The Study On Quasisymmetric Minimal Sets And Conformal Dimensions

In this dissertation,we study the quasisymmetric minimality of sets in the Eu-clidean space and the conformal dimension of connected planar self-similar sets.In addition,we show the upper bound of the conformal dimension of the Sierpinski carpet Sp.In the first part,we prove that[0,1]× Z is quasisymmetrically minimal for any nonempty B orel set Z(?)Rd-1,where d>1 is an integer.We say that such a set is of Tyson type because Tyson got the same conclusion for compact Z(?)Rd-1[56],[66].As applications,we obtain the quasisymmetrically minimality of the other three versions of sets of Tyson type,which are list as follows:(1)E={rx:r E Z,x?M},where Z is a nonempty Borel set in(0,?),M is a k-dimensional smooth surface in Sd-1 and k?d-1.(2)E={rx:r?[0,1],x?Z},where Z is a nonempty Borel set in Sd-1.(3)G(h,Z)={(z,y):z:?Z,y?[0,h(z)]},where h:Rd-1?R1 is a Borel function and Z is a nonempty Borel set in Rd-1.In the second part,we prove that the conformal dimension of a class of connected planar self-similar sets X? is 1,whilst X? is not quasisymmetrically equivalent with any metric space of Hausdorff dimension 1.Here X? is the attractor of the iterated function system {f1?,f2?,f3?,f4?},0<?<1/2,f1?(z)=z/2,f2?(z)=z+1/2,f3?(z)=1/2+?iz,f4?=1/2-?iz,and i is the imaginary unit.In the third part,we give the upper bound of the conformal dimension of the Sier-pinski carpet Sp,where p?3 is odd.We shall equip Sp with a new class of metrics d?A such that these metrics d?A are quasisymmetrically equivalent with the Euclidean metric of Sp.By selection of A,we prove that the conformal dimension of Sp is?log((p2-1)4-8)/4 log p.This results implies that Sp is not quasisymmetrically minimal set.The idea of Kigami[43]is adopted here.