Lipschitz Equivalence Of Self-similar Sets,Neighbour Automaton And Related Problems

Lipschitz equivalence is a kernel problem on geometric measure theory and fractal geometric.This problem is initiated from the works of K.Falconer[31,32],G.David and S.Semmes[12].In recent years,the study of this area is very active,and there are many important progresses.These already of the research are required to have a relatively simple fractal topology,that is,the fractals are totally disconnected,and even satisfy strong separation conditions.The main goal of this paper is to study the Lipschitz problem of not totally disconnected self-similar sets.We study a special kind of fractal sets,its connected components is a single point or parallel lines.This type of fractals is homcomor-phic,thus can not use conventional topological methods to illustrate their Lipschitz equivalence.Our first method is to introduce the neighbor automaton and the separation number,and the Lipschitz equivalence relation between self-similar sets and symbol spaces is established.Thus transforming the problem into two metricized symbol spaces Lipschitz equivalence.Particularly,if two self-similar sets are equivalent to the same symbol space,they are also equivalent.A subset of a fractal is called a 'big piece' if it has a positive Hausdroff measure.For two totally disconnected self-similar sets E and F,if a big piece of E and a big piece of F are Lipschitz equivalence,then E and F are also equivalent[27].We define a class of p-uniform fractals whose neighbor automaton has only two states.We prove that for the p-uniform fractals,the above-mentioned 'big piece' property is still true.Moreover,two p-uniform fractals are Lipschitz equivalence if and only if they have the same Hausdroff dimension.Finite type property of fractals play a key role in the Lipschitz equivalence study of all disjoint fractal sets.But the fractals we study are longer of a finte type property.In order to overcome the difficulties cansed by infinites,we through the decomposition of the sequence,constructed a transducer to establish the symbol space between the bi-Lipschitz mapping.Using this method,we prove the Lipsehitz equivalence of a fractal square of a single parameter class E? self-similar sets,and we hope that this method can be developed into a universal method.In addition,we generalize Fekete's lemma of sub-additive,that is,let {bn}n>1 be a sequence of real numbers such that bn+bm?bn+m+f(n+m),(?)m,n>1,where the map f:(0,+?)?(0,?)is called perturbation term.If f(n)satisfies such conditions,then the conclusion of Fekete's lemma is still satisfied.Especially,f(n)=n? with 0<?<1.