3 problems

Let $(X=G/\Gamma,\mu,T_1,\ldots,T_d)$ be an ergodic $\mathbb{Z}^d$-nilsystem. Its Koopman representation on $L^2(X,\mu)$ decomposes as … Here…

Let $(X,T)$ be a minimal nilsystem and let $d\in\mathbb N$. For $k\geq 2$, write $X_k$ for the maximal $k$-step nilfactor of $(X,T)$, $N_d(X,T)$ for the associated orbit-closure sy…

Let $(X,T)$ be a minimal topological dynamical system, and let ${\rm Ind}_{fip}(X,T)$ denote its set of finite-independent pairs. Write $\Delta_X=\{(x,x):x\in X\}$. Unique ergodici…