2 problems

For a topological dynamical system $(X,T)$, let $\textbf{AP}^{[2]}(X,T)$ denote its order-two approximate proximal relation and let $\Delta_X=\\{(x,x):x\in X\\}$. Approximate-proxi…

Let ${{\mathbb{I}}}=[0,1)$ be the unit interval, and let $T\colon{{\mathbb{I}}}\to{{\mathbb{I}}}$ be measurable. The map $T$ is 1-collapsing when $\phi_1(x,y)=0$ for all…