SPECTRAL CLASSES AND THE PARAMETER DISTRIBUTION SET

Abstract

Abstract. The natural projection of a parameter lower (upper) distri-bution set for a self-similar measure on a self-similar set satisfying theopen set condition is the cylindrical lower or upper local dimension setfor the Legendre self-similarmeasure which is derived from the self-similarmeasure and the self-similar set. 1. IntroductionRecently, we [1] investigated the relation between spectral classes of a self-similar Cantor set in a set theoretical sense. More recently, using the parameterdistribution, we find the parallel results for the self-similar set (attractor of theIFS consisting of n(≥ 2) similitudes satisfying the OSC (open set condition))instead of the self-similar Cantor set (attractor of the IFS consisting of 2 simil-itudes satisfying the SSC (strong separation condition)), which leads to a gen-eralization of [1]. In this paper, we define the Legendre self-similar measureson the self-similar set which is derived from the self-similar measure and theself-similar set. Using the Legendre self-similar measures on the self-similar set,we give full relationship between the natural projection of a parameter lower(upper) distribution set for a self-similar measure on a self-similar set and thecylindrical lower or upper local dimension set for the Legendre self-similar mea-sures.2. PreliminariesLet N and R be the set of positive integers and the set of real numbersrespectively. An attractor K in the d-dimensional Euclidean space R