On the dimension of deterministic and random Cantor-like sets

this paper we unify and extend many of the known results on the Hausdorff and box dimension of deterministic and random Cantor-like sets in R determined by geometric constructions (see [PW] for the complete description of results and detailed proofs). Most authors have considered similarity processes which impose a strong restriction on the geometry of the construction. Moreover, these constructions were modeled by either the full shift, or subshifts of finite type. In this paper we weaken these restrictions significantly and consider geometric constructions which need not be self-similar and have more complicated geometry. Our constructions are also modeled by arbitrary symbolic dynamical systems. Symbolic dynamics and the thermodynamic formalism thus become essential tools in our analysis. We also introduce two new fundamental classes of geometric constructions: asymptotic constructions and random constructions determined by an arbitrary ergodic stationary process