Fractal Dimensions And Classification Of One-dimensional Homogeneous Moran Sets

Moran sets play an important role in fractal geometry.In this paper,we mainly study one-dimensional Moran sets,including two problems:fractal dimensions of one-dimensional Moran sets and the fatness and thinness of one-dimensional Moran sets for doubling measures.For studying the fractal dimensions of Moran sets,we construct a class of special homogeneous Moran sets:{mk}-quasi-homogeneous perfect set-s(which contain homogeneous perfect sets in chapter 3 basing on the con-nected components and their gaps,and obtain that the Hausdorff dimension of the sets is dimH(?)under some conditions.Wealso prove that under the condition supk?1{mk}<?,the upper box dimension and the packing dimension of the sets can get the minimum value of the one-dimensional homogeneous Moran sets,which is (?).We also obtain that the upper box dimension of the sets is(?) in two special cases.For studying the fatness and thinness of Moran sets for doubling measures,we construct a class of good Moran sets in one-dimensional case in chapter 4,and obtain the necessary and sufficient conditions for the six classifications of fat and thin sets of the sets for doubling measures under the condition supk?1{nk}<?.If E is a fat good Moran set,we have:(1)E is very fat if and only if(?)for all 0<p<1;(2)E is minimally fat if and only if(?)for al-1 0<p<1;(3)E is fairly fat if and only if there exist 0<p<q<1,such that(?)Similarly,if E is a thin good Moran set,we have:(4)E is very thin if and only if(?)for all 1<p<?;(5)E is minimal-ly thin if and only if(?)for all 1<p<?;(6)E is fairly thin if and only if there exist 1<p<q<?,such that(?),(?).We generalize the conclusions to a larger class of one-dimensional Moran sets.