3
votes
2answers
134 views

Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
2
votes
1answer
162 views

Angle between two subspaces

Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space $T_xM=E^s(x)\oplus ...
18
votes
5answers
778 views

Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given point $p$ such that it gets "stuck" reflecting between two congruent mirror-disks. For why there is such a ray, see the (amazing!) answer ...
2
votes
0answers
79 views

Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...
9
votes
2answers
344 views

How to draw a Zoll surface?

I take into account that lots of questions on Zoll surfaces have already been asked on the forum. But I will stubbornly continue asking. Are there any chances to draw explicitely at least one Zoll ...
2
votes
1answer
171 views

Is it possible to approximate an area-preserving diffeomorphism by a sequence of conjugates of periodic rotations?

Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ ...
9
votes
0answers
342 views

Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...
1
vote
0answers
96 views

Extension of diffeomorphisms preserving bilateral bounds of the derivatives

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
2
votes
3answers
220 views

Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
7
votes
2answers
218 views

Transitive geodesics on closed surfaces of genus greater than one

A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose ...
1
vote
1answer
148 views

Relation between volume entropy and Hausdorff dim of limit set?

I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Patterson sullivan ...
1
vote
1answer
327 views

Ricci flow as a gradient flow and its Lyapunov function

In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then ...
19
votes
4answers
935 views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
12
votes
1answer
1k views

The Dedekind Eta Function in Physics

This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind $\eta$-function makes a cameo ...
6
votes
1answer
453 views

Topological equivalence of homotopic vector fields

Two (tangent) vector fields $X$ and $Y$ on oriented differentiable manifolds $M$ and $N$, respectively, are topologically equivalent, if there is an orientation-preserving homeomorphism $M \to N$, ...
6
votes
3answers
477 views

Given a vector field all of whose integral curves are closed, is the period a smooth function?

Disclaimer: The original question consisted of two parts. The first one has been answered negatively (see below the answers of Sam Lisi and Alejandro). It remains the second one. Background ...
6
votes
1answer
253 views

Ruelle inequality on a noncompact space

Does someone have a reference where the Ruelle inequality would be proved in the following context. Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, ...
3
votes
3answers
1k views

Flow of a Hamiltonian vector field

Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$, $$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$ and by the symplectic form also with ...
1
vote
1answer
349 views

How to deduce the existence of stationary points from fixed points of evolution maps?

This is probably a very elementary question. Nevertheless I decided to post it on MO. Consider a smooth manifold $M$ and a smooth complete vector field $v:M\rightarrow TM$. Consider an autonomous ...
6
votes
2answers
577 views

Linearly independent vector fields

Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$. Question: Is it possible to choose a cube $C$ around ...
2
votes
0answers
279 views

Partial feedback linearization (Control theory)

Greetings, I'm trying to understand a theorem about partial feedback linearization from a paper "On the largest feedback linearizable subsystem" by R.Marino (you can find it here: ...
12
votes
4answers
1k views

What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics. What are good examples to illustrate him of the usefulness of contact geometry in this context? On one hand the Hamiltonian ...
3
votes
5answers
1k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
4
votes
1answer
251 views

volume entropy for manifolds with boundary?

Recently I learn the volume entropy: http://en.wikipedia.org/wiki/Volume_entropy I wonder if one can define volume entropy for a manifold with boundary. Does anyone come up with a reasonable ...
13
votes
3answers
596 views

Codimension zero immersions

Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk? Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
22
votes
3answers
991 views

square root of diffeomorphism of R: does it always exist?

Let $f:\mathbb R\to\mathbb R$ be a smooth, orientation preserving diffeomorphism of the real line. Is it the case that there always exist another diffeomorphism $g:\mathbb R\to\mathbb R$ such that ...
3
votes
1answer
246 views

Injective flow lines of a vector field near a closed orbit

Suppose $v$ is a vector field on a manifold $X$ with flow $\phi^t$. Suppose $v$ carries a first integral $f:X \rightarrow \mathbb{R}$ (i.e. $f$ is constant on the orbits of $v$). Suppose ...
3
votes
3answers
566 views

Geometry and Integrability in Other Bundles

Background: Suppose $E=TM$ is the tangent bundle to some differentiable manifold $M^n$. If we specify some subbundle $D\subset TM$ (distribution of $k$-planes) then there are two natural situations ...
8
votes
5answers
773 views

What are the zero entropy invariant measures for an Anosov geodesic flow?

Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed ...
22
votes
3answers
2k views

Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
53
votes
6answers
5k views

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
4
votes
2answers
626 views

Birkhoff conjecture about integrable billiards

There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse. Integrability here might be ...
7
votes
1answer
351 views

Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
6
votes
1answer
411 views

Estimating the flow when we know the vector field

Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...
3
votes
1answer
646 views

How to shown that the Tangent Bundle of a vector space is a Vector Bundle

Hello, I have the following question about the tangent bundle $T_M = \bigcup_{p \in M} \{p\} \times T_p M$ defined on a manifold $M$ of class $C^r$ modeled on a normed space $X$. My problem is ...
5
votes
5answers
1k views

How can generic closed geodesics on surfaces of negative curvature be constructed?

As far as I understand it the closing lemma implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general? A concrete answer that dovetails ...
4
votes
1answer
311 views

Is there a name for this differential operator and/or its corresponding spectrum?

Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the ...