We examine various types of $ℱ$-hypercyclic ($ℱ$-topologically transitive) and disjoint $ℱ$-hypercyclic (disjoint $ℱ$-topologically transitive) properties of binary relations over topological spaces. We pay special attention to finite structures like simple graphs, digraphs and tournaments, providing a great number of illustrative examples.

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space ${\ell }^1\left(\mathbb{N}\right)$, any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_0\left(\mathbb{N}\right)$ or ${\ell }^p\left(\mathbb{N}\right)$, $1\<p\<\infty$. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.