ABSTRACT
The so-called multifractal theory was introduced by the theoretical physicists Frisch & Parisi [Fr] and Halsey et al. [Ha] in 1986. Recently much physics literature has been devoted to the study of mult ¡fractals, cf. e.g. [Bo,Co1,Grl,Gr2,He,Pa, Te1,Te2]. The first rigorous results on self-similar multifractals were obtained by Cawley & Mauldin [Ca], Edgar & Mauldin [Ed] and Falconer [Fa4] during the period 1991 through 1992. Multifractals have subsequently been studied by a large number of mathematicians [Av,Bo,Br,Co1,Co2,Fen,Ho,Kah5,Kil,Ki2,Lo1,Lo2,Lo3,Ol,Pel,Pe2,Pe3,Pey3,Ra,Str]. The purpose of this exposition is to present a rigorous foundation for the multifractal structure of random geometrically graph directed self-similar measures along the lines introduced and developed by Olsen in [Ol]. The random graph directed self-similar measures that we study are natural measure-theoretical extensions of the random self-similar sets that appear in Graf [Gra1], and the results that we obtain are natural multifractal extensions of the main results in Graf [Gra1].
