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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*The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as the open problem in General Relativity? I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper, "Die ...
Joseph O'Rourke's user avatar
18 votes
3 answers
2k views

Connection between properties of dynamical and ergodic systems

While studying topological and ergodic dynamics, I've got quite perplexed by the different properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, uniquely ...
amir sagiv's user avatar
18 votes
2 answers
2k 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 ...
Vesselin Dimitrov's user avatar
16 votes
3 answers
2k views

Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system. fact 1 Consider the "tent map" f:[0,1]→[...
Pietro Majer's user avatar
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16 votes
2 answers
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Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I found this in claim a ...
Ilya Grigoriev's user avatar
15 votes
3 answers
1k views

Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...
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15 votes
3 answers
773 views

Neural networks over gadgets other than $\mathbb{R}$

Recently, I learned that neural networks (NN) can be defined over fields other than $\mathbb{R}$: for example, Khrennikov and Tirozzi wrote a paper in 1999 (!) on $p$-adic neural networks, or neural ...
xuq01's user avatar
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15 votes
2 answers
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Are rounded rectangle billiard dynamics ergodic?

Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.             (Image from Microwave_billiards_and_quantum_chaos.)...
Joseph O'Rourke's user avatar
15 votes
1 answer
1k views

In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem. Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
Sebastien Palcoux's user avatar
14 votes
2 answers
1k views

Polygonal billards programs

I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure. (source) It was a good exercise, but at this point I ...
john mangual's user avatar
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14 votes
5 answers
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Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in http://...
Jorge Vitório Pereira's user avatar
13 votes
1 answer
1k views

Conjectures on iterated polynomial maps on finite fields

Let $p$ be a prime, and consider the sequence $x_0, x_1, \dots$ of elements of the finite field $\mathbf F_p$ given by $x_0 = 0$ and $x_{i+1} = x_i^2 + 1$ for all $i \ge 0$. This sequence must ...
Mark Dickinson's user avatar
13 votes
1 answer
586 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 ...
Sebastien Palcoux's user avatar
13 votes
2 answers
2k views

Simple example of renormalization

As far as I understand, the RG theory, or functional RG theory is a mathematical tool for moving in the "scale dimension". The tool can be used for calculation of Feigenbaums constant (e.g. mentioned ...
science.nest's user avatar
13 votes
3 answers
865 views

Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$. Does anyone know of an ...
12 votes
5 answers
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Computing the centers of Apollonian circle packings

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to ...
john mangual's user avatar
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12 votes
3 answers
384 views

Dynamics in one matrix variable

Are dynamical systems $$X \mapsto F(X)$$ studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined noncommutative)...
Qfwfq's user avatar
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12 votes
2 answers
1k views

Rational numbers with dense orbits in [0,1] under iteration by f(x)=4x(1-x)

Let $f(x)=4x(1-x)$. For which rational numbers $r\in [0,1]$ is the sequence $f^n(r)$, $n\in \mathbb N$, dense in $[0,1]$ ? $(f^n(r)=f\circ f\circ ...\circ f(r)$ n times) I would be happy to find a ...
coudy's user avatar
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12 votes
2 answers
931 views

Birkhoff ergodic theorem and the measure of the bad points

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My ...
user39115's user avatar
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12 votes
1 answer
470 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
13829's user avatar
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11 votes
4 answers
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Routh-Hurwitz for eigenvalues

The Routh-Hurwitz criterion provides a convenient test, even for hand calculation, of whether a polynomial with real coefficients has all its roots in the left half plane. I'm wondering about a ...
David Schaeffer's user avatar
11 votes
2 answers
885 views

Random circle rotations

Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever $\...
Vaughn Climenhaga's user avatar
11 votes
1 answer
267 views

Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geodesic flow is recurrent?

Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = ...
italo lira's user avatar
11 votes
1 answer
3k views

Understanding the application of two inequalities?

I am reading the paper "The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps" by Driss Kiouach and Yassine Sabbar. I have two ...
Math's user avatar
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11 votes
1 answer
665 views

Do quantum "Sure-Shor separators" have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)

Aram Harrow asked: "Is there any place this is written up?" Update  Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...
John Sidles's user avatar
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11 votes
0 answers
212 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}(...
Marc's user avatar
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11 votes
1 answer
351 views

Nonperiodic points of piecewise-linear homeomorphisms

Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $T^n(x)=x$...
James Propp's user avatar
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11 votes
2 answers
2k views

Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process of creating a midpoint piecewise geodesic polygon converges on a surface $S \subset \mathbb{R}^3$. $S$ may be assumed smooth, closed, and ...
Joseph O'Rourke's user avatar
11 votes
1 answer
2k views

Conley Theorem (or fundamental theorem of dynamical systems)

Notations: $\mathcal{R}(f)$ denotes the chain recurrent set of $f$ $NW(f)$ denotes the non wandering set of $f$ $R(f)$ denotes the recurrent set of $f$ ($x: x\in \omega(x)$) Given compact ...
user avatar
10 votes
1 answer
486 views

Ping-pong progress through a quincunx

A quincunx or Galton board consists of staggered pegs from which ping-pong balls bounce and eventually display a binomial / normal distribution in catch-bins. I am wondering if the downward progress ...
Joseph O'Rourke's user avatar
10 votes
1 answer
551 views

Periodic billiard paths in hyperbolic triangles

It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path, one obeying the reflection law that the angle of incidence equals the angle of reflection. ...
Joseph O'Rourke's user avatar
10 votes
0 answers
532 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 $r$,...
alvarezpaiva's user avatar
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10 votes
1 answer
661 views

Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
Ethan Splaver's user avatar
9 votes
0 answers
365 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 ...
Joseph O'Rourke's user avatar
9 votes
1 answer
781 views

Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field. Does there exist a conformal factor $c$ ...
Ali's user avatar
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9 votes
1 answer
614 views

Reeb flows on $S^3$ versus volume preserving flows

Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field? Recall that $v$ is a Reeb vector field if there exists a contact $...
aglearner's user avatar
  • 14k
9 votes
1 answer
335 views

Relaxation of notion of positive definite function

A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
Hans's user avatar
  • 2,863
9 votes
1 answer
364 views

Integrality of iterates of rational functions

Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of ...
Kimball's user avatar
  • 5,709
9 votes
1 answer
696 views

When does a Lagrangian dynamical system have an equivalent Hamiltonian description?

Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$ (i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
Konstantinos Kanakoglou's user avatar
9 votes
1 answer
773 views

Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure. EDIT: The answer below is unsatisfactory, and possibly ...
Douglas Sirk's user avatar
8 votes
0 answers
151 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 ...
Dmytro Taranovsky's user avatar
8 votes
0 answers
497 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
Ali Taghavi's user avatar
8 votes
1 answer
411 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 ...
Sebastien Palcoux's user avatar
8 votes
3 answers
777 views

What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
Saal Hardali's user avatar
  • 7,549
8 votes
1 answer
466 views

Metrics on torus without closed contractible geodesics

It is easy to see that any closed geodesic on a flat 2-torus is noncontractible. Further the same holds true for a torus of revolution. Indeed either a closed geodesic is a meridian and therefore ...
Anton Petrunin's user avatar
8 votes
3 answers
5k views

Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...
camomille's user avatar
  • 541
8 votes
1 answer
1k views

Beautiful examples of arc-like continua

A continuum is a nonempty compact, connected metric space. A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
8 votes
1 answer
226 views

Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?

Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...
Will Sawin's user avatar
  • 135k
8 votes
1 answer
623 views

Aproximating dynamical systems by intrinsically ergodic systems

Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that $(X,f)$ is approximated from below by a sequence of compact metric spaces $(X_i)_{i \geq 1}$ and a sequence of ...
Rafael Alcaraz Barrera's user avatar
8 votes
2 answers
399 views

Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations"

Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a ...
Charles Staats's user avatar