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We examine various types of -hypercyclic (-topologically transitive) and disjoint -hypercyclic (disjoint -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 investigate the relations between local α-times integrated semigroups and (α + 1)-times integrated Cauchy problems, and then the relations between global α-times integrated semigroups and regularized semigroups.
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