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Questions tagged [ds.dynamical-systems]

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

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771 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 ...
23
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0answers
644 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 ...
22
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0answers
1k views

Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only. A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$, $$ C^{-1} \...
18
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0answers
586 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 $n=...
18
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0answers
796 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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397 views

Trapping lightrays with segment mirrors

Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? I posed this question in several forums before (e.g., here and in an ...
13
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0answers
338 views

Diffeomorphisms of $\mathbf R^n$

Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies. Question: Is there an example to ...
12
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0answers
215 views

If two group actions lead to the same orbifold, are they conjugate?

In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...
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413 views

Strange formula in arithmetic dynamic

Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two. We discovered the following operator which acts on the space of polynomials (or ...
11
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0answers
211 views

Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
11
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197 views

Upper bounds for lattice points in orbits, and representations of binary quadratic forms

Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $n\geq 3$ and $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{n\times 2}$, ...
11
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163 views

Shift invariant measurable selection theorem

Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
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175 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 ...
11
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0answers
368 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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0answers
345 views

horocycle flow and the prime number theorem

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this. ...
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0answers
463 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 $r$,...
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406 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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372 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 ...
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460 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 ...
9
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0answers
144 views

Fixed-points of a topological circle action

Suppose the circle group $G = S^1$ acts on $X$. If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
9
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0answers
217 views

Mackey's Program on Algebraic Ergodic Theory

I knew about Mackey's Program from Arnold's book Random Dynamical Systems and it referred to K. Schmidt's book Algebraic Ideas in Ergodic Theory, which was published in 1990. However, that is the ...
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334 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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379 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 $g_1, \ldots g_n$....
9
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0answers
364 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 ...
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188 views

Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary. Let $M^2$ be a compact connected 2-manifold and $\...
8
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0answers
364 views

Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
8
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0answers
696 views

Weak$^*$ convergence of measures vs. convergence of supports

Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
8
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0answers
238 views

when is the Brun continued fraction periodic?

I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun: $$ (x,y,z) \mapsto \text{sort}...
8
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0answers
440 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 $x_k, |k|>n$. ...
8
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0answers
116 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, ...
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0answers
187 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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0answers
230 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 counter-examples:...
8
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0answers
441 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 ...
8
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0answers
229 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 $Q$ ...
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175 views

Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
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0answers
147 views

Partition the rationals with respect to a multivariate polynomial which sends classes to classes

Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial. Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
7
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0answers
231 views

The graph of Rule 110 and vertices degree

Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete): It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...
7
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0answers
471 views

Limit cycles as closed geodesics(2)

Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
7
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0answers
184 views

Smooth dependence on parameters of invariant manifolds

This may be considered a followup question to smooth dependence of stable manifold on parameters. In particular, I would like to know in more detail how smooth parameter dependence of a (generalized) ...
7
votes
0answers
210 views

integrality of a Riccati-type equation

The following is a problem we were unable to prove and left stated in the paper "Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846. Define ...
7
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0answers
125 views

Is there any example of a endomorphism of a Lie group that has recurrent points with non-compact orbit closure?

Is there any example of a continuous endomorphism of a Lie group that has recurrent points with non-compact orbit closure?
7
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0answers
283 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
0answers
155 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 ...
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0answers
1k 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
342 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 ...
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0answers
212 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
0answers
313 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 t);...
7
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0answers
173 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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0answers
163 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 *?
6
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0answers
58 views

Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$  ($r>0$ is distance from the origin). If you start at the origin ...