Quasisymmetric Rigidity Of Square Carpets Of Measure Zero In Rectangle Rings

Carpet plays an important role in the study of quasisymmetry mapping.A set S(?)C is called a carpet,if it can be written as S=D0\∪i=1∞ Di,where for each i ≥ 0 the set Di(?) C is a Jordan region and the following conditions are satisfied:Di (?) D0 for each i≥ 1;Di ∩ Dj=(?) for i≠j,i,j≥ 1;diam(Di)→ 0 as i→∞;S is nowhere dense.In this paper,the quasisymmetric rigidity of generalized square carpet is studied.We review the definitions and properties of quasisymmetry,quasiconformality and conformal modules.The main result is as follows:Let Q=[0,1]2 (?) C,let T be a carpet of measure zero which can be written as T=(Q\R)\(?)Qi,where R is a rectangle of sides parallel to the axes and R is contained in the interior of Q,Qi is a square of sides parallel to the axes and Qi (?) int(Q)\R for each i ∈ N.We proved that,if f is an orientation-preserving quasisymmetric map from T onto T that sends the four vertices of Q to the four vertices of Q and satisfies f(0)=0 and f((?)R)=(?)R,then f is the identity on T.Bonk-Merenkov proved that the standard square carpet T=Q\(?)Qi has quasisymmetric rigidity.Our result generalizes their result.