Typical Renyi dimensions of measures. The cases: q=1 and q=infinity

We study the typical behaviour (in the sense of Baire's category) of the q-Renyi dimensions (D) under bar mu (q) and (D) over bar mu(q) of a probability measure mu on R-d for q is an element of [-infinity, infinity]. Previously we found the q-Renyi dimensions (D) under bar mu (q) and (D) over bar mu (q) of atypical measure for q is an element of (0, infinity). In this paper we determine the q-Renyi dimensions (D) under bar mu(q) and (D) over bar mu(q) of a typical measure for q = 1 and for q = infinity. In particular, we prove that a typical measure mu is as irregular as possible: for q = infinity, the lower Renyi dimension (D) under bar mu (q) attains the smallest possible value, and for q = 1 and q = infinity the upper Renyi dimension (D) over bar mu (q) attains the largest possible value. (c) 2006 Elsevier Inc. All rights reserved.