7
votes

Accepted

### Examples of ODEs with complex constant coefficients and applications to physics?

A standard/classic example is a model for tippe top inversion. This model is a linear ODE with constant complex coefficients: $$
\ddot{z} + i \alpha \dot{z} + \beta \dot{z} + i \gamma z + \delta z = ...

- 5,570

5
votes

Accepted

### Does locally nilpotent imply nilpotent for continuous self-maps of intervals?

Yes, this implies that $f$ is nilpotent.
As explained in my comment, $f(0)=0$ because otherwise Sarkovski's theorem would give us other periodic orbits which, of course, won't visit $x=0$. We also ...

5
votes

### Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Here's a very partial answer. Let $R_G\subset G^2$ be the set of recurrent points of $F$.
Then for a compact group $G$, we have $R_G=G^2$ if and only if $G$ is profinite.
Proof. First, for $G$ the ...

- 52.8k

3
votes

Accepted

### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

This question was answered recently by Geschke, Grebík, and Miller:
S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," Proceedings of the AMS 149 (link to the arxiv version)
...

- 15.1k

2
votes

### What is the name for a point that is periodic to within $\varepsilon$?

TL;DR: There is no standard name in the topological dynamics literature for the property in question; however currently "recurrent point" seems to be the most common.
As Ville Salo wrote ...

- 228

2
votes

Accepted

### Borel summation and the Abel function of $e^z-1$

So, there's been some developments on the tetration forum, and I came across two papers:
Introduction to 1-summability and the resurgence theory, David Sauzin
https://hal.archives-ouvertes.fr/hal-...

- 306

1
vote

Accepted

### Lyapunov exponents of convergent sequence of matrices

So the answer is yes. A brief version is that it suffices to check this for the top exponent and then to use exterior powers to deduce the result for subsequent exponents.
(Slightly modified to make $\...

- 21.7k

1
vote

### A process of repeated convolution and conditioning and the resulting sequence of probability distributions

$\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}\newcommand{\Ga}{\Gamma}$We have
\begin{equation*}
f_{j+1}(x)=(f_j*f_j)(x)\,1(\|x\|<1)/A_j
\end{equation*}
for $j=0,1,\dots$ and $x\in\R^n$, ...

- 82.3k

1
vote

Accepted

### Infinite composition of continuous functions

Since you’re searching for names: there is a rich literature on discrete non-autonomous dynamical systems generated by an infinite family of continuous maps. Here "non-autonomous" means of ...

- 3,414

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