Double minimality, entropy and disjointness with all minimal systems (Q1633131)

A new sufficient condition is proposed for disjointness with all minimal systems. The following are the main results of this paper. \par Theorem 1. Let $(X,T)$ be weakly mixing. Suppose that, for every minimal system $(Y,S)$ there exists a countable set $D\subset X$ such that, for every nonempty open set $U\subset X$ the following condition holds: for each $y\in Y$ and each open neighborhood $V$ of $y$, there exists $x\in U\cap D$ such that the set of transfer times $N_{T\times S}((x,y),U\times V)$ is syndetic. Then, $(X,T)$ is disjoint with every minimal system. \par Theorem 2. Suppose that $(X,T)$ is transitive with at least two points, and $(\omega_T(x),T)$ is weakly mixing for each minimal point $x$. Then, for every transitive point $q$ there exists an open set $U\ni q$, such that $N_T(z,U)$ is not an $IP^*$-set for each $z\in X$.