On Quasisymmetric Rigidity Of Rectangular Square Carpets

Let S(?)R2.We say that S is a carpet,if it is homeomorphic to the standard Sierpinski carpet.Thus,a carpet S can be written as (?)where Di(?)S2(i ∈ N)are Jordan regions satisfying the following conditions:S has no interior points,diam(Di)→0 as i→∞,and Di ∩ Dj=(?)whenever i≠j.In this case,we call(?)Di a peripheral circle of the carpet S.We say that a carpet S is a square carpet,if all but one peripheral circles of S are squares.In this thesis,we discuss quasisymmetric maps between square carpets.We are motivated by Bonk-Merenkov’s work on quasisymmetric rigidity of standard Sierpin-ski carpets and Bonk-Kleiner-Merenkov’s work on quasisymmetric rigidity of round carpets.The former proved that every quasisymmetric automorphism of a standard Sierpinski carpet is an isometric map.The latter proved that every quasisymmetric homeomorphism between two round carpets of measure zero is the restriction of a Mobius transformation.We only consider rectangular square carpets,where a rectangular square carpet is a square carpet whose exceptional peripheral circle in its definition is a rectangule.We prove that every rectangular square carpet is quasisymmetrically rigid,i.e.every quasisymmetric map from a rectangular square carpet onto another rectangular square carpet is a similarity map.