Abstract
We study the typical behaviour (in the sense of Baire's category) of the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a probability measure μ on R d for q ∈ [ − ∞ , ∞ ] . Previously we found the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a typical measure for q ∈ ( 0 , ∞ ) . In this paper we determine the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a typical measure for q = 1 and for q = ∞ . In particular, we prove that a typical measure μ is as irregular as possible: for q = ∞ , the lower Rényi dimension D ̲ μ ( q ) attains the smallest possible value, and for q = 1 and q = ∞ the upper Rényi dimension D ¯ μ ( q ) attains the largest possible value.
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