Posts Tagged ‘generic’

The moving needle problem

November 22, 2010

First, let’s clarify that this post has nothing to do with the Kakeya conjecture (except for the word ‘needle’ in it). Anyways, I was asked the following question via an e-mail from Charles this summer: (It turns out that the question was invented by Jonathan King and then communicated to Morris Hirsch and that’s where Charles heard about it from) In any case, I find the problem quite cute:

Problem: Given a smoothly embedded copy of \mathbb{R} in \mathbb{R}^3 containing \{ (x,0,0) \ | \ x \in (-\infty,-C] \cup [C, \infty) \}. Is it always possible to continuously slide a unit length needle lying on the ray (-\infty, -C] to the ray [C, \infty), while keeping the head and tail of the needle on the curve throughout the process?

i.e. the curve is straight once it passes the point (-C, 0) and (C,0), but can be bad between the two points:

We are interested in sliding the needle from the neigative x-axis to the positive x-axis:

Exercise: Try a few examples! It’s quite amusing to see that sometimes both ends of the needle needs to go back and forth along the curve many times, yet it always seem to get through.

One should note that this is not possible if we just require the curve to be eventually straight and goes to infinity at both ends. As we can see on a simple ‘hair clip’ curve:

The curve consists of two parallel rays of distance <1 apart, connected with a semicircle. A unit needle can never get from one ray to the other since the needle would have to rotate 180 degrees and hence it has to be vertical at some point in the process, but no two points on the curve has vertical distance 1.

After some thought, I think I can show for each given curve, 'generic' needle length can pass through:

Claim: For any C^2 embedding as above, there is a full measure and dense G_\delta set \mathcal{L} \subseteq \mathbb{R} of lengths where the needle of any length L \in \mathcal{L} can slide through the curve.

Proof: Let \gamma: \mathbb{R} \rightarrow \mathbb{R}^3 be a smooth parametrization of the curve s.t. \gamma(t) = t for t \in (-\infty, -C] \cup [C, \infty).

Define \varphi: \mathbb{R}^2 \rightarrow \mathbb{R} where \varphi: (s, t) \mapsto d(\gamma(s), \gamma(t)).

Hence \varphi vanishes on the diagonal and takes positive value everywhere else.
Since \gamma(t) = t for t \notin (-C, C), hence \varphi(s,t)=|s-t| for |s|, |t|>C. \varphi^{-1}(L) contains four rays \{ |s-t| = L \ | \ s, t \notin (-C, C) \}:

Observation: a needle of length L can slide through the curve iff there is a continuous path in the level set \varphi(p) = L connecting the two rays above the diagonal. (This is merely projection onto the x and $larex y$-axis.)

We also have \varphi is C^2 other than on the diagonal. (It behaves like the absolute value function near the diagonal). By Sard’s theorem, since \varphi is C^2 on \{s < t\}, the set of critical values is both measure 0 and first category.

Let \mathcal{L} be the set of regular values of \varphi. For any L \in \mathcal{L}, by implicit function theorem, the level set \varphi^{-1}(L) is a C^2 sub-manifold.

Since the arc \gamma([-C, C]) is compact, we can find large R where \gamma([-C, C]) \subseteq B(\bar{0}, R).

Hence for |t| > R + L and s \in [-C, C], we have

\varphi (s, t) = d(\gamma(s), \gamma(t)) > d(\gamma(t), B(\bar{0}, R)) > L

The same holds with |s| > R + L and t \in [-C, C]

i.e. \varphi takes value >L in the shaded region below:

Hence for L \in \mathcal{L}, \varphi^{-1}(L) \cap \{x \leq y\} is a 1 dimensional sub-manifold, outside a bounded region it contains only two rays. We also know that the level set is bounded away from the diagonal since \varphi vanishes on the diagonal. By an non-ending arc argument, one connected component of \varphi^{-1}(L) must be a curve connecting the end points of the two rays. Establishes the claim.

Remarks: This problem happens to come up at the very end (questio/answer part) of Charles’s talk in the midwest dynamics conference last month (where he talked about our joint work about funnel sections). A couple weeks later Michal Misiurewicz e-mailed us a counter-example when the curve is not smooth (only continuous).

Initially I tried to use the above argument to get the length 1 needle. Everything works fine until a point where one has a continuum in the level set connecting the end-points of the two rays. We want the continuum to be path connected. I got stuck on that. In the continuous curve case, Michal’s counter-example corresponds to the continuum containing a \sin(1/x) curve, hence is not path connected.

I believe such thing cannot happen for smooth. The hope would be that the length 1 needle can slide through any C^2 (or C^1) curve. (Note that once the length 1 needle can pass through, then all length can pass through just by rescaling the curve.) In any case, still trying…

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! ^^

Types of hyperbolicity

January 12, 2010

Axiom A

1. Nonwandering set is hyperbolic

2. Periodic points are dense in the nonwandering set

Kupka-Smale

1. All periodic points are hyperbolic

2. For each pair of periodic points p, q of f, the intersection between the stable manifold of $p$ and the unstable manifold of q is transversal

Kupka-Smale theorem

The set of Kupka-Smale diffeomorphisms is residual in \mbox{Diff}^r(M) under C^r topology.

Morse-Smale

1.Axiom A with only finitely many periodic points (hence \Omega(f) is just the set of periodic points)

2.For each pair of periodic points p, q of f, the intersection between the stable manifold of p and the unstable manifold of q is transversal.

Anosov

All points are hyperbolic, i.e. there is a splitting of the whole tangent bundle such that under the diffeo, stable directions are exponentially contracted and unstable directions are exponentially expanded.

Relations:

Morse-Smale \subseteq Axiom A

Morse-Smale \subseteq Kupka-Smale

Anosov \subseteq Axiom A