jesus' problems

Conjecture 1.1: Let $\mathbb{T}^{2}$ be a closed two-dimensional box with periodic boundary conditions. Consider $\Omega \subsetneq  \mathbb{T}^{2}$ a closed domain occupied by a two-dimensional radioactive source and $\mathcal{C}$ a two-dimensional cat embedded in $\mathbb{T}^{2} \setminus \Omega$ .

Apply, now, Arnol'ds cat map $\Gamma$ to $\mathbb{T}^{2} \setminus \Omega \rightarrow \mathbb{T}^{2}$. Then, the probability that the cat dies goes very rapidly to $1$.

We have a beautiful proof of the above statement, but this internet is too small for the proof to fit in.