# A report of my Princeton generals exam

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 $50$ $0$s 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.

# A survey on ergodicity of Anosov diffeomorphisms

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

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

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

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! ^^