A Class Of Bi--Lipschitz Transformation Problem On Self-similar Sets

This paper aims to study the bi-Lipschitz transformations on a class of strongly separated self-similar sets,which maps one point of the self-similar set to another.Firstly,the middle-third Cantor set C is discussed.By using the symbolic space to construct a bi-Lipschitz transformation on C,we prove that for any two points in C,there exists a bi-Lipschitz transformation on C which can exchange these two points,and generalized to any finite number of points in C.Then these conclusions are ex-tended to the homogeneous Cantor set C_?=???.Next,the self-similar set C_?,?generated by contractive similarities f₀?x?=?x,f₁?x?=?x+1-?is discussed,where 0<?<?<1,?+?<1,log?/log??Q.By redefining the basic intervals and their layers in which the length of basic intervals of the mth layer is between?^m+1and?^m.Based on this,the types of basic intervals are defined.It is proved that for two points in C_?,?that satisfy a certain condition about type,there is a bi-Lipschitz transformation on C_?,?that can exchange these two points.