Some remarks about self-products and entropy

Let $(X,\mathcal{B},\mu ,T)$ be a probability-preserving system with $X$ compact and $T$ a homeomorphism. We show that if every point in $X\times X$ is two-sided recurrent, then $h_{\mu }(T)=0$, resolving a problem of Benjamin Weiss, and that if $h_{\mu }(T)=\mathrm{\infty }$ then every full-measure set in $X$ contains mean-asymptotic pairs (i.e. the associated process is not tight), resolving a problem of Ornstein and Weiss.