^^ ) The conference was fun! Here I want to sketch a few cool items I picked up in the past two days:

Definition:A Borel measure \mu on [0,1]^n is said to be an Oxtoby-Ulam measure (OU for shorthand) if it satisfies the following conditions:
i) \mu([0,1]^n) = 1
ii) \mu is positive on open sets
iii) \mu is non-atomic
iv) \mu(\partial [0,1]^n) = 0

Oxtoby-Ulam theorem:
Any Oxtoby-Ulam measure is the pull-back of the Lebesgue measure by some homeomorphism \phi: [0,1]^n \rightarrow [0,1]^n.

i.e. For any Borel set A \subseteq [0,1]^n, we have \mu(A) = \lambda(\phi(A)).

It’s surprising that I didn’t know this theorem before, one should note that the three conditions are clearly necessary: A homeo has to send open sets to open sets, points to points and boundary to boundary; we know that Lebesgue measure is positive on open sets, 0 at points and 0 on the boundary of the square, hence any pull-back of it must also has those properties.

Since I came across this at such a late time, my first reaction was: this is like Moser’s theorem in the continuous case! But much cooler! Because measures are a lot worse than differential forms: many weird stuff could happen in the continuous setting but not in the smooth setting.

For example, we can choose a countable dense set of smooth Jordan curves in the cube and assign each curve a positive measure (we are free to choose those values as long as they sum to 1. Now we can define a measure supported on the union of curves and satisfies the three conditions. (the measure restricted to each curve is just a multiple of the length) Apply the theorem, we get a homeomorphism that sends each Jordan curve to a Jordan curve with positive n dimensional measure and the n dimensional measure of each curve is equal to our assigned value! (Back in undergrad days, it took me a whole weekend to come up with one positive measured Jordan curve, and this way you get a dense set of them, occupying a full measure set in the cube, for free! Oh, well…>.<)

Question: (posed by Albert Fathi, 1970)
Does the homeomorphism \phi sending \mu to Lebesgue measure depend continuously on \mu?

My first thought was to use smooth volume forms to approximate the measure, for smooth volume forms, Moser’s theorem gives diffeomosphisms depending continuously w.r.t. the form (I think this is true just due to the nature of the construction of the Moser diffeos) the question is how large is the closure of smooth forms in the space of OU-measures. So I raised a little discussion immediately after the talk, as pointed out by Tim Austin, under the weak topology on measures, this should be the whole space, with some extra points where the diffeos converge to something that’s not a homeo. Hence perhaps one can get the homeo depending weakly continuously on \mu.

Lifted surface flows:

Nelson Markley gave a talk about studying flows on surfaces by lifting them to the universal cover. i.e. Let \phi_t be a flow on some orientable surface S, put the standard constant curveture metric on S and lift the flow to \bar{\phi}_t on the universal cover of S.

There is an early result:

Theorem: (Weil) Let \phi_t be a flow on \mathbb{T}^2, \bar{\phi}_t acts on the universal cover \mathbb{R}^2, then for any p \in \mathbb{R}^2, if \displaystyle \lim_{t\rightarrow \infty} ||\bar{\phi}_t(p)|| = \infty then \lim_{t\rightarrow \infty} \frac{\bar{\phi}_t(p)}{||\bar{\phi}_t(p)||} exists.

i.e. for lifted flows, if an orbit escapes to infinity, then it must escape along some direction. (No sprial-ish or wild oscillating behavior) This is due to the nature that the flow is the same on each unit square.

We can find its analogue for surfaces with genus larger than one:

Theorem: Let \phi_t be a flow on S with g \geq 2, \bar{\phi}_t: \mathbb{D} \rightarrow \mathbb{D}, then for any p \in \mathbb{D}, if \displaystyle \lim_{t\rightarrow \infty} ||\bar{\phi}_t(p)|| = \infty then \lim_{t\rightarrow \infty} \bar{\phi}_t(p) is a point on the boundary of \mathbb{D}.

Using those facts, they were able to prove results about the structure of \omega limiting set of such orbits (those that escapes to infinity in the universal cover) using the geometric structure of the cover.

I was curious about what kind of orbits (or just non-self intersecting curves) would ‘escape’, so here’s some very simple observations: On the torus, this essentially means that the curve does not wind around back and forth infinitely often with compatible magnitudes, along both generators. i.e. the curve ‘eventually’ winds mainly in one direction along each generating circle. Very loosely speaking, if a somewhat similar thing is true for higher genus surfaces, i.e. the curve eventually winds around generators in one direction (and non-self intersecting), then it would not be able to have very complicated \omega limiting set.

Measures on Cantor sets

In contrast to the Oxtoby-Ulam theorem, one could ask: Given two measures on the standard middle-third Cantor set, can we always find a self homeomorphism of the Cantor set, pushing one measure to the other?

Given there are so many homeomorphisms on the Cantor set, this sounds easy. But in fact it’s false! –There are countably many clopen subsets of the Cantor set (Note that all clopen subsets are FINITE union of triadic copies of Cantor sets, a countable union would necessarily have a limit point that’s not in the union), a homeo needs to send clopen sets to clopen sets, hence for there to exist a homeo the countably many values the measures take on clopen sets must agree.

So a class of ‘good measures’ on Cantor sets was defined in the talk and proved to be realizable by a pull back the standard Hausdorff measure via a homeo.

I was randomly thinking about this: Given a non-atomic measure \mu on the Cantor set, when can it be realized as the restriction of the Lebesgue measure to an embedding of the Cantor set? After a little bit of thinking, this can always be done. (One can simple start with an interval, break it into two pieces according to the measure \mu of the clopen sets before and after the largest gap, then slightly translate the two pieces so that there is a gap between them; iterate the process)

In any case, it’s been a fun weekend! ^^

On C^1 closing lemma

May 25, 2010

Let f: M \rightarrow M be a diffeomorphism. A point p is non-wandering if for all neighborhood U of p, there is increasing sequence (n_k) \subseteq \mathbb{N} where U \cap f^{n_k}(U) \neq \phi. We write p \in \mathcal{NW}(f).

Closing lemma: For any diffeomorphism f: M \rightarrow M, for any p \in \mathcal{NW}(f). For all \varepsilon>0 there exists diffeomorphism g s.t. ||f-g||_{C^1} < \varepsilon and g^N(p) = p for some N \in \mathbb{N}.

Suppose p \in \mathcal{NW}(f), \overline{\mathcal{O}(p)} is compact, then for any \varepsilon>0, there exists x_0 \in B(p, \varepsilon), k \in \mathbb{N} s.t. f^k(x) \in B(p, \varepsilon).

First we apply a selection process to pick an appropriate almost-orbit for the closing. Set x_i = f^i(x_0), \ 0 \leq i \leq k.

If there exists 0 < j < k where

\min \{ d(x_0, x_j), d(x_j, x_k) \} < \sqrt{\frac{2}{3}}d(x_0, x_k)

then we replace the origional finite sequence by (x_0, x_1, \cdots, x_j) or (x_j, \cdots, x_k). Iterate the above process. since the sequence is at least one term shorter after each shortening, the process stops in finite time. We obtain final sequence (p_0, \cdots, p_n) s.t. for all 0 < i < n,

\min \{ d(p_0, p_i), d(p_i, p_n) \} \geq \sqrt{\frac{2}{3}}d(p_0, p_n).

Since the process is applied at most k times, x_0, x_k \in B(p, \varepsilon), after the first shortening, d(p, x_{i_1}) \leq \max \{d(p, x_0), d(p, x_k) \} + \sqrt{\frac{2}{3}}d(x_0, x_k) \leq \varepsilon +  2 \sqrt{\frac{2}{3}} \varepsilon.

i.e. both initial and final term of the sequence is at most (\frac{1}{2}+ \sqrt{\frac{2}{3}}) 2 \varepsilon. Along the same line, we have, at the i-th shortening, the distance between the initial and final sequence and p is at most (\frac{1}{2} + \sqrt{\frac{2}{3}} + (\sqrt{\frac{2}{3}})^2 + \cdots (\sqrt{\frac{2}{3}})^i) 2 \varepsilon. Hence for the final sequence p_0, p_n \in B(p, 1+2 \sqrt{\frac{2}{3}}/(1-\sqrt{\frac{2}{3}}) \varepsilon) \subseteq B(p, 10 \varepsilon).

There is a rectangle R \subseteq M where p_0, p_n \in \sqrt{\frac{3}{4}}R
(i.e. shrunk R by a factor of \sqrt{\frac{3}{4}} w.r.t. the center) and for all 0 < i < n, \ p_i \notin R.

Next, we perturb f in R i.e. find h: M \rightarrow M with ||h||_{C^1} < \delta and h|_{M \backslash R} = id. Hence ||h \circ f - f ||_{C^1} < \delta.

Suppose R = I_1 \times I_2; L_1, L_2 are the lengths of I_1, I_2, L_1 < L_2.
By main value theorem, for all x \in M, \ d(x, h(x)) < \delta L_1.
On the other hand, since p_0 \in \sqrt{\frac{3}{4}}R, it's at least \frac{1}{2}(1-\sqrt{\frac{3}{4}})L_1 away from the boundary of R. i.e. there exists bump function h satisfying the above condition and d(p_0, h(p_0)) > \frac{\delta}{8}(1-\sqrt{\frac{3}{4}})L_1.

Hence in order to move a point by a distance L_1, we need about 1/ \delta such bump functions, to move a distance L_2, we need about \frac{L_2}{\delta L_1} bumps.

For simplicity, we now suppose M is a surface. By starting with an \varepsilon (and hence R) very small, we have for all 0 \leq i \leq N+M, \ f^i(R) is contained in a small neighbourhood of p_i. Hence on f^i(B), f^i is C^1 close to the linear map p_i + Df^i(p_0)(x-p_0). Hence mod some details we may reduce to the case where f is linear in a neighborhood of \mathcal{O}(p_0).

By choosing appropiate coordinate system in R, we can have f preserving the horizontal and vertical foliations and the horizontal vectors eventually grow more rapidly than the vertical vectors.

It turns out to be possible to choose R to be long and thin such that for all i \leq 40 / \delta, f^i(R) has height greater than width. (note that M = \lfloor 40/ \delta \rfloor bumps will be able to move the point by a distance equal to the width of the original rectangle R. Since horizontal vectors eventually grow more rapidly than the vertical vectors, there exists N s.t. for all N \leq i \leq N+M, f^i(R) has width greater than its height.
For small enough \epsilon, the boxes f^i(R) are disjoint for 0 \leq i \leq N+40/ \delta. Construct h to be identity outside of

\displaystyle \bigsqcup_{i=0}^M f^i(R) \sqcup \bigsqcup_{i=N}^{N + M} f^i(R)

For the first M boxes, we let h preserve the horizontal foliation and move along the width so that g = h \circ f has the property that g^M(p_n) lies on the same vertical fiber as f^M(p_0).

On the boxes f^{N+i}(R), \ 0 \leq i \leq M, we let h pushes along the vertical direction so that

g^{N+M}(p_n) = f^{N+M}(p_0)

Since iterates of the rectangle are disjoint, for N+M \leq i \leq n, \ h(p_i) = p_i, g(p_i) = f(p_i).

Hence g^n(p_n) = g^{n-(N+M)} \circ g^{N+M}(p_n) = g^{n-(N+M)} f^{N+M}(p_0) = g^{n-(N+M)} (p_{N+M}) = p_n.

Therefore we have obtained a periodic point p_n of g.

Since p_n \in B(p, 10 \varepsilon), we may further perturb g to move p_n to p. This takes care of the linear case on surfaces.

On compact extensions

May 10, 2010

This is again a note on my talk in the Szemerédi’s theorem seminar, going through Furstenberg’s book. In this round, my part is to introduce compact extension.
Let \Gamma be an abelian group of measure preserving transformations on (X, \mathcal{B}, \mu), \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') be an extension map.
i.e. \alpha: X \rightarrow Y s.t. \alpha^{-1} sends \nu-0 sets to \mu-0 sets;

\gamma'\circ \alpha (x) = \alpha \circ \gamma (x)

Definition: A sequence of subsets (I_k) of \Gamma is a Folner sequence if |I_k| \rightarrow \infty and for any \gamma \in \Gamma,

\frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0

Proposition: For any Folner sequence I = (I_k) of \Gamma, for any f \in L^1(X), \displaystyle \frac{1}{|I_k|} \sum_{\gamma \in I_k} \gamma f converges weakly to the orthogonal projection of f onto the subspace of \Gamma-invariant functions. (Denoted P(f) where P: L^2(X) \rightarrow L^2_{inv}(X).

Proof: Let \mathcal{H}_0 = P^{-1}(\bar{0}) = (L^2_{inv}(X))^\bot
For all \gamma \in \Gamma,

\gamma (L^2_{inv}(X)) \subseteq L^2_{inv}(X)

Since \Gamma is \mu-preserving, \gamma is unitary on L^2(X). Therefore we also have \gamma( \mathcal{H}_0) \subseteq \mathcal{H}_0.

For f \in \mathcal{H}_0, suppose there is subsequence (n_k) where \displaystyle \frac{1}{|I_{n_k}|} \sum_{\gamma \in I_{n_k}} \gamma (f) converges weakly to some g \in L^2(X).

By the property that \frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0, we have for each \gamma \in \Gamma, \gamma(g) = g, \ g is \Gamma-invariant. i.e. g \in (\mathcal{H}_0)^\bot

However, since f \in \mathcal{H}_0 hence all \gamma(f) are in \mathcal{H}_0 hence g \in  \mathcal{H}_0. Therefore g \in \mathcal{H}_0 \cap (\mathcal{H}_0)^\bot, g=\bar{0}

Recall: 1)X \times_Y X := \{ (x_1, x_2) \ | \ \alpha(x_1) = \alpha(x_2) \}.

i.e. fibred product w.r.t. the extension map \alpha: X \rightarrow Y.

2)For H \in L^2(X \times_Y X), \ f \in L^2(X),

(H \ast f)(x) = \int H(x_1, x_2) f(x_2) d  \mu_{\alpha(x_1)}(x_2)

Definition: A function f \in L^2(X) is said to be almost periodic if for all \varepsilon > 0, there exists g_1, \cdots g_k \in L^2(X) s.t. for all \gamma \in \Gamma and almost every y \in Y,

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

Proposition: Linear combination of almost periodic functions are almost periodic.

Proof: Immediate by taking all possible tuples of g_i for each almost periodic function in the linear combination corresponding to smaller \varepsilonl.

Definition: \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') is a compact extension if:

C1: \{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), f \in L^2(X) \} contains a basis of L^2(X).

C2: The set of almost periodic functions is dense in L^2(X)

C3: For all f \in L^2(X), \ \varepsilon, \delta > 0, there exists D \subseteq Y, \ \nu(D) > 1- \delta, \  g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma and almost every y \in Y, we have

\displaystyle \min_{1 \leq i \leq k} || \gamma (f)|_{f^{-1}(D)} - g_i||_y < \varepsilon

C4: For all f \in L^2(X), \ \varepsilon, \delta > 0, there exists g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma, there is a set D \subseteq Y, \ \nu(D) > 1- \delta, for all y \in D

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

C5: For all f \in L^2(X), let \bar{f} \in L^1(X \times_Y X) where

\bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2)

Let I=(I_k) be a Folner sequence, then \bar{f}=\bar{0} iff P \bar{f} = \bar{0}.

Theorem: All five definitions are equivalent.

Proof: “C1 \Rightarrow C2″

Since almost periodic functions are closed under linear combination, it suffice to show any element in a set of basis is approximated arbitrarily well by almost periodic functions.

Let our basis be as given in C1.

For all H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), the associated linear operator \varphi_H: L^2(X) \rightarrow L^2(X) where \varphi_H: f \mapsto H \ast f is bounded. Hence it suffice to check H \ast f for a dense set of f \in L^2(X). We consider the set of all fiberwise bounded f i.e. for all y \in Y, ||f||_y \leq M_y.

For all \delta > 0, we perturb H \ast f by multiplying it by the characteristic function of a set of measure at least 1- \delta to get an almost periodic function.

“C2 \Rightarrow C3″:

For any f \in L^2(X), there exists f' almost periodic, with ||f-f'||< \frac{\epsilon \sqrt{\delta}}{2} . Let \{ g_1, g_2, \cdots, g_{k-1} \} be the functions obtained from the almost periodicity of f' with constant \varepsilon/2, g_k = \bar{0}.

Let D = \{ y \ | \ ||f-f'||_y < \varepsilon/2 \}, since

|| f - f'||^2 = \int ||f-f'||_y^2 d \nu(y)

Hence ||f-f'||< \frac{\varepsilon \sqrt{\delta}}{2} \ \Rightarrow \ ||f-f'||^2 < \frac{\varepsilon^2 \delta}{4}, \{ y \ | \ ||f-f'||_y > \varepsilon/2 \} has measure at most \delta/2, therefore \nu(D) > 1- \delta.

For all \gamma \in \Gamma, ify \in \gamma^{-1}(D) then

|| \gamma f|_{\alpha^{-1}(D)} - \gamma f'||_y  = ||f|_{\alpha^{-1}(D)} - f'||_{\gamma(y)} < \varepsilon /2

Hence \displaystyle \min_{1 \leq i \leq k-1} ||\gamma f|_{\alpha^{-1}(D)} - g_i||_y < \varepsilon /2 + \varepsilon /2 = \varepsilon

If y \notin \gamma^{-1}(D) then f|_{\alpha^{-1}(D)} vanishes on \alpha^{-1}(\gamma y) so that || \gamma f|_{\alpha^{-1}(D)} - g_i||_y = 0 < \varepsilon.

Hence \alpha satisfies C3.

“C3 \Rightarrow C4″:

This is immediate since for all y \in \gamma^{-1}(D), we have \gamma f = \gamma f|_{\alpha^{-1}(D)} on \alpha^{-1}(y) hence

\displaystyle \min_{1 \leq i \leq k} ||\gamma f - g_i||_y < \min_{1 \leq i \leq k-1} ||\gamma f_{\alpha^{-1}(D)} - g_i||_y < \varepsilon

\nu(\gamma^{-1}(D)) = \nu(D) > 1-\delta. Hence \alpha satisfies C4.

“C4 \Rightarrow C5″:

For all f \in L^2(X), \ \varepsilon, \delta > 0, by C4, there exists g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma, there is a set D \subseteq Y, \ \nu(D) > 1- \delta, for all y \in D

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

W.L.O.G. we may suppose all g_i are bounded since by making \delta slighter larger we can modify the unbounded parts to be bounded.

\bar{g_j} \otimes g_j \in L^\infty(X \times_Y X), suppose P(\bar{f}) = 0.

Recall in C5 we have \bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2), and \displaystyle P_I \bar{f}(x_1, x_2) = \lim_{k \rightarrow \infty} \frac{1}{|I_k|} \sum_{\gamma \in I+k} f(\gamma x_1) \bar{ f(\gamma x_2)}.

For each 1 \leq j \leq k, we have \int (\bar{g_j} \otimes g_j) \cdot P \bar{f} d(\mu \times_Y \mu) = 0

Hence we have \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int (\bar{g_j(x_1)} g_j(x_2)) \cdot \gamma f(x_1) \bar{\gamma f(x_2)} d\mu \times_Y \mu(x_1, x_2) = 0

\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) = 0

\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \{ \sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) \} = 0

Hence for large enough i, there exists \gamma \in I_i s.t. \sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) is as small as we want.

We may find D' \subseteq Y with \nu(D) > 1-\delta s.t. for all y \in D' and for all j, we have

| \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 < \varepsilon^2

On the other hand, by construction there is j with || \gamma f - g_j||^2_y < \varepsilon^2 for all y \in D, with \nu(D) > 1-\delta .

Hence for y \in D \cap D', \ ||f||_{\gamma'^{-1}(y)}^2 = || \gamma f||_y^2 < 3 \varepsilon^2.

Let \varepsilon \rightarrow 0, \ \delta \rightarrow 0 we get f = \bar{0}. Hence C5 holds.

“C5 \Rightarrow C1″

Let f \in L^2(X) orthogonal to all of such functions. Let (I_k) be a Folner sequence.

Define \displaystyle H(x_1, x_2) := \lim_{i \rightarrow \infty} \frac{1}{|I_i|}\sum_{\gamma \in I_i} \gamma f(x_1) \cdot \gamma f(x_2) = P \bar{f}(x_1, x_2)

Let H_M(x_1, x_2) be equal to H whenever H(x_1, x_2) \leq M and 0 o.w.

H is \Gamma-invariant \Rightarrow \ H_M is \Gamma-invariant and bounded.

Therefore f \bot H_M \ast f, i.e.

\int \bar{f(x_1)} \{ \int H_M(x_1, x_2) d \mu_{\alpha(x_1)}(x_2) \} d \mu(x_1) = 0 <\p>

Since \mu = \int \mu_y d \nu(y), we get

\int \bar{f} \otimes f \cdot H_M d \mu \times_Y \mu = 0 <\p>

Hence H_M \bot (\bar{f} \otimes f). For all \gamma \in \Gamma, \ \gamma (\bar{f} \otimes f) \bot \gamma H_M = H_M.

Since H = P \bar{f} is an average of \gamma (\bar{f} \otimes f), \ \Rightarrow \ H \bot H_M.
0 = \int \bar{H} \cdot H_M = \int |H_M|^2 \ \Rightarrow \ H_M = \bar{0} for all M

Hence H = \bar{0}. By C5, we obtain f = \bar{0}. Hence \{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), f \in L^2(X) \} contain a basis for L^2(X).

Definition: Let H be a subgroup of \Gamma, \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') is said to be compact relative to H if the extension \alpha: (X, \mathcal{B}, \mu, H) \rightarrow ( Y, \mathcal{D}, \nu, H') is compact.

On plaque expansiveness

May 5, 2010

This note is mostly based on parts of (RH)^2U (2006) and conversations with R. Ures while he was visiting Northwestern.

Let \mathcal{F} be a foliation of the manifold M, for p \in M, a plaque in of \mathcal{F} through p is a small open neighborhood of p in the leaf \mathcal{F}_p that’s pre-image of a disc via a local foliation chart. (i.e. plaques stuck nicely to make open neighborhoods where the foliation chart is defined.) For \varepsilon small enough, whenever the leaves of \mathcal{F} are C^1, the path component of B(p, \varepsilon) containing p is automatically a plaque, we denote this by \mathcal{F}_\varepsilon(p).

Given a partially hyperbolic diffeomorphism f: M \rightarrow M, suppose the center integrates to foliation \mathcal{F}^c.

Definition: An \varepsilon-pseudo orbit w.r.t. \mathcal{F}^c is a sequence (p_n) where for any n \in \mathbb{Z}, f(x_n) \in \mathcal{F}^c_\varepsilon(x_{n+1}).

i.e. p_{n+1} is the f-image of p_n except we are allowed to move along the center plaque for a distance less than \varepsilon.

Definition: f is plaque expansive at \mathcal{F}^c if there exists \varepsilon>0 s.t. for all \varepsilon-pseudo orbits (p_n),  (q_n) w.r.t. \mathcal{F}^c, d(p_i, q_i)<\varepsilon for all i \in \mathbb{Z} then p_0 \in \mathcal{F}^c_\varepsilon(q_0).

i.e. any two pseudo-orbits in different plagues will eventually (under forward or backward iterates) be separated by a distance \varepsilon.

In the book Invariant Manifolds (Hirsch-Pugh-Shub), it’s proven that

Theorem: If a partially hyperbolic system has plaque expansive center foliation, then the center being integrable and plaque expansiveness are stable under perturbation (in the space of diffeos). Furthermore, the center foliation of the perturbed system g is conjugate to the center foliation of the origional system f in the sense that there exists homeomorphism h: M \rightarrow M where

1) h sends leaves of \mathcal{F}^c_f to leaves of \mathcal{F}^c_g i.e. for all p \in M,

h(\mathcal{F}^c_f(p)) = \mathcal{F}^c_g(p)

2) h conjugates the action of f and g on the set of center leaves i.e. for all p \in M,

h \circ f \ (\mathcal{F}^c_f(p)) = g \circ h \ ( \mathcal{F}^c_f(p))

(both sides produce a \mathcal{F}^c_g leaf)

Morally this means plaque expansiveness implies structurally stable in terms of permuting the center leaves.

It’s open whether or not any partially hyperbolic diffeomorphism with integrable center is plaque expansive w.r.t. its center foliation.

Another problem, stated in HPS about plaque expansiveness is:

Question: If f is partially hyperbolic and plaque expansive w.r.t. center foliation \mathcal{F}_c, then is \mathcal{F}_c the
unique f−invariant foliation tangent to E^c?

(RH)^2U has recently gave a series of super cool examples where the 1-dimensional center bundles of a C^1 partially hyperbolic diffeomorphism 1) does not integrate OR 2) integrates to a foliation but leaves through a given point is not unique (there is other curves through the point that’s everywhere tangent to the bundle). I will say a few words about the examples without spoil the paper (which is still under construction).

Start with the cat map on the 2-torus (matrix with entries ( 2, 1, 1, 1), take the direct product with the source-sink map on the circle, we obtain a diffeo on the 3 torus. For the purpose of our map, we make the expansion in the source-sink map weaker than that of the cat map and the contraction stronger.

Then we perturb the map by adding appropriate small rotations to the system, the perturbation vanish on the \mathbb{t}^2 fibers corresponding to the two fixed points in the source-sink map. This will make our system partially hyperbolic, with center bundles as shown below:

To construct a non-integrable center, we make a perturbation that gives center boundle (inside the unstable direction of the cat map times the circle):

For intergrable but have non-unique center leaves, we simply rotate the upper and bottom half in opposite directions and obtain:

Note that in this case, all center leaves are merely copies of S^1. The example is plaque expansive due to to fact that all centers leaves are compact (and of uniformly bounded length). However, although the curve through any given point tangent to the bundle is non-unique, there is only one possible foliation of the center. Hence this does not give a counter example to the above mentioned question in HPS.

I think there are hopes to modify the example and make one that has similar compact leafs but non-unique center foliation, perhaps by making the unique integrability fail not only on a single line.

Fundemental Theorem of Dynamical Systems (Part 2)

April 15, 2010

Now we begin to prove the theorem.

    4.Attractor-repeller pairs

Definition: A compact set A \subseteq X is an attractor for f if there exists U open, f(\bar U) \subseteq U and \displaystyle \bigcup_{i=0}^\infty f^i(U) = A. U is called a basin of attraction.

For any attractor A \subseteq X, U be a basin for A, let U^\ast = X \backslash \bar U, \ U^\ast is open. By definition, f(A) = A and f(A^\ast) = A^\ast. We also have

f^{-1}(\overline{U^\ast}) = f^{-1}(\bar{X \backslash \bar U}) \subseteq X \backslash f^{-1}(U) \subseteq X \backslash U \subseteq \overline{U^\ast}

Definition: A repeller for f is an attractor for f^{-1}. A basin of repelling for f is a basin of attracting for f^{-1}.

Hence A^\ast is a repeller for f with basin U^\ast.

It’s easy to see that A^\ast is defined independent of the choice of basin for A. (Exercise)

We call such a pair A, \ A^\ast an attracting-repelling pair.

The following two properties of attracting-repelling pairs are going to be important for our proof of the theorem.

Proposition 1: There are at most countably many different attractors for f.

Proof: Since X is compact metric, there exists countable basis \mathcal{U} = \{U_i\}_{i \in \mathbb{N}} that generates the topology.

For any attractor A, any attracting basin \mathcal{B} of A is a union of sets in \mathcal{U}, i.e. \displaystyle \mathcal{B} = \bigcup_{i=1}^\infty U_{n_i}latex for some subsequence (n_i) of \mathbb{N}.

Since A is compact, U_{n_i} is an open cover of A, we have some \{ m_1, \ m_2, \ \cdots \ m_k \} \subseteq \{n_i\}_{i \in \mathbb{N}} s.t. \{ U_{m_1}, \ U_{m_2}, \ \cdots, \ U_{m_k} \} covers A.
Let B' = U_{m_1} \cup U_{m_2} \cup \cdots \cup U_{m_k} hence A \subseteq B' \subseteq B. We have

\displaystyle A \subseteq \bigcap_{n \in \mathbb{N}} f^n(B') \subseteq \bigcap_{n \in \mathbb{N}} f^n(B)

Since B is an attracting basin for A, all three sets are equal. Hence A = \bigcap_{n \in \mathbb{N}} f^n(B'). i.e. any attractor is intersection of foreward interates of come finite union of sets in \mathcal{U}. Since \mathcal{U} is countable, the set of all finite subset of it is coubtable.

Hence there are at most countably many different attractors. This establishes the proposition.

By proposition 1, we let (A_n)_{n \in \mathbb{N}} be a list of all attractors for f. Now we are going to relate the arrtactor-repeller pairs to the chain recurrent set and chain transitive components.

Proposition 2:

\mathcal{R}(f) = \displaystyle \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n)

Proof: i) \mathcal{R}(f) \subseteq \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n)

This is same as saying for any attractor A, \mathcal{R}(f) \subseteq (A \cup A^\ast).

For all x \notin (A \cup A^\ast), let B be a basin of A, then there is N \in \mathbb{N} for which x \notin (f^N(B) \cup f^{-N}(B^\ast)) (recall that B^\ast is the dual basin of B for A^\ast). Since B^\ast = X \backslash \overline{B} and f(\overline{B}) \subseteq B we conclude

X \backslash f^{-N}(B^\ast) = f^{-N}(\overline{B}) \subseteq f^{-N-1}(f(\overline{B})) \subseteq f^{-N-1}(B)

Hence x \in f^{-N-1}(B). Let M be the smallest integer for which x \in f^{-M}(B). Hence x \in f^{-M}(B) \backslash f^{-M+1}(B). Let U = f^{-M}(B) is also a basin for A.

Now we show such x cannot be chain recurrent: Since X \backslash f(U) and \overline{f^2(U)} are compact and disjoint, we may let

\varepsilon_1 = \frac{1}{2} \min\{ d(a, b) \ | \ a \in X \backslash f(U), \ b \in \overline{f^2(U)} \}

Since f(x) \in f(U), there exists some \varepsilon_2 s.t.

\overline{B(f(x), \varepsilon_2)} \subseteq f(U)

f(\overline{B(f(x), \varepsilon_2)}) \subseteq f^2(U) so there exists \varepsilon_3 s.t.

N(f(\overline{B(f(x), \varepsilon_2)}), \varepsilon_3) \subseteq f^2(U)

(Here B(p, r) denotes the ball of radius r around p and N(C, r) denotes the r-neignbourhood of compact set C)

Now set \varepsilon = \min\{ \varepsilon_1, \ \varepsilon_2, \ \varepsilon_3\}, for any \varepsilon-chain x, x_1, x_2, \cdots, we have: Since \varepsilon < \varepsilon_2 and \varepsilon_3, x_1 \in B(f(x), \varepsilon_2)\subseteq f(U) and x_2 \in B(f(x_1), \varepsilon_3) \subseteq N(f(\overline{B(f(x), \varepsilon_2)}), \varepsilon_3) \subseteq f^2(U). Hence the third term of any such chain must be in f^2(U). Since \varepsilon < \varepsilon_1, no \varepsilon-chain starting at x_2 can reach X \backslash f(U), in particular, the chain x, x_1, x_2, \cdots does not come back to x. Hence we conclude that x is not chain recurrent. ii) \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n) \subseteq \mathcal{R}(f) Suppose not, there is x \in \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n) and x \notin \mathcal{R}(f). i.e. for some \varepsilon > 0 there is no \varepsilon-chain from x to itself. Let U be the open set consisting all points that can be connected from x by an \varepsilon-chain.

We wish to generate an attractor by V, to do this all we need to check is f(\overline{V}) \subseteq V:
For any y \in \overline{V} there exists y' \in V with d(f(y), f(y')) < \varepsilon. Since y' \in V, there is \varepsilon-chain x, x_1, \cdots, x_n, y' which gives rise to \varepsilon-chain x, x_1, \cdots, x_n, y', f(y). Therefore f(\overline{V}) \subseteq V.

Hence \displaystyle A = \bigcap_{n \in \mathbb{N}} f^n(V) is an attractor with V as a basin.

By assumption, x \in A \cup A^\ast, since A \in V and there is no \varepsilon-chain from x to itself, x cannot be in A. i.e. x \in A^\ast Take any limit point y of (f^n(x))_{n\in \mathbb{N}}, since A^\ast is compact f-invariant we have y \in A^\ast. But since we can find N where d(f^N(x), y)<\varepsilon, x, f(x), \cdots, f^{N-1}(x), y gives an \varepsilon-chain from x to y, hence y \in V.

Recall that V is a basin of A hence A^\ast \cap V is empty. Contradiction. Establishes proposition 2.

This proposition says that to study the chain recurrent set is the same as studying each attractor-repeller pair of the system. But the dynamics is very simple for each such pair as all points not in the pair will move towards the attractor under foreward iterate. We can see that such property is goint to be of importantce for our purpose since the dynamical for each attractor-repeller pair is like the sourse-sink map.

5. Main ingredient

Here we are going to prove a lemma that’s going to produce for us the ‘building blocks’ of our final construction. Namely a function for each attracting-repelling pair that strickly decreases along the orbits not in the pair. In light of proposition 2, we should expect to put those functions together to get our complete Lyapunov function.

Lemma1: For each attractor-repeller pair A, A^\ast there exists continuous function g: X \rightarrow [0,1] s.t. g^{-1}(0)=A, \ g^{-1}(1)=A^\ast and g(f(p)) < g(p) for all p \notin (A\cup A^\ast).

Proof: First we define \varphi: X \rightarrow [0,1] s.t.

\varphi(x) = \frac{d(x,A)}{d(x, A)+d(x, A^\ast)}

Note that \phi takes value 0 only on A and 1 only on A^\ast. However, \phi can’t care less about orbits of f.

Define \bar\varphi(x) = \sup\{\varphi(f^n(x)) \ | \ n\in\mathbb{N} \}. Hence automatically for all x, \bar \varphi(f(x)) \leq \bar \varphi(x). Since no points accumulates to A^\ast under positive iterations, we still have the \bar \varphi^{-1}(0)=A and \bar \varphi^{-1}(1) = A^\ast.

We now show that \bar\varphi is continuous:

For x \in A^\ast and (x_i)\rightarrow x, \varphi(x_i) \leq \bar\varphi(x_i) \leq 1 and (\varphi(x_i)) \rightarrow 1 hence \bar\varphi(x_i)\rightarrow 1 i.e. \bar\varphi is continuous on A^\ast.

For x \in A we use the fact that A is attracting. Let B be a basin of A. For all (x_i) \rightarrow x, for any \varepsilon>0, there is N \in \mathbb{N} s.t. f^N(B) \subseteq N(A, \varepsilon). Therefore for some x_i \in f^N(B), all f^n(x_i) are in N(A, \varepsilon) i.e. \varphi(f^n(x_i))\leq \frac{\varepsilon}{\varepsilon+C} hence \bar\varphi(x_i)\leq \frac{\varepsilon}{\varepsilon+C}. But for some M \in \mathbb{N} and all m>M, x_m \in B. Therefore \bar\varphi(x_i) \rightarrow 0. \bar\varphi is continuous on A.

Let T = \overline{B} \backslash f(B), for any \displaystyle x \in T, \ r=\inf_{x \in T} \varphi(x), since f^n(T) \subseteq f^n(\overline{B}), there exists N>0 s.t. for all n>N \varphi(f^n(T))\subseteq [0,r/2]. i.e. \displaystyle \bar\varphi9x) = \max_{0\leq n\leq N} \varphi(f^n(x)) which is countinous. Since those ‘bands’ T partitions the whole X (by taking B_n to be f^n(B) \backslash f^{n+1}(B)), hence we have proven \bar\varphi is continuous on the whole X.

Finally, we define

\displaystyle g(x) = \sum_{n=0}^\infty \frac{\bar\varphi(f^n(x))}{2^{n+1}}

We check that g is continuous since \bar\varphi is. g takes values 0 and 1 only on A and A^\ast, respectively. For any x \notin (A \cup A^\ast),

\displaystyle g(f(x))-g(x) = \sum_{n=0}^\infty \frac{\bar\varphi(f^{n+1}(x))-g(f^n(x))}{2^{n+1}}

therefore g(f(x))-g(x) = 0 iff \bar\varphi(f^{n+1}(x)) = \bar\varphi(f^n(x)) for all n i.e. \bar\varphi is constant on the orbit of x. But this cannot be since there is a subsequence of (f^n(x)) converging to some point in A, continuity of \bar\varphi tells us this constant has to be 0 hence x \in A.

Therefore g is strictly decreasing along orbits of f not in (A \cup A^\ast).

Establishes lemma 1.

6.Proof of the main theorem

The proof of the main theorem now follows easily from what we have established so far.

First we restate the fundamental theorem of dynamical systems:

Theorem: Complete Lyapunov function exists for any homeomorphisms on compact metric spaces.

Proof: First we enumerate the countably many attractors as (A_n)_{n \in \mathbb{N}}. For each A_n, we have function g_n: X \rightarrow R where g_n is 0 on A_n, 1 on A_n^\ast and is strictly decreasing on X \backslash ( A_n \cup A_n^\ast).
Define g: X \rightarrow \mathbb{R} by

\displaystyle g(x) = 2 \cdot \sum_{n=1}^\infty \frac{g_n(x)}{3^n}

Since each g_n is bounded between 0 and 1, the sequence of partial sums converge uniformly. Hence the limit function g is continuous.

For points p \in \mathcal{R}(f), we have p \in (A_n \cup A_n^\ast) for all n \in \mathbb{N}. i.e. $latex  \ \forall n \in \N, \ g_n(p) = 0$ or g_n(p) = 1. Hence we have

g(p) = \sum_{n=1}^\infty \frac{2 \cdot g_n(p)}{3^n} = \sum_{n=1}^\infty \frac{a_n}{3^n}

where each a_n is in \{0, 2 \}. This is same as saying the base-3 expansion of g(p) only contains digits 0 and 2. We conclude g(\mathcal{R}(f)) \subseteq \mathcal{C} where \mathcal{C} is the standard middle-third Cantor set in [0, 1]. i.e. g(\mathcal{R}(f)) is compact and nowhere dense in \mathbb{R}.

For p \notin \mathcal{R}(f), there exists n \in \mathbb{N} such that p \notin (A_n \cup A_n^\ast), hence g_n(f(p)) < g_n(p). This implies g(f(p)) < g(p) since g_i(f(p)) \leq g_i(p) for all i \in \mathbb{N}. i.e. g is strictly decreasing along orbits that are not chain recurrent.

To show g is constant only on the chain-transitive components, we need the following lemma:

Claim: p, \ q \in \mathcal{R}(f) are in the same chain-transitive component iff there is no attracting-repelling pair A, \ A^\ast where one of p, \ q is in A while the other in A^\ast.

Proof (of claim):\Rightarrow” Suppose x, y \in \mathcal{R}(f) and x \sim y, for any attractor A, if x \notin A and y \notin A, then x, y are both in A^\ast and we are done. Hence suppose at least one of x, y is in A. W.L.O.G. suppose x \in A. Let B be a basin of A. Since \overline{f(B)}, \ X\backslash B are closed and disjoint, we may choose \varepsilon<\min\{ d(a, b) \ | \ a \in (X\backslash B), \ b \in \overline{f(B)} \}. By the same arguement as in proposition 2, there are no \varepsilon/2-chain (with length >1) from any point in f{B} to any point in X \backslash B. Hence there is also no \varepsilon/2-chain from any point in A to any point in A^\ast. Hence y \notin A^\ast i.e. y \in A.

\Leftarrow” Suppose for any attractor A, x \in A iff y \in A. For any \varepsilon>0, let U be the set of all points y for which there is an \varepsilon-chain from x to y, as defined in proposition 2. We have showed in proposition 2 that U is a basin of some attractor A'. Since x \in \mathcal{R}(f) \subseteq (A' \cup {A'}^\ast) and x \in U, hence x \in A'. Hence by our assumption, y must be also in A'. Hence y \in U i.e. there is an \varepsilon-chain from x to y. Since the construction is symetric, we may also show there is an \varepsilon-chain from y to x. i.e. x\sim y.

Establishes the claim.

Finally, for p, q \in \mathcal{R}(f), g(p) = g(q) means g(p) and g(q) has the same base-3 expansion in the Cantor set. This is same as saying \forall i \in \mathbb{N}, \ g_i(p) = g_i(q) \in \{0, 2\}, which is to say there is no i \in \mathbb{N} for which one of p, \ q is in A_i while the other in A_i^\ast. Hence by Lemma, we conclude that g(p) = g(q) iff p, \ q are in the same chain transitive component.

This establishes our theorem.

Fundemental Theorem of Dynamical Systems (Part 1)

April 15, 2010

This article was written as a homework of professor Wilkinson’s dynamical systems course. Since the content is expository and detailed presentation of the theorem is missing from many books, I decided to post it here. I have mostly followed a set of notes by John Franks, with additional discussions on the intuition and ideas behind the statement and the proof.

1.Introduction

So far we have discussed various different kinds of dynamical systems ranging from topological, smooth to hyperbolic and partially hyperbolic. One might wonder if there is a united theme to the subject as a whole. Indeed, as in many other subjects, there is a so-called fundamental theorem of dynamical systems. This theorem is first stated and proved by Conley in where he studies attractor and repellers. The theorem, loosely speaking, gives a universal decomposition of any systems on compact metric spaces into invariant compact sets wandering orbits that travels between such sets.

To state this more precisely we make an analogy with Morse theory: When looking at the gradient flow on a compact embedded manifold, we ‘decompose’ the manifold into critical points and orbits that originates and ends at critical points. In a similar spirit, given any homeomorphisms on a compact metric space, we may look at it’s ‘indecomposible’ compact invariant sets and how they are ‘connected’ by wandering orbits, we then ‘place’ those compact sets on different ‘heights’ and have all other point going between the minimal sets they originates and ends at. The theorem guarantees that we can ‘place’ the space in a way that all wandering orbits are going ‘down’ at all times.

In light of the theorem, we have descried the global structure of the system except for what happens on the ‘indecomposible’ sets. i.e. The problem of understanding general topological systems on compact manifolds is reduced to understanding ‘transitive’ homeomorphisms on compact sets. The latter, unfortunately, could still be quite complicated as we have seen in the Horseshoe example.

The theorem is proposed to be the Fundemental theorem of dynamical systems because of its nature in giving concise description of all possible behaviors of a system in the given setting. In some sense, dynamics is the study of limiting behrviors of all points under interation, the theorem breaks the system down into a recurrent part and a wandering part where the behavior of the wandering part is gradient-like. Since we have developped sets of different tools for studying systems that exhibits a lot of recurrence as well as for studying gradient-like systems, this allows us to connect combine the tool sets and treat any systems in the setting.

2.Background

In this section, we define \varepsilon-chains, chain recurrent sets and chain transitive components for a homeomorphism on a compact metric space. Those concepts will come up in the statement of the fundamental theorem. In fact, those are going to be the ‘minimal compact sets’ we decompose our metric space into.

Given compact metric space X and homeomorphism f: X \rightarrow X,
Definition: Given two points p, q \in X, an \varepsilon-chain from p to q is a sequence x_1, x_2, \cdots x_n, n>1 where x_1 = p, \ x_n = q and for all 1 \leq i \leq n-1, d(f(x_i), x_{i+1}) < \varepsilon.

i.e. we take a point and start applying f to it, but at each iterate, we are allowed to perturb the resulting point by \varepsilon. Such ‘pseudo-orbits’ are in general much easier to obtain than true orbits.

More generally, \varepsilon-chains can be taken infinite. i.e. if we have a (possibly infinite) subinterval I \subseteq \mathbb{Z}, an \varepsilon-chain indexed by I is a set of points (p_i)_{i \in I} s.t. d(f(p_i), p_{i+1}) < \varepsilon whenever i, i+1 are both in I.

Definition:p \in X is chain recurrent if for all \varepsilon > 0, there exists an \varepsilon-chain from p to itself. The set of all chain recurrent points in X is called the \textbf{chain recurrent set}, denoted by \mathcal{R}(f).

Note that non-wandering points are necessarily chain recurrent: If p \in X is non-wandering, we may take the neighborhood to be the \varepsilon-ball around p, since p is non-wandering, we have some n>1 where f^n(B_\varepsilon(p)) \cap B_\varepsilon(p) \neq \phi, we pick q in the intersection and define \varepsilon-chain p, f^{-n}(q), f^{-n+1}(q), \cdots, q, p.

At this point, it’s perhaps worthwhile to mention our completed ordering of different notions of recurrence:

\overline{\mbox{Per}(f)} \subseteq \mbox{Rec}(f) \subseteq \mbox{NW}(f) \subseteq \mathcal{R}(f)

Each of the above inclusion can be made strict (see Exercises). Chain recurrence is perhaps the weakest notion I’ve seen for a point to be, in any sense, recurrent. A Friendly challenge to the reader: think of a case where you feel confortable calling a point ‘recurrent’ while it’s not in the chain recurrent set of the system.

We now define equivlence relation on \mathcal{R}(f) as follows:

For p, q in \mathcal{R}(f), p \sim q iff for all \varepsilon > 0, there are \varepsilon-chains from p to q and from q to p. \sim is reflexive since all points in \mathcal{R}(f) are chain recurrent; symmetric by definition and transitive by the obvious composition of \varepsilon-chains.

Definition: The equivalence classes in \mathcal{R}(f) for \sim are called chain transitive components.

It’s easy to check that chain transitive components are compact and f-invariant. Those are components that’s transitive in a very weak sense. i.e. any two points are connected by a ‘pseudo-orbit’, or equivalently, there is a dense (infinite) pseudo-orbit. (see exercises)

As mentioned above, throughout the rest of the chapter, we will consider chain transitive components as ‘indecomposible parts’ of our system. Those are the parts for which all points are ‘recurrently’ and each component is ‘transitive’, both in a very weak sense. We further specify how does the points that are not in the chain-recurrent set iterates between those components.

3.Statement of the theorem

Given compact metric space X and homeomorphism f: X \rightarrow X,

Definition: g: X \rightarrow \mathbb{R} is a complete Lyapunov function for f if:

\forall \ p \notin \mathcal{R}(f), \ g(f(p)) < g(p)

\forall \ p, q \in \mathcal{R}(f), \ g(p) = g(q) \ \mbox{iff} \ p \sim q

g(\mathcal{R}(f)) \ \mbox{is compact and nowhere dense in} \ \mathbb{R}

Hence this is a function that stays constant only on the chain transitive components and strictly decreases along any orbit not in \mathcal{R}(f). We also require the image of \mathcal{R}(f) to be compact and nowhere dense which cooresponds to the ‘critical values’ of a gradient function being compact nowhere dense as a result of Sard’s theorem.

Fundemental theorem of dynamical systems:

Complete Lyapunov function exists for any homeomorphisms on compact metric spaces.

As a historical remark, the theorem first appeared in Charles Conley’s CBMS monograph Isolated Invariant Sets and the Morse Index in 1978 [C]. In the book he developed the theory of attractor-repeller pairs in relation to Morse decomposition and index theory. The above theorem was one of the major results. Although Conley was originally more focused on the setting where instead of a homeomorphism, we have a continuous flow on the manifold (which makes it even more similar to the gradient flow), but this discrete formulation became more popular as the theory develops. The theorem is later proposed by D. Norton as the fundamental theorem of dynamical in 1995.

The proof is going to be a specific construction: First, we define a family of partitions of the chain recurrent set, each divides the set into two pieces (i.e. a attractor-repeller pair intersected with \mathcal{R}). Then we prove the family is countable and points in the same chain transitive component are not separated by any partition in the family. Furthermore, each chain-transitive component is uniquely defined by specifying which set does it belong to in each partition. i.e. the smallest common refinement for the family exactly partitions \mathcal{R} into chain-transitive components. (section 4)

Next, for each attractor-repeller pair, we prove the existence of a function that takes value 0 on the attractor and 1 on the repeller and strictly decreases along orbits of points that’s not in \mathcal{R}(f). We should also mention the fact that all points that are contained in one of the sets in each pair must be in chain recurrent. (section 5)

The complete Lyapunov function is then constructed by taking an appropriate infinite sum of such functions. This way we get a function that separates all chain transitive components, stays constant on each component and strictly decreases along all orbits which are not in \mathcal{R}. The image of the chain recurrent set will be contained in the middle-third Cantor set. (section 6)

(see part 2 for sections 4-6)