Typical Rényi dimensions of measures. The cases: q = 1 and q =∞

We study the typical behaviour (in the sense of Baire’s category) of the q-Rényi dimensions Dμ(q) and Dμ(q) of a probability measure μ on Rd for q ∈ [−∞,∞]. Previously we found the q-Rényi dimensions Dμ(q) and Dμ(q) of a typical measure for q ∈ (0,∞). In this paper we determine the q-Rényi dimensions Dμ(q) and Dμ(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 Dμ(q) attains the smallest possible value, and for q = 1 and q =∞ the upper Rényi dimension Dμ(q) attains the largest possible value. © 2006 Elsevier Inc. All rights reserved.