**84**

votes

**0**answers

6k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**21**

votes

**0**answers

615 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 ...

**19**

votes

**0**answers

563 views

### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel ...

**18**

votes

**0**answers

706 views

### Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...

**16**

votes

**0**answers

443 views

### Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...

**15**

votes

**0**answers

471 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 ...

**11**

votes

**0**answers

327 views

### Does Langton's ant cover every n by 6 gridded torus?

This post follows this other post about times cover by Langton's ant of $n$ by $n$ gridded torus.
For $n$ by $n$ gridded torus, I've checked for $n \le 1000$ that the ant covers all. This fact needs ...

**11**

votes

**0**answers

162 views

### Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...

**10**

votes

**0**answers

348 views

### flexibility of almost contact ``Reeb'' vector fields

New version of the question:
Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form ...

**10**

votes

**0**answers

338 views

### Rational maps whose complex conjugate equals a PGL conjugate

Let $f(z)\in\mathbb{C}(z)$ be a rational function, and let $\bar{f}(z)$ denote the function obtained by taking the complex conjugate of the coefficients of $f$. I am interested in maps $f$ for which ...

**10**

votes

**0**answers

347 views

### Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing

Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...

**9**

votes

**0**answers

271 views

### Periodic orbits of a spinning ball in a square

Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a rough ball, one
whose high friction causes it to pick up ...

**9**

votes

**0**answers

392 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 ...

**9**

votes

**0**answers

312 views

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

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 ...

**9**

votes

**0**answers

343 views

### How aspherical can a Gömböc be?

A Gömböc is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the ...

**9**

votes

**0**answers

404 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**8**

votes

**0**answers

173 views

### Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...

**8**

votes

**0**answers

302 views

### Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...

**8**

votes

**0**answers

199 views

### Billiards with incompatible regions

An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong ...

**8**

votes

**0**answers

388 views

### What is the “category of bifurcations”?

While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added):
In this paper we show that every bifurcation set contains a copy of the boundary of the ...

**7**

votes

**0**answers

232 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

**7**

votes

**0**answers

120 views

### Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...

**7**

votes

**0**answers

160 views

### Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...

**7**

votes

**0**answers

169 views

### A model of self-organizing behavior

I'd just like to know if the following model has received any attention:
A state at discrete time $t$ consists of a function $S_t:{\Bbb Z}^2\rightarrow S^1$.
So view each cell $c$ (element of ${\Bbb ...

**7**

votes

**0**answers

218 views

### Monotone invariants of braid forcing

Let $\phi$ be a diffeomorphism of the unit disk $D^2$, fixed on the boundary, and suppose that $Q$ is a finite subset of the interior permuted by $\phi$. The isotopy class of $\phi$ relative to $Q$ ...

**6**

votes

**0**answers

77 views

### How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
...

**6**

votes

**0**answers

113 views

### How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?

Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the ...

**6**

votes

**0**answers

231 views

### Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...

**6**

votes

**0**answers

171 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 ...

**6**

votes

**0**answers

165 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 ...

**6**

votes

**0**answers

255 views

### Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and
$T:X\rightarrow X$ be an ergodic measure preserving isometry.
Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...

**6**

votes

**0**answers

252 views

### Invariant curves of rational functions

Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function
of degree at least 2 which
maps $\gamma$ onto itself homeomorphically. The following examples of such ...

**6**

votes

**0**answers

310 views

### “topological” conjugacy of group automorphisms

In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and ...

**6**

votes

**0**answers

563 views

### Reference request: natural extensions of topological dynamical systems

I am currently writing a paper in which I need to use the following fact: if $T \colon X \to X$ is a uniquely ergodic transformation of a compact metric space, and $\mathcal{A}$ is a continuous ...

**6**

votes

**0**answers

416 views

### Differential equation of line tangent to caustics

This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...

**6**

votes

**0**answers

155 views

### Unbounded energy growth in a Hamiltonian system

Does there exist an orbit with unbounded velocity in the system
$\ddot x = (-1)^{[t]+[x]}$, where $[*]$ denotes the integer part of *?

**5**

votes

**0**answers

65 views

### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.
I have been interested in some technical properties of the ...

**5**

votes

**0**answers

90 views

### Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...

**5**

votes

**0**answers

63 views

### Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims

I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...

**5**

votes

**0**answers

73 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**5**

votes

**0**answers

92 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

**5**

votes

**0**answers

55 views

### Is there a name for a bifurcation where a stable fixed point bifurcates into three fixed points: stable, unstable and saddle?

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by
$$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & ...

**5**

votes

**0**answers

106 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**5**

votes

**0**answers

222 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**5**

votes

**0**answers

90 views

### Quantitative approximation of invariant measures by periodic ones

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...

**5**

votes

**0**answers

295 views

### Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n ...

**5**

votes

**0**answers

112 views

### Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup:
Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.
For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} ...

**5**

votes

**0**answers

288 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**5**

votes

**0**answers

120 views

### Unit eigenvalue of the linearized Poincare return map

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...

**5**

votes

**0**answers

185 views

### Is Akcoglu's theorem for power bounded positive operators still an open problem?

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...