Consider the function $f:\mathbb{R}\to\mathbb{R}$ where for the base-3 expansion of $x\in\mathbb{R}$, we take "pseudo-random" iterations of a function that in the first iteration, replaces the base-3 digits zero with one; one with two, and two with zero.

In mathematical terms, if $\mathscr{F}:\left\{0,1,2\right\}\to\left\{0,1,2\right\}$ where $\mathscr{F}:=\left\{(0,1),(1,2),(2,0)\right\}$, such that equation:
$$\mathscr{F}^{k}=\underset{k\text{ times}}{\underbrace{\mathscr{F}(\mathscr{F}(\cdots\mathscr{F}(x)))}}$$ is the $k$-th iterations of $\mathscr{F}$, and $\left[\cdot\right]$ rounds to the nearest integer, we want:

$${f(x)=\left\{\sum\limits_{k=-\infty}^{\infty}{\mathscr{F}^{\big[|k\sin((a+1)k)|\big]}(a)}/{3^k}:a\in\left\{0,1,2\right\}\right\}}$$

I'm not sure if this covers an "infinite expanse of space". (This would be interesting to graph).

I assume none of the expected values in the **preliminaries** give this function a unique and finite expected value. (Hopefully, someone can check.)