| In this paper,we mainly study the conformal dimension of two kinds of plane self-similar sets.First,we introduce the preliminary knowledge of this paper,including some definitions and properties related to Hausdorff dimension,self-similar set and quasisymmetric mapping.Secondly,we define a kind of planar "antenna type" self-similar set,and discuss its conformal dimension and quasi-symmetric minimality.Then,we define a class of planar"closed loop" self-similar sets,and discuss their conformal dimensions and quasi-symmetric minimality.Finally,based on the research methods and results of this paper,the problems to be further studied are proposed.The structure and specific contents of the full text are as follows:In Chapter 1,we describes the background of the research and puts forward the research content of this paper.In Chapter 2,we mainly introduces the relevant preliminary knowledge involved in the research,such as Hausdorff measure,Hausdorff dimension,self-similar set,symbol space,Quasi-conformal mapping,quasi-symmetric mapping,definition of conformal dimension,etc.In Chapter 3,we define a kind of planar"antenna type" self-similar set Xa,and give its Hausdorff dimension.By constructing the Gibbs measure μ on Xa,and by virtue of μ’s pull measure μ Ο f-1 under the quasi-symmetric mapping f and the principle of mass distribution,we prove that the conformal dimension of Xa is 1,and then show that this kind of self-similar set is not quasi-symmetrical minimal.In chapter 4,we define a class of planar "closed loop" self-similar set Xa,give the properties of Xa under the Euclidean metric,and prove that the conformal dimension of Xa is 1 by constructing a new metric dm on Xa that is quasi-symmetric equivalent to the original Euclidean metric and combining the structural properties of(Xα,dm),and then show that this kind of self-similar set is not quasi-symmetrical minimal.In chapter 5,we summarize the conclusions of this paper and put forward some problems to be further studied. |
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