2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/19698We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs--Shannon entropy into more general framework - Renyi's information entropy. We address the renormalization issue for Renyi's entropy on (multi)fractal sets and consequently show how Renyi's parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between Renyi's information entropy and thermodynamics on multifractals. Important issues as, for instance, Renyi's entropy versus Tsallis--Havrda--Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.24 pages, 2 figures, RevTex, thoroughly revised, accepted to Annals of PhysicsStatistical MechanicsAstrophysicsHigh Energy Physics - TheoryMathematical PhysicsData Analysis, Statistics and Probabilitytext