# Baker map-like problem

Let $$S =[0, 1]^2$$ denote the unit square in $$\mathbb R^{2}$$. For any subset $$A$$ of $$S$$ let $$A^{c}$$ denote its complement in $$S$$, and $$\overline{A}$$ its closure in $$S$$. Given a measurable map $$g: W \to V$$ between measure spaces $$(W, \mu)$$, $$(V, \phi)$$, we say it is equally scaling if there exists some $$r$$ in $$\mathbb R$$ such that for any measurable set $$K$$ in $$V$$, we have $$\phi(K) = r\mu(g^{-1}(K))$$.

Given any measurable bounded function $$f: S \to [0, \infty]$$ such that $$\int_{S} fd\mu= 1$$, does there exist

• a sequence of measurable sets $$A_n$$ in $$S$$ such that for every $$n$$ in $$\mathbb N$$, the closures $$\overline{A_n}$$, $$\overline{A_n ^{c}}$$ have Lebesgue measure $$\frac{1}{2}$$ $$\overline{A_n}$$, $$\overline{A_n ^{c}}$$ and $$S$$ are homeomorphic;

• and two sequences of equally scaling homeomorphisms $$h_{n}:$$ $$\overline{A_n} \to$$ $$\overline{A_n ^c}$$, $$s_n:A^c \to S$$ such that $$\lim_{n \to \infty} F_{n}...F_{0}(f) \to 1$$ uniformly a.e.?

Here $$F_{n}: M(S) \to M(S)$$ is defined by $$F_{n}(f) (x) = \frac{[fs_n^{-1}(x) + fh_n^{-1}s_n^{-1}(x)]}{2}$$ and $$M(S)$$ is the set of measurable functions on $$S$$.

• This is a pretty notation-heavy question. You might want to consider adding a simple example, e.g. with $f$ constant equal to $1$, to illustrate this. – Wojowu Jan 31 at 13:55