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

AbstractWe study the typical behaviour (in the sense of Baire's category) of the q-Rényi dimensions D̲μ(q) and D¯μ(q) of a probability measure μ on Rd for q∈[−∞,∞]. Previously we found the q-Rényi dimensions D̲μ(q) and D¯μ(q) of a typical measure for q∈(0,∞). In this paper we determine the q-Rényi dimensions D̲μ(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 D̲μ(q) attains the smallest possible value, and for q=1 and q=∞ the upper Rényi dimension D¯μ(q) attains the largest possible value