## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 1425-1439 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 331 |

DOIs | |

Publication status | Published - 15 Jul 2007 |

## Keywords

- multifractals
- Renyi dimensions
- baire category
- residual set