Posts Tagged ‘flow’

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.

Anosov flows

January 19, 2010

Amie told me today about their new result on perturbation of a volume-preserving Anosov flow in three dimensions. In order to not forget what it’s about, I decided to write a sketch of what I still remember here:

So, you are given a volume preserving Anosov flow in some three-manifold (and since it’s volume preserving and Anosov and three dimensional, of course we have one dimensional stable and unstable manifolds), let \varphi_1: M \rightarrow M be its time-1 map. Consider a C^\infty perturbation of \varphi_1. We are interested in when is the perturbed map still a time-1 map of a flow.

Note that we know partial hyperbolicity is an open property, our perturbed map will still be a partially hyperbolic diffeo. However in general it would no longer be a time 1 map of a flow. It turns out that we can tell whether or not it’s a time-1 map just by looking at the center foliation. (some condition to do with whether some measure on the center is atomic…I can’t recall)

Furthermore this infact don’t have much to do with the fact it’s a perturbation of the Anosov flow: we may start with any volume-preserving partially hyperbolic diffeomorphism in three-manifold M, assuming the diffeo preserves its center foliation (or more generally if it permutes each center leaf peroidically), then it’s time-one map of a flow precisely when their condition on the center foliation holds. Note that the center leaves are automatically preserved if the map was a perturbation of the Anosov flow.

Note that restriction our attention to volume preserving flows is essential in obtaining any of such results since in part it guarantees a dense set of periodic orbits. I’m suppose to check Franks and William’s paper on “Anomolous Anosov Flows” in which they gave many examples of different non-volume-preserving Anosov flows. The question of whether or not it’s possible to classify all Anosov flows (in the sense presented in the paper) is still open.