I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$ that has these properties:

(1) When iterated $n$ times starting from some $p$, connecting the points in order with segments and closing last to first,

$$(p, f(p), f^2(p), \ldots, f^n(p), p)$$

results in a simple (non-self-intersecting) closed polygonal cycle $K$.

(2) When $K$ is viewed as a knot, it is highly tangled, e.g., it has large crossing number, or large unknotting number. The tangledness, however defined, should increase with $n$, the faster the better.

(3) These properties should hold for infinitely many $n$.

Expressed differently, I would like a way to generate
an infinite variety of increasingly tangled stick knots
via a simple function iteration.
My requirements are a bit loose, as I just want to
simply generate knotty examples.
Likely some weaving is known to accomplish this...?

(This question is related to an earlier question,
"Complexity of random knot with vertices on sphere.")