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 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-...
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 $x\in\R^n$, ...
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 ...

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