4

A sequence is called lacunary if, in your terminology, its minimum growth rate is strictly greater than $1$. The following articles prove that every lacunary sequence is remote. If I understand your question correctly, this means that the (only) maximal $M$ you seek is the interval $(1,\infty)$.
Pollington, Andrew D., On the density of sequences $\{n_k\xi\}...

3

The definition which you wrote can be translated in words like this.
You evaluate the solution at time $t$. Then you fix a small distance $\epsilon>0$. Then you can find a time length $\ell(\epsilon)$ (independent of $t$) having the following property: within $\ell$ time units from whatever instant $a$, you will find an instant $\tau$ such that the ...

ap.analysis-of-pdes ds.dynamical-systems sg.symplectic-geometry integrable-systems almost-periodic-function

2

Let $\mu$ be Lebesgue measure on $S^1$, and $\delta_P$ be a point-mass at a point $P \in S^1$.
Then there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the orbit of $P$.)
This distribution seems like it should count as "well-behaved." Its support is connected, and both $\mu$ and $\delta_P$ themselves arise ...

1

I suspect that your confusion stems from misinterpreting the norm on $L^2(X|Y)$ in your definition 2. You should note that $\mathbb{E}_Y$ takes $L^\infty(X)$ to $L^\infty(Y)$ and the new norm taken on $L^\infty(X)$ is with respect to the $L^\infty$-norm on $L^\infty(Y)$, not the $L^2$-norm. That is $\|f\|:=\|\mathbb{E}_Y(f\cdot\overline{f})\|_\infty^{1/2}$ ...

1

I'm not sure about what your requirements are. In particular, I didn't get whether you need that the phenomenon arises from a chaotic oscillator, or it was just your try to search among those.
However, if I understood well your requirements, the phenomenon is quite mundane in the solutions of nonlinear PDEs exhibiting soliton character. For instance, if you ...

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