Dynamical Properties For Feigenbaum's Attractors And Substitution Systems

In1978,Feigenbaum proposed well-known Feigenbaum functional equationwhich played an important role in explaining an universal metric property ofperiod-doubling bifurcations in transition to chaotic behavior(i.e.the so-calledFeigenbaum phenomenon). Feigenbaum phenomenon appeared in physics, bi-ology,chemistry and any other fields,so it attracted many people's attention.Inthe following thirty years,people did further research on the solutions of Feigen-baum functional equations and obtained some rich fruits in this respect,such as,the existence,dynamical behaviors and structure of the solutions.The corresponding functional equation in this paper is as follows:where p≥2is an integer.For p=4, the solutions of equation(2)can be classified to two types.Thesolutions of one type have0topological entropy,and the others' topological en-tropies are larger than0.For p=5, the solutions of equation(2)can be classifiedto three types,which have diferent dynamical behaviors. The research of attractor is one of the interesting topics in dynamical system. Milnor introduced the concept of attractor in1985:Let M be a smooth compact manifold,possibly with boundary,and let f be a continuous map from M into itself.The Milnor attractor Λ of f is defined as the smallest invariant closed subset of M with the property that w(x)(?)Λ for every point x∈M except a set of measure zero.First,we introduce the recent results of the research on Feigenbaum's maps,and give some important properties of them.We pay more attention to some p-order non-univallecular Feigenbaum's map g,and prove the restriction g|E have the same properties.Let g be a p-order(p≥2)non-univallecular Feigenbaum's map.If there exists an α∈(λ,1),such that g(α)=0,and for any x∈[λ,α],g'(x)<-(p-1)；for any x∈[α,1],g'(x)≥1,then according to the properties of Feigenbaum' maps given by theorem3.1.6,we defie:φi:I→I and φ=∪i=1pφi.Where g0=g|α,1].We prove the following conelusions:(1)for all x∈I,(2) for any is an invariant set of g,i.e.,(3)for any subsets φi1…ik(I)and φj1…jk(I),there is an n>0such that Then the set E satisfied φ(E)=E is the attractor of Feigenbaum's map,which is a minimal set attracts almost every point in [0,1],and g|E～τ,(τis a p-adic system).The dynamical behaviors of adic system(∑k+,τ)are simple:(1) τ is minimal;(2) the topological entropy of τ is0;(3) τ is strictly ergodic；(4) τis not Li-Yorke chaotic(5) τ is not Schweizer-Smital chaotic;(6)if(K(∑k|),τ)is the hyperspace system if(∑k|,τ),then h(τ)=0.If two systems are topologically conjugate to each other,they have the same dynamical behaviors.So the map g|E has the properties(1)(6).In addition,we prove that for any0<t<1,there always exists a p-order Feigenbaum's map g0which has an attractor with Hausdorff dimension l.At the same time,we construct an example of Feigenbaum's map,and prove it has positive entropy and is Schweizer-Smital chaotic.But according to the above results,this map has simple dynamical behaviors restricted on its attractor.Substitution system is a kind of sub-system of symbolic system.Let η be a constant-length substitution on∑2such that η(0)=a=a0a1…an-1,η(1)=b=b0b1…bn-1,and satisfies condition (H):(1)a0=0;(2)for some i>0,a0=1.Under the hypothesis(H),η has in∑2a fixed point beginning with0,denoted Let Xη be the closure of the orbit of u, Let f=σ|Xη,f:Xη→Xη a subshift of σ,called constant-length substitution subshift induced by η.If0(?)η(1),we call η primitive,and non-primitive otherwise.We prove the following results for non-primitive constant-length substitution:(1)if there exist some distinct s1,s2,t∈{0,1,…,n-1},such that as1≠bs1,as2≠bs2,and at=bt.then f has a Li-Yorke pair,and vice verse；(2)f has no Schweizer-Smital pairs；(3)if(K(Xη),f)is the hyperspace system and η satisfies the condition of result(1),then f is Li-Yorke chaotic.