# Tagged Questions

**4**

votes

**3**answers

298 views

### Generic absence of non-trivial first integrals of geodesic flows

Is it true that given a smooth manifold M (with or without boundary), a "generic" metric g on M does not possess any non-trivial (non-constant) first integral for the geodesic flow induced by g on the ...

**0**

votes

**0**answers

62 views

### Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:
$X$ has a finite number of singularities on $S^{2}$
The equator is invariant under flow of $X$
3.$g_{*}X=X\; ...

**2**

votes

**0**answers

57 views

### Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.
Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of ...

**7**

votes

**1**answer

92 views

### Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of ...

**1**

vote

**0**answers

105 views

### Why Poincare sphere compactification and not torus compactification

The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to ...

**5**

votes

**1**answer

113 views

### “The” kronecker foliation or “a” kronecker foliation?

Consider the following two foliations of torus:
1)The Kronecker foliation with slope $\sqrt{2}$
2)The Kronecker foliation with slope $\pi$
As I learn from the literature, these two foliations are ...

**1**

vote

**0**answers

106 views

### The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...

**1**

vote

**0**answers

63 views

### Smooth normal forms of vector fields (the path method)

I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of ...

**14**

votes

**3**answers

1k views

### Prime factorization “demoted” leads to function whose fixed points are primes?

Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \,$$
i.e., exponentiation is ...

**44**

votes

**5**answers

3k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

**0**

votes

**0**answers

48 views

### Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution
$\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$?
...

**4**

votes

**0**answers

98 views

### First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...

**11**

votes

**1**answer

256 views

### A strengthening of base 2 Fermat pseudoprime

If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$,
$k$ divides ${n-1 \choose 2k-1}$ because of the identity
${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether
an ...

**15**

votes

**0**answers

432 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

**14**

votes

**3**answers

965 views

### Not-lonely runners

The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...

**15**

votes

**2**answers

784 views

### If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...

**1**

vote

**1**answer

107 views

### Generic path in the space of vector fields on the orientable surface

Who knows, does a theorem of such form exist:
Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that
1) all fields in the family except finite number ...

**1**

vote

**2**answers

201 views

### A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows
For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...

**1**

vote

**0**answers

95 views

### Positively invariant set

In my research I need to show that the set $$S:=\left\{x_{i}\in [0,1] , y_{i} \in [0,c_{i}] ; i = 1,\dots , n; \sum_{i=1}^{n}(x_{i}+y_{i}) =1\right\},$$ is positively invariant with respect the ...

**3**

votes

**1**answer

141 views

### Are there $0$ entropy non-atomic invariant measures for $2x$ and $3x$ modulo $1?$

This question appears for first time (to my knowledge) in
×2 and ×3 invariant measures and entropy
Daniel J. Rudolph
Ergodic Theory and Dynamical Systems / Volume 10 / Issue 02 / June 1990, pp 395 - ...

**0**

votes

**0**answers

186 views

### A derivational approach to the Poincare Bendixson Theorem

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a smooth vector field on the plane. Assume that $K\subset \mathbb{R}^{2}$ is a compact subset (not necessarily invariant under ...

**1**

vote

**0**answers

109 views

### Name/terminology for a relationship between group actions

Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this ...

**6**

votes

**0**answers

137 views

### Measure theoretic entropy

I don't know if this is an elementary question or not. In what follows all maps are continuous
Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let ...

**23**

votes

**2**answers

655 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**6**

votes

**0**answers

148 views

### The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...

**18**

votes

**0**answers

526 views

### Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers.
Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...

**4**

votes

**1**answer

125 views

### Jacobian has small eigenvalues everywhere in an open set. Does this imply globally attracting fixed point in that set?

Let $U$ be an open subset of $\mathbb{R}^2$. For my purposes, we can assume that $U$ is just a rectangle. I have an infinitely differentiable map $M:U\to U$ that has a unique fixed point $p$ in $U$. ...

**6**

votes

**1**answer

109 views

### Angles and proportions occurring in L-system fractals

This is about properties of certain fractals defined by Lindenmayer systems, a.k.a. L-systems. Unlike “classical” fractals like Julia sets or the Mandelbrot set (the name “set” says it all), these ...

**18**

votes

**2**answers

722 views

### Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...

**1**

vote

**0**answers

209 views

### The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**3**

votes

**0**answers

116 views

### On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...

**2**

votes

**1**answer

141 views

### On the Birkhoff ergodic theorem for geodesic flows

Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$.
Let $f: ...

**6**

votes

**3**answers

155 views

### Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...

**0**

votes

**0**answers

56 views

### Does the number of zero eigenvalues correspond to the dimension of equilibrium manifold for nonlinear system?

Consider nonlinear dynamical system with $m \times m$ Jacobian matrix $J(x)$ that has $k$ zero eigenvalues for all $x$. The rest $m-k$ eigenvalues have negative real part for all $x$. Is that true ...

**17**

votes

**2**answers

2k views

+200

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see this related post
Added : According to their method, what of the following ...

**0**

votes

**1**answer

241 views

### Heisenberg group acts on the circle

Let $H$ be a Heisenberg group, i.e.
$$
H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.
$$
$H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of ...

**-4**

votes

**1**answer

231 views

### Proof of im/possibility of constructing any fractal by iterated function systems? [closed]

Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...

**0**

votes

**0**answers

102 views

### Regarding a paper relating surfaces and integrable mappings

This question has been posted on stackexchange, without answers.
I'm reading the paper "A classification of two-dimensional integrable mappings and rational elliptic surfaces". I have two questions:
...

**3**

votes

**0**answers

74 views

### Regarding the definition of S-flows over a category (given a monoid S)

(This question was originally directed to Simone Virili, referring to the answer http://mathoverflow.net/a/103840/2926, but could also be addressed to the greater community.)
I was wondering if you ...

**1**

vote

**0**answers

130 views

### The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
...

**6**

votes

**1**answer

135 views

### Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...

**5**

votes

**1**answer

224 views

### “strongly mixing” action on dimers?

In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity.
His paper is going to discuss the frequency of various "motifs" in ...

**0**

votes

**1**answer

97 views

### Non-convergence of ergodic measures with positive entropy

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let ...

**1**

vote

**1**answer

84 views

### Example of non-convergence of iteration of measures

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes ...

**0**

votes

**1**answer

110 views

### Is the following conjecture regarding rotational iterates of collection of points on a circle true?

Let $\varphi_1,\dots,\varphi_k$ be a set of angles in $[0,\pi)$. For each $n\in\mathbb{N}$, let
$$ \rho(n):=\min_{j=1,\dots,k} \{n\cdot \varphi_j \mod \pi \}. $$
Is it true that for infinitely many ...

**8**

votes

**2**answers

246 views

### The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...

**4**

votes

**0**answers

126 views

### Kopell's lemma in higher dimensions

Kopell's lemma states the following: suppose f and g are commuting $C^2$ diffeomorphisms of $[0, \infty)$ such that $f(x) < x$ for every $x > 0$ and $g(p) = p$ for some $p > 0$. Then $g$ is ...

**7**

votes

**2**answers

273 views

### Wait time to grid network disconnection with failing edges

Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...

**1**

vote

**1**answer

97 views

### Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities".
Let $f\colon \mathbb{C}^n\to ...

**0**

votes

**0**answers

52 views

### Is an odometer action on a product space always conjugate to its inverse by an involution?

This is a follow on question from
Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse?
Given a measure $\mu$ on the product space $X = \prod_{i=1}^\infty ...