A Non-additive Thermodynamic Formalism and Dimension Theory of Hyperbolic Dynamical Systems

Introduction We begin with the description of a geometric construction in the real line. We consider p positive numbers λ1, . . . , λp < 1, and choose p disjoint closed intervals ∆1, . . . , ∆p with length λ1, . . . , λp. For each k = 1, . . . , p, we choose again p disjoint closed intervals ∆k1, . . . , ∆kp ⊂ ∆k with length λkλ1, . . . , λkλp. Iterating this procedure, for each integer n we obtain p disjoint closed intervals ∆i1···in with length ∏n k=1 λik . We define the set