On the Tukey types of Fubini products

We extend the class of ultrafilters $U$ over countable sets for which $U\cdot U\equiv_T U$, extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal $\alpha\geq 2$, the generic ultrafilter $G_\alpha$ forced by $P(\omega^\alpha)/\text{fin}^{\otimes\alpha}$ satisfy $G_\alpha\cdot G_\alpha\equiv_T G_\alpha$. This answers a question posed in \cite[Question 43]{Dobrinen/Todorcevic11}. Additionally, we establish that Milliken-Taylor ultrafilters possess the property that $U\cdot U\equiv_T U$.