This is a continuation of this question.
Let $f: [0,1]^2 \to [0,1]^2$ be a homeomorphism of the unit square. By a nonuniform $n$-grid I mean a collection of smaller squares with corners of the form $a/2^n$ for $a = 0,1,\ldots, 2^{n}$ such that
- The smaller squares cover the unit square
- The smaller squares intersect only along their edges
I would like to take two nonuniform grids fine enough that $f$ can be approximated by swapping the grid elements
Formally, let $\epsilon >0$ be given. Does there exist some $n \in \mathbb N$ and nonuniform $n$-grids $F,G$ and bijection $\sigma:G \to F$ such that for each $x \in [0,1]^2$ we have $d(f(x),y) < \epsilon$ for every grid element $A$ with $x \in A$ and $y \in \sigma(A)$?
Edit: This used to read: "$d(f(x),\sigma(A)) < \epsilon$ for every grid element $A$ with $x \in A$" which can be ensured by taking a single grid element.
This looks a little like simplicial approximation but I can't find anything about simplicial approximations of homeomorphisms being bijective.