Lipschitz Classification Of Fractal Squares And Quasisymmetric Rigidity Of Triangular Sierpi(?)ski Carpets

In this paper,we mainly discusses the Lipschitz classification of Fractal squares and quasisymmetric rigidity of the triangular Sierpi（?）ski carpets.（1）The Lipschitz classification of Fractal squaresTwo classical sets from the self-similar sets are the Cantor Middle Third set and the standard Sierpi（?）ski carpet.The fractal squares are shown as their generalization,and are defined as follows:Let n>2 be an integer and numbers set D = {d₁,d₂,…,d_m)（?）{0,1,2…,n—1}² be nonempty.The fractal square F:= F（n,D）of data（n,D）is defined to be the unique nonempty compact subset of the unit square[0,1]2 satisfying F = Ud∈D（F +d）/n.For each d in D,let fd（z）=（z+d）/n,z ∈[0,1]2.Then F is the self-similar set induced by the iterated function system（IFS）{fd:d E D}.We call{fd:d ∈ D} the standard IFS of F[1].To form a self-similar fractal square,one will take the intersection of a sequence of nested sets,as in the definition of the Cantor Middle Third set.According to a given set of numbers set D,we divide a single unit square into n2 subsquares.leave m of these subsquares.The second step,each rest of the small squares in the first step divided into n2 squares,leaving m;As such,the limit set is the fractal square F determined by n and D.Different numerical sets D get different fractal squares.When m,n are giving,Fn,m represent the fractal cube families.The Hausdorff dimension of all fractal squares in Fn,m is log m/log n[1].The Lipschitz equivalent classification of fractal square in F3.6 is solved in this paper.Main method is:if two fractal squares are Lipschitz equivalent,we will construct it;if not,then observe some topological invariants exist or not to prove two fractal squares are not homeomorphic,also are not Lipschitz equivalent.（2）Quasisymmetric rigidity of the triangular Sierpi（?）ski carpetIn the work of Bonk and Merenkov[2-4],it was proved that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi（?）ski carpet S3 is a Euclidean isometry.For the standard 1/p-Sierpi（?）ski carpets Sp,p>3 odd,they showed that the groups QS（Sp）of quasisymmetric self-maps are finite dihedral.They also established that Sp and Sq are quasisymmetrically equivalent if only if p = q.The main tool in their proof is the carpet modulus,which is a certain discrete modulus of a path family and is preserved under quasisymmetric maps of carets.Zeng and Su[5]study a new class of square Sierpi（?）ski carpets,they prove that the group of quasisymmetric self-maps of this carpet is the Euclidean isometry group.We construct a new Sierpi（?）ski carpet—triangular carpet,and use the tools of carpet modulus and weak tangent spaces to prove that every quasisymmetric self-map of this carpet is a Euclidean isometry.