**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 ...

**4**

votes

**0**answers

106 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

281 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

464 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

1k 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 ...

**16**

votes

**2**answers

896 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

122 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

221 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

138 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

169 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 - ...

**1**

vote

**0**answers

112 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

164 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

725 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

159 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 ...

**19**

votes

**0**answers

594 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

144 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 $U$. ...

**7**

votes

**2**answers

147 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 ...

**19**

votes

**3**answers

927 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 = ...

**2**

votes

**0**answers

440 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

124 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 ...

**3**

votes

**1**answer

202 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

222 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

88 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 ...

**25**

votes

**2**answers

4k views

### 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 and also the following post.
Added : According to their ...

**0**

votes

**1**answer

273 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

275 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 ...

**3**

votes

**1**answer

150 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 ...

**2**

votes

**0**answers

183 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

159 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

252 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 ...

**1**

vote

**1**answer

109 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

92 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

113 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

267 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

145 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

284 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

101 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

62 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 ...

**4**

votes

**1**answer

618 views

### The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...

**2**

votes

**2**answers

113 views

### Original article about a theorem of Cartan on iterations of analytic functions

I'd like to know in which paper of H. Cartan I could find the following theorem :
Let $\Omega$ be a connected, open and bounded subset of $\mathbb{C}$. Let $a \in \Omega$ and $f \in ...

**1**

vote

**0**answers

62 views

### Single parameter bifurcations caused by a simple additive term

Note: I asked this question on Math.SE over two months ago, and it has not received any answers.
Motivation: A practical dynamical system is often described by an ODE that has a parameter that ...

**4**

votes

**3**answers

407 views

### Precise location of the Mandelbrot Bulb Attachment to the main Cardioid

Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...

**0**

votes

**1**answer

185 views

### Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ ...

**1**

vote

**2**answers

379 views

### $\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$

Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any ...

**6**

votes

**1**answer

135 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...

**2**

votes

**0**answers

91 views

### Automorphisms of Nilmanifolds

Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...

**1**

vote

**1**answer

74 views

### A question for the inverse orbit in the construction of conformal measure

Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset ...

**14**

votes

**2**answers

560 views

### Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...

**1**

vote

**1**answer

223 views

### Fixed point of a function on the circle

Consider a circle $C$ with radius of $r$, we place $m$ balls(treated as point) randomly on it, and each ball $i$ has the mass $m_i$. We define a function $\varphi:C\rightarrow C$ which maps $x\in C$ ...

**1**

vote

**0**answers

136 views

### Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there.
For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...