# Tagged Questions

**3**

votes

**2**answers

140 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

**1**answer

167 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

**5**answers

782 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

**0**answers

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

**2**answers

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

**1**answer

172 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

**0**answers

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

**0**answers

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

**3**answers

222 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

**2**answers

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

**1**answer

149 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

**1**answer

328 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

**4**answers

940 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

**1**answer

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

**1**answer

457 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 $M \to N$, ...

**5**

votes

**3**answers

478 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

**1**answer

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

**3**answers

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

**1**answer

354 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

**2**answers

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

**0**answers

280 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

**4**answers

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

**5**answers

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

**1**answer

252 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

**3**answers

599 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

**3**answers

992 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

**1**answer

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

**3**answers

567 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

**5**answers

781 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

**3**answers

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

**6**answers

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

**2**answers

628 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

**1**answer

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

**1**answer

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

**1**answer

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

**5**answers

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

**1**answer

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 ...