User olga - MathOverflow

Reference question: Poncelet theorem

2013-02-07T11:14:08Z

A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in Russian jail so he didn't write it down. Could anybody give me a reference on a first book or article where the proof was actually given?

I want to make a correct reference in the article I am writing. Moreover, I am also thinking about mentioning a greek mathematician Proclus who lived in the 5th century BC and proved that once a line in an ellipse is tangent to some other smaller ellipse, its reflection is tangent to the smaller ellipse too.

Bessel and Neyman functions:ordering the zeroes

2012-10-12T09:42:40Z

I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could give me a good reference, the question is quiet classical.

Let us consider a Bessel equation $$ r^2 R'' + r R' + (r-k^2) R = 0 $$

with the conditions on the boundaries $R(\sqrt{\lambda})=R(2\sqrt{\lambda})=0$. This comes from the equation of oscillation of an annulus with fixed boundaries. The question is to find the frequencies of oscillation in an increasing order, that is just find such $\sqrt{\lambda}$ that the system above holds.

We know that the general solution of the equation is the linear combination between Bessel and Neyman functions $J_k$ and $Y_k$, $k \in \mathbb{N}\cup {0}$ so the question is reduced to ordering the zeroes of $J_k$ and $Y_k$. Is there an answer in a form of explicit sequence or, at least, how many first terms do we know?

For a group with one end does the property of connected spheres follow?

2012-07-04T05:23:26Z

One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group $G$ with generators $g_1, \ldots g_n$. The notion of a length of an element $g$ can be given as a length of a minimal representation of $g$ in terms of generators.

Let me recall a classical definition of a group with one end. If $G$ is a connected, locally path connected, locally compact topological space. Then $G$ has one end if given a compact subset $K \subset G$, there is a compact set $L$: $K \subset L$ such that for any $x,y \in G \setminus L$ there is a path in $G \setminus K$ joining $x$ and $y$.

For a group with one end let us define a property, that rises from the studies of group actions on the circle, namely a property of connected spheres: For any ball $B_R$ let us take a nonbounded component of its compliment $(B_R)^c_{\infty}$: this component is unique since our group has one end.

Then we say, that a group has a property of connected spheres if there exists $C>0$ such that for any ball $B_R$ of radius $R$ the points in the fiber $(B_R)^c_{\infty} \cap B_{R+C} $ could be connected by the path in the group, i.e. for any $x,y$ in the fiber $(B_R)^c_{\infty} \cap B_{R+C} $ there exists a finite number of group elements, such that $x=gy$ and $g$ is a word in the alphabet $g_1, \ldots g_n$ and all the steps still lie in the fiber considered between the spheres.

The question is if for a group with one end the property of connected spheres holds automatically or not, and what are the examples in the case?

Looking for at least one beautiful and not too technical result in asymptotic group theory

2012-04-18T17:08:33Z

We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this beautiful area, and the thematic is considerably wide: the main idea is that somebody presents a theorem, which is proved beautifully and has a lot of applications. For example, we have proved de Finetti theorem (see Wikipedia article on it), some ergodic theorems for group actions, and Wigner's semi-circle law. I'm looking for something as beautiful and as useful, and not very difficult for being able to talk about it for 2 hours on our seminar. I hope you could give me some hints.

I repeat, that the area of problems we discuss is wide and I am not able to define it strictly, so any ideas are welcome. But generally, the methods we use are either dynamical either probabilistic. I had a thought to tell some of Grigorchuk's ideas of constructing the groups of intermideate growth, but I found those proofs technical.

A concept of dynamical coherence

2012-02-29T11:30:05Z

I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo Carrasco, Compact Dynamical Foliations 2010), some call it strong and unnatural (see the work of Amy Wilkinson and Keith Burns Dynamical coherence, accessibility and center bunching). The definition which is the most common is that local cental-unstable $E^{cu}$ and center-stable $E^{cs}$ bundles integrate to foliations $W^{cu}$ and $W^{cs}$. Let us suppose, that in a normally hyperbolic case, i.e. when we already have the $E^c$ that integrates to a foliation F, at which some diffeomorphism is hyperbolic.

My question is how the normally hyperbolic (i.e. partially hyperbolic on foliation) system could be dynamically incoherent and is this concept somewhat related to the concept of local product structure?

My question is, what is a simplest example of a normally hyperbolic foliation when $E^cu$ and $E^cs$ do not integrate to foliations? And how "often" does it happen in the world of normally hyperbolic foliations?

PS. Updated after a useful remark of Rafael Potrie, the definition of a dynamical coherence is now more precise.

A measure theory question

2012-03-29T06:34:30Z

Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:

On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does every measure zero set equal a countable union of the sets of less than full Hausdorff dimension?

For a diffeomorphism $f$ of $M$ and a continuous function $\varphi$ on $M$, define $$\overline \varphi = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_0^{n-1} \varphi \circ f^k(x).$$ Then the Birkhoff theorem asserts that for almost all $x$, $\overline \varphi \rightarrow \int_M \varphi, n \rightarrow \infty$. But consider the set $K_{\alpha}$ of $x$ where $$\alpha \leq |\overline \varphi - \int \varphi|.$$ So Birkhoff says $\mu(K_\alpha)=0$, but what about the Hausdorff dimension of $\mu(K_\alpha)=0$? For some diffeomorphisms, for example hyperbolic maps, it was proven that $\dim_H K_\alpha < \dim X$. That fact gives a rise to my question. I expect a negative answer, but I can not find a counterexample.

Comment by Olga

2013-03-20T14:21:23Z

The manifold should be smooth, otherwise there is an example in Besse book with closed geodesics, all of the same length except one ("an equator").

Comment by Olga

2013-02-15T18:34:59Z

Yes, you are right, I wanted to ask about this theorem.

Comment by Olga

2013-01-24T21:15:34Z

I love these jokes because they connect mathematics to culture.

Comment by Olga

2012-10-30T07:14:26Z

This is a question for math.stackexchange.com

Comment by Olga

2012-10-08T08:15:52Z

You should at least verify your problem: we know that the order of any element of the group is a divisor of $|G|$. So maybe you mean $|G|=n$? Otherwise, there is no sense.

Comment by Olga

2012-07-14T10:41:49Z

Thank you very much for your valuable comments - if the property holds for at least the finitely presented groups, it's already very well.

Comment by Olga

2012-07-04T07:53:23Z

Thank you very much for your comment: for sure, $z_k$ have to lie in the fiber considered. Secondly, what is important, a case of dead ends is not an issue here - I corrected a definition of connected spheres after your remark, see a new one above.

Comment by Olga

2012-05-14T21:21:54Z

I did it with 10+3+2 and I know the answer in this case.=)

Comment by Olga

2012-05-05T19:24:53Z

Thank you for your answer: so at this point the DC concept is still intriguing and not completely studied.

Comment by Olga

2012-04-25T07:12:47Z

I can not say that I got more information following the link..

Comment by Olga

2012-04-18T19:54:34Z

Thank you for your interesting comment, and the proof that Lueck gives of this theorem, is short and beautiful, and uses basic concepts in topology. But I do not see here the big and interesting story for a talk. I won't be able to make a survey on all the geometrical group theory in 2 hours, and also I think the article is in the style of survey, I'm searching for something more short, not as global, but beautiful. Just for everybody not to forget my talk.

Comment by Olga

2012-04-12T20:42:16Z

I'm sorry for the impatience - but did anything change in these 4 months? That's a theme of a great interest!

Comment by Olga

2012-04-03T19:54:15Z

Thank you so much for your answer, as far as I understand, for $\mathbb{R}^n$ we have to consider $f_n(x)=x^n \log (e/x)$ and everything will work as it has to. And the number e in the definition of f doesn't value much - we can take any positive number we want.

Comment by Olga

2012-03-29T08:49:16Z

I mean expresing the set of measure zero as a countable (or less) union of the sets of a Hausdorff dimension less than full. In the remark, $\alpha$ could be taken $\frac{1}{n}$. So I do not accept such an easy solution. I hope, now it's more precise.

Comment by Olga

2012-03-28T17:46:12Z

Maybe I didn't write explicitly. The integrability of $E^c$ is not the definition I wanted to give of dynamical coherence: the one Hirsch, Pugh, Shub give is that the local center unstable manifold of one plaque and local center stable manifold of another plaque intersect in a subplaque of a foliation. It's equivalent to the fact that W^cu and W^cs are foliated by W^c, i.e. by our leaves. As far as I understand, you gave an explicit answer to the question of the problem of the integrability of a center bundle. My question is, whether DC holds supposing that the center foliation exists.