Suppose I have a discrete dynamical system defined as a map $f: X \to X$ where $X = [0,1]^N$ i.e. the $N-$dimensional unit hypercube.
The map $f$ is discontinuous (specifically it is piecewise affine, as it includes a Heaviside function), and I am attempting to show that this map is not chaotic in the sense that there does not exist a scrambled set $S$ as defined as in Li-Yorke that is dense in a region of $X$. An uncountable set $S$ is scrambled iff for any pair $x, y \in S$:
- $\lim_{n\to\infty}\sup |f^{(n)}(x) - f^{(n)}(y)| > 0$
- $\lim_{n\to\infty}\inf |f^{(n)}(x) - f^{(n)}(y)| = 0$
So far I have shown $f$ does not exhibit any periodic orbits in $S$. That is, there is no $p > 1 \in \mathbb{N}$ and $x \in X$ such that $f^{(p)}(x) = x$.
It seems there should be an easy step to show that scrambled sets are impossible, since Li-Yorke Chaos by definition includes both an uncountable scrambled set and arbitrarily many periodic orbits, so there's clearly an intimate link between periodic orbits and scrambledness (i.e. example).
However everything I have seen so far offers sufficient conditions for Li-Yorke and not necessary conditions for scrambled sets only.
TLDR: I was hoping to find some reference that shows scrambled sets imply the existence of at least one period, in which case I have my result by contradiction.