**0**

votes

**0**answers

196 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**2**

votes

**1**answer

306 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**13**

votes

**2**answers

530 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 ...

**42**

votes

**10**answers

6k views

### The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...

**23**

votes

**3**answers

2k views

### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

**10**

votes

**1**answer

744 views

### Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...

**25**

votes

**4**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**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 ...

**11**

votes

**3**answers

612 views

### How to draw a Zoll surface?

I take into account that lots of questions on Zoll surfaces have already been asked on the forum. But I will stubbornly continue asking. Are there any chances to draw explicitely at least one Zoll ...

**14**

votes

**1**answer

334 views

### Nonperiodic points of homeomorphisms of a ball

Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...

**5**

votes

**2**answers

528 views

### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**4**

votes

**1**answer

179 views

### Under what conditions can interval exchanges be approximated by periodic maps?

Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ).
Are there non-trivial examples of periodically ...

**1**

vote

**0**answers

129 views

### Why Poincare sphere compactification and not torus compactification

The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to ...

**1**

vote

**2**answers

214 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 $\chi^{\infty}(M)$ ...

**67**

votes

**11**answers

14k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**98**

votes

**1**answer

9k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**60**

votes

**6**answers

6k views

### Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual ...

**36**

votes

**6**answers

2k views

### Can we actually find any fixed points with Brouwer's theorem?

Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...

**21**

votes

**1**answer

1k views

### Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...

**16**

votes

**4**answers

2k views

### Is there a categorical treatment of dynamical systems?

Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$?
More precisely, is there a category whose ...

**26**

votes

**3**answers

885 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

**30**

votes

**6**answers

2k views

### Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been
studied?
What I have in mind is something like this:
Still Snell's ...

**19**

votes

**5**answers

854 views

### Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...

**14**

votes

**3**answers

1k views

### Prime factorization “demoted” leads to function whose fixed points are primes?

Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \,$$
i.e., exponentiation is ...

**12**

votes

**9**answers

3k views

### Book recommendation for ergodic theory and/or topological dynamics?

Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...

**5**

votes

**3**answers

1k 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 $F(r)=ar$ ...

**20**

votes

**2**answers

3k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this apparently 2 dimensional amazing problem . Best wishes for ...

**16**

votes

**2**answers

864 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 ...

**9**

votes

**0**answers

385 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

**2**answers

960 views

### sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...

**18**

votes

**6**answers

1k views

### Quantitative versions of ergodic theorem

Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...

**16**

votes

**2**answers

556 views

### “Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence.
Start with a polygon, and replace it by locating a point on every edge
a fraction $\alpha$ between the two endpoints. For ...

**15**

votes

**3**answers

1k views

### Do complex iterates of functions have any meaning?

Using a method explained in this answer it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th iterate, where $i=\sqrt{-1}$. Here ...

**14**

votes

**1**answer

673 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) = ...

**12**

votes

**2**answers

991 views

### 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 ...

**11**

votes

**0**answers

301 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 ...

**9**

votes

**4**answers

965 views

### Computing the centers of Apollonian Circle Packings

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-1, 28, 27, 23) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to ...

**8**

votes

**6**answers

1k views

### Examples of transformations which are weak-mixing but not strong-mixing

I was reminded of this topic by some of the answers to this question, where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of ...

**6**

votes

**4**answers

731 views

### Lie algebra admitting some hyperbolic automorphism is nilpotent

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $\mathbb{R}$ and $\phi:\mathfrak{g}\to\mathfrak{g}$ be a Lie algebra automorphism.
Viewing $\mathfrak{g}$ as a linear space and $\phi$ a ...

**19**

votes

**0**answers

444 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 ...

**13**

votes

**1**answer

441 views

### A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...

**12**

votes

**5**answers

2k views

### Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...

**11**

votes

**1**answer

715 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 ...

**7**

votes

**2**answers

188 views

### Well-definedness of single-particle smooth billiards flow

Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well ...

**5**

votes

**3**answers

719 views

### Simultaneous diophantine approximation

Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over ...

**2**

votes

**0**answers

426 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 ...

**2**

votes

**2**answers

323 views

### Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

**19**

votes

**2**answers

1k views

### How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. ...

**15**

votes

**3**answers

691 views

### Codimension zero immersions

Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...

**10**

votes

**2**answers

502 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 ...