Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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86
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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
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630 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 ...
20
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598 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
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468 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 ...
18
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723 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
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485 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
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337 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
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164 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
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398 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 ...
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351 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
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343 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
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349 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
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282 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
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322 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
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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
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409 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
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174 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
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309 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
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202 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
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395 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
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91 views

Volume preserving conjugacy in Hartman-Grobman theorem

According to Hartman-Grobman theorem, a $C^1$ germ of diffeomorphism $f$ on $\mathbb{R}^n$ at a fixed point $x$ whose differential $Df(x)$ is hyperbolic is always $C^0$-conjugated to its differential, ...
7
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248 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
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124 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
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817 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 ...
7
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0answers
162 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
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0answers
170 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
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219 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
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205 views

Topological entropy and periodic sequences of a subshift

Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$. ...
6
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0answers
121 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
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0answers
251 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
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177 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
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169 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
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264 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
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0answers
253 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
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278 views

The Arnol'd family of circle maps - origins and density of hyperbolicity

$\newcommand{\R}{\mathbb{R}}\newcommand{\Z}{\mathbb{Z}}$ The Arnol'd family or standard family of circle maps is defined by $$F_{\mu_1,\mu_2}:\R/\Z\to\R/\Z;\quad t\mapsto t + \mu_1 + \mu_2\sin(2\pi ...
6
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0answers
313 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
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0answers
582 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
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0answers
419 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
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0answers
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
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88 views

The autonomous diameter of the group of Hamiltonian diffeomorphisms of the standard symplectic space

The autonomous norm of a Hamiltonian diffeomorphism $h$ of a symplectic manifold $(M,\omega)$ is the smallest number $n\in \mathbf N$ such that $h=a_1\dots a_n$, where $a_i$ are autonomous ...
5
votes
0answers
102 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
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0answers
67 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
0answers
74 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
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95 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
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61 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
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0answers
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
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0answers
227 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
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0answers
98 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
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0answers
308 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
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114 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}} ...