  We 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.
