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...?
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Tangled Knot][1]<br />
(This question is related to an earlier question,
"[Complexity of random knot with vertices on sphere][2].")


  [1]: https://i.sstatic.net/z38fk.jpg
  [2]: https://mathoverflow.net/questions/54412/