Define $\xi\in C^1([-1,1]\times[-1,1])$ such that $ \int\limits_{-1}^1 \xi(x,y)\,dy = 1 $ for all $x\in[-1,1]$ and $\xi\geq 0$. Put $A_0 = [0,1]$ and $A_{n+1} = \left\{x\in A_n:\int\limits_{A_n}\xi(x,y)\,dy = 1\right\}.$
Are there methods to find the rate of convergence of $\lambda(A_{n}\setminus A_{n+1})$ where $\lambda$ is a Lebesgue measure? Maybe you can refer me to the relevant literature?
There are at least two kinds of situation:
for some $N$ we have $A_N = \emptyset$, then $A_{N+1} = A_N$ and $\lambda(A_n\setminus A_{n+1}) = 0$ for $n\geq N$.
for some $N$ we have $A_N = A_{N+1}$, then again $\lambda(A_n\setminus A_{n+1}) = 0$ for $n\geq N$.
Can you help me to construct an example for the third case, namely when $A_n\neq A_{n+1}$ for all $n\geq0$ (of course if such an example exists)? We can also assume for this example that $\xi\in \operatorname{Lip}([-1,1]\times[-1,1])$, not necessary differentiable.
I am mostly interested if it is possible to find a Lipschitz $\xi$ such that $A_n$ coincide with iterations of Smith-Volterra-Cantor set (Fat Cantor Set)?