Fractal Dimensions And Arithmetic Operations Of Some Fractal Sets

This thesis is devoted to studying some topics in Fractal Geometry and Measure Theory.It contributes to calculating the fractal dimension of some fractal sets with graph-directed construction and the arithmetic operations on such sets.Besides,we give some notes on two packing measures constructed by method Ⅰ and method Ⅱ.In the first object,we give a uniform method to calculating the Hausdorff dimension of the univoque sets u of a class of self-similar sets with complete overlaps.Fix an integer m≥ 3 and fix a λ ∈(0,1).Let A be the collection of all self-similar sets K generated by the IFSs {fi(x)=λx+bi}i=1n,where n≥3 and bi ∈ R for every 1≤i≤n,satisfying the following conditions:(Ⅰ)0=b1<b2<…<bn=1-λ;(Ⅱ)fi([0,1])∩ fj([0,1])=? for any 1≤i<j≤n with j-i≥2;(Ⅲ)There exist i,j ∈ {1,…,n-1} such that fi([0,1])∩ fi+1([0,1])=? and fi([0,1])∩fj+1([0,1])≠?;(Ⅳ)If fi([0,1])∩fi+1([0,1])≠?,then |fi([0,1])∩fi+1([0,1])|=λj with j ∈{2,3,…,m},where |·| stands for the length of an interval.For K∈A,for every x∈K,there exists at least one sequence(ik)k=1∞∈{1,...,n｝N such thatThus,Π:｛1,…,n}N→K is surjective and continuous.We call such a sequence a coding of x.A point x ∈ K is called a univoque point if its coding is unique.Denote by u the set of all univoque points in K.We define a subset K*of K with a graph-directed construction such that K*differs from u by at most a countable set.In other words,dimH K*=dimHu.Then,we prove that K*is a configuration set of finite patterns,leading to an effective method of calculating dimH K*.Secondly,we specify the sufficient conditions for which f(u1,u2)contains some interior in R,where f:U→R is a continuous function defined on an open set U C R2 and u1,u2 are two univoque sets corresponding to the self-similar sets K1,K2 in the following class:Fix λ ∈(0,1)and let A be the collection of all selfsimilar sets K generated by the IFSs {fi(x)=λx+ai}i=1n,where n≥3 and ai∈R for every 1≤i≤n,satisfying the following conditions:(Ⅰ)a1<a2<…<an with f1(a)=a:=a1/1-λ,fn(b)=b:=an/1-λ,and fi(a)<fi+1(a)for every i=1,2,…,n-1;(Ⅱ)fi([a,b])∩ fj([a,b])=? for any 1≤i<j≤n with j-i≥2;(Ⅲ)There exist i,j ∈ {1,…,n-1} such thatfi([a,b])∩ fi+1([a,b])=? and fi([a,b])∩fj+1([a,b])≠?;(Ⅳ)If fi([a,b])∩ fi+1([a,b])≠?,then |fi([a,b])∩fi+1([a,b])|=λ2(b-a),where|·| stands for the length of an interval;(Ⅴ)α=fs0+1(a)-fs0(b)and β=ft0(a)-ft0-1(b),where s0=min{1≤i≤n-1:fi([a,b])∩fi+1([a,b])=?},t0=max{2≤i≤n:fi([a,b])∩fi-1([a,b])=?}and s0≠t0-1.We can similarly apply the above study on other graph-directed sets.Finally,let Ps and τs be the two s-dimensional packing measures constructed by method Ⅰ and method Ⅱ,respectively,defined as follows:Let 0 ≤s<∞.For F?Rn and 0<δ<∞ put where {Bi} is a collection of disjoint balls of radii at most δ with centers in F.We call such collection {Bi} a δ-packing of the set F.Then,is a measure on Rn known as the s-dimensional packing measure constructed by method Ⅰ.For each 0<δ ≤+∞,letτs is a measure on Rn known as the s-dimensional packing measure constructed by method Ⅱ.We call such collection {Ei} a δ-covering of the set E.The corresponding packing dimensions are defined as follows:dimp E=inf{s:Ps(E)=0}=sup{s:Ps(E)=+∞},dimτE=inf{s:τs(E)=0}=sup{s:τs(E)=+∞}.We get τs(E)=Ps(E)and dimp(E)=dimτ(E),for any subset E of Rn.