Further Study On Dimensions Of Multitype Moran Sets

Multitype Moran sets were firstly founded in the study of the structure of the quasi-crystal spectrum.They generalize some well-known fractal structures such as self-similar sets,graph-directed sets and Moran sets and so on.We introduce them in this thesis,and consider their dimensional properties with much care.As we know that for any Moran set E satisfying a bounded condition on contracting ratios,the following inequality series holddimH E=s*≤s*= dime E=(?)B E,in which s*and s*are the lower and upper pre-dimension with respect to the natural cover-ings.We attempt to obtain the similar results on multitype Moran sets.Firstly,natural covering structure is introduced.We go further enough to get the upper bound estimation of Hausdorff dimension and lower bound estimation of upper box counting dimension on this fractal structure.Moreover,additional bounded conditions on infimum and supremum of contracting ratios justify the precise estimation of upper box counting dimension accordingly.Besides,the notion of net measure and the definition of weak equivalence on net mea-sure are used to prove the equivalence of Hausdorff dimension on net measure and Hausdorff dimension which is commonly defined for sets given by natural covering structures under specific condition.More importantly,based on the preceding arguments,associating with weakened condition on nonnegative infimum of contraction and strengthened condition on primitivity,we finally obtain that dimH E=s*remain holds for any multitype Moran set E.At last,we conclude the whole work of this thesis and deliver some ideas which may be feasible for our further study.