We give general theorems which assert that divergence and universality of certain limiting processes are generic properties. We also define the notion of algebraic genericity, and prove that these properties are algebraically generic as well. We show that universality can occur with Dirichlet series. Finally, we give a criterion for the set of common hypercyclic vectors of a family of operators to be algebraically generic.
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We compute the typical (in the sense of Baire's category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^d$. Our results are new even in the context of box dimensions of measures.
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