Recently, new topological properties that an operator acting on a topological vector space can have were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity, and we compare it to the other common topological properties, including the new ones. For the family of weighted backward shifts on $c_0$ and $\ell^p$, we prove the existence of a strongly hypercyclic weighted backward shift which is not ultra hypercyclic.

arXiv admin note: text overlap with arXiv:2304.02073