We study dynamical notions lying between \(\mathcal{U}\)-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply \(\mathcal{U}\)-frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if T is \(\mathcal{U}\)-frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of T is still U-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered \(\mathcal{U}\)-frequent hypercyclicity notions.

通过研究未加权上限密度和上Banach密度之间的加权上限密度，我们研究了\（\ mathcal {U} \）-频繁超循环性和迭代超循环性之间的动力学概念。尽管混沌暗示了迭代超循环性，但我们证明，对于任何加权上限密度，混沌并不意味着\（\ mathcal {U} \）-频繁的超循环性。此外，我们表明，如果T为\（\ mathcal {U} \）-经常超循环（相对于迭代超循环），则T的n乘积仍然是U-频繁超循环（相对于迭代的超循环），并且这意味着对每个考虑的\（\ mathcal {U} \）也都满足-频繁的超循环性概念。