Yali DUDepartment of Mathematics,South China University of Technology,Guangzhou 510641,China E-mail:Duyali 123@126.com

Junjie MIAOShanghai Key Laboratory of PMMP;Department of Mathematics,East China Normal University, Shanghai 200241,China E-mail:jjmiao@math.ecnu.edu.cn

Min WU†Department of Mathematics,South China University of Technology,Guangzhou 510641,China E-mail:wumin@scut.edu.cn

LEVEL SETS AND EQUIVALENCES OF MORAN-TYPE SETS∗

In the paper,we consider Moran-type sets Eagiven by sequencesand.we prove that Eamay be decompose into the disjoint union of level sets.Moreover, we define three type of equivalence between two dimension functions associated to two Morantype sets,respectively,and we classify Moran-type sets by these equivalent relations.

Moran-type sets;dimension function;level set;logarithmical equivalence

2010 MR Subject Classification28A78;28A80

1　Introduction

Fractal sets may be obtained by removing a sequence of disjoint regions from a given set, which are named cut-out sets.Obviously,all compact subsets in ℝ can be obtained in this manner.For example,the middle-third Cantor set in ℝ.Similarly,in the plane,the Sierpinski triangle is obtained by removing a sequence of equilateral triangles from an initial equilateral triangle(see[6]).

Let A be a compact interval in ℝ andbe the disjoint open subintervals of A withdenotes the length of Ak.Then E is a compact set with Lebesgue measure zero an.We call E a cut-out set and Akthecomplementary intervals.Clearly the set E is determined by the intervals Ak,some geometric information can be obtained from only knowing the lengths and independent of the position of Ak.Note that cut-out sets are strongly connected with gap sequences of fractals which often characterise the geometric properties of fractal sets,and we refer the readers to[4,21]for details.

In this paper,we will study another class of cut-out sets which is a generalization of C.Let{a}be a positive non-increasing sequence withakk≥1akconvergent and{nk}beWe first remove n1−1 open intervals from I with lengths a1,a2,···,an1−1from left to right,and there remain n1intervals, indexed by I1,I2,···,In1.On the second step,we remove n2−1 open intervals from each Ij

(1≤j≤n1)with lengths an1+(n2−1)(j−1)+1,···,an1+(n2−1)(j−1)+n2−2,and there remain n2 closed subsets,indiced by Ij1,Ij2,···,Ijn2.Suppose Ii1i2···ikis the j-th interval in step k,we remove nk+1−1 open intervals from interva lContinue this process,we call the limit set

Moran-type set which is compact and perfect.

Note that the construction uniquely determines the location of gap at each step.For instance,the location of the first gap in Cantor set Cais determined by the interval whose length is equal to a2+a4+a5+a8+···.Such Moran-type sets can be thought as generalized Moran sets(see[11])and need not be central or(quasi-)self-similar.When{nk}≡2,Ea,{nk}is the Cantor set Cadefined in[1].Due to the sequence{nk}k≥1,the structure of Ea,{nk}is more general and more complicated than Ca.For example,the lengths of intervals in Cahave“decreasing property”in some sense,that is,for each k≥1,the lengths of all intervals in kth step are longer than in(k+1)th step.But this“decreasing property”does not hold for Ea,{nk}if{nk}is not a constant sequence.

Furthermore,we will explore the level sets of Eaand the complement of the union of level sets.Since level sets of Eamay be viewed as a partial decomposition of the measureµ into a family of subfractals,it is also an important topic in Fractal geometry,see[11,26]for further information.The complement of the setε(µ,α)is the collection of divergent pointsof measureµ,it has attracted an enormous interest in mathematical literature.We refer to [3,5,13,15–17,19,24]for relevant studies.In our paper,we show that in the definition of general level sets the balls can be replaced by the intervals in Ea,and Eacan be written as the union of the general level sets.

This paper is organized as follows.In Section 2,we introduce notations and some basic relevent properties.In Section 3,we decompose Eαinto level sets.In Section 4,We define three equivalent relations on Moran-type set and prove that these relations can be characterized by the sequence a={ak}k≥1(see Theorems 4.1,4.2 and 4.3 in Section 4).We also classify Morantype sets by these equivalence relations which is finer than Hausdorff dimension(see Corollary 4.5).In the end,we characterize the logarithmical equivalence in terms of the probability measure supported on a Moran-type set(see Theorem 4.7).

2　Notations and Properties

2.1h-Measures and Equivalence

First,we recall some basic definitions and notions.

Definition 2.1We say a mapping h:ℝ→ℝ is doubling if there exists a constant c≥1 such that h(2t)≤ch(t)for all t∈ℝ.

Definition 2.2We say a mapping h:[o,∞)→[o,∞)is dimension function with h(o)=o if h is continuous,non-decreasing and doubling.

We denote the set of all dimension functions by D.Now we recall h-Hausdorff measure and h-packing measure as well as their corresponding dimensions,see[2o]for details.These quantities are frequenctly used in studying the properties of fractals,and we refer the readers to[14,18,25]for related studies.

Definition 2.3Given a set A⊂ℝ and δ>o,the set{Ai}i≥1⊂ℝ is called a δ-cover of A if A⊆∪iAiand|Ai|≤δ.For each h∈D,the h-Hausdorff measure of A is defined as

If ot and Hs(A)=∞if s

Definition 2.4A family of disjoint open balls with centres in A and diameters smaller than δ is called a δ-packing of a set A.The h-packing pre-measure of A is defined as

The packing dimension dimP(A)is the unique number satisfying Ps(A)=o for s>dimP(A) and Ps(A)=∞for s

To study dimension function,it often requires certain equivalent properties.In[12],they studied the equivalence of quasisymetric mappings on non-connected sets.

Definition 2.5Two non-negative functions f and g are equivalent,denoted by f～g,if there exists a constant c≥1 such that

Definition 2.6Two non-negative functions f and g are logarithmically equivalent,denoted by f◇g,if logf～logg,that is to say,there exists a constant c≥1 such that

Definition 2.7Two non-negative functions f and g are strong-logarithmicallyequivalent, denoted by f◇◇g,if

2.2Properties of Moran-Type Sets

We adopt the following notations throughout our paper.We write B(x,r)or Br(x)for the ball centered at x with radius r.Let{an}n≥1be a positive non-increasing sequence with Σanconvergent and{nk}k≥1be a positive integer sequence with nk≥2,for all k≥1 andand we write

Since the set Eamay be different by changing the sequence{nk},see Example 2.8,we write Ea,{nk}to emphasize the dependence on{nk}.If there is no ambiguity,we write Eafor Ea,{nk}.

For each integer n≥o,let Ωn={(i1i2···in):1≤ij≤nj,1≤j≤n}be the set of sequences with length n with Ωocontaining only the empty wordbe the set of all finite sequences and Ωω={(i1i2···):1≤ij≤nj,j=1,2,···}be the corresp onding set of infinite sequences.Let ω=i1i2···∈Ωω,for each integer k≥1,we write ω|k=i1i2···ik.to be the conjunction of two finite words

Fig.1Middle-Cantor set Ea,{nk}

Fig.2Moran-type set Ea,{nk′}

There exists a natural projection from the Moran-type set to the sequence space Ωω,hence every element in Moran-type set corresponds a unique infinite word in Ωω,i.e.,x∈Eaif and only if there is a unique ω∈Ωωsuch that xfor all k≥1.For each ω∈Ωω,we denote ωabe the unique point in Easuch that ωafor all k≥1,and denote bythe unique interval of step k containing x∈Ea.Note thatBy the construction of Moran-type set Ea,we have the following simple facts.

Property 2For all k≥1 and all ω=i1i2···ik∈Ωk,

Property 3For all k≥1 and all ω=i1i2···ik∈Ωk,

By the extension theorem of measureis a probability measure supported on the Moran-type sets Ea.

Let a={a}be a positive non-increasing sequence withΣaconvergent and h∈D.We

kkwrite

These two quantities play important roles in the estimation of the h-Hausdorff and h-packing measures.

Let a={a}be a positive non-increasing sequence withnconvergent.We define two sequences

It is obvious that an≤an≤¯anfor all n≥1,and

The following two lemmas are required in our proofs,and the proof can be found in[1,1o].

Lemma 2.10Let a={an}be a positive non-increasing sequence withanconvergent.

Let Eabe the Moran-type set.Then for every h∈D,

that

For the h-packing measure,we have the similar results.

Lemma 2.11Let a={an}be a positive non-increasing sequence withanconvergent.

Let Eabe the Moran-type set.Then for every h∈D,

Further,suppose c∗<∞,then for every h∈D,there exists a constant C2>o such that

3　Multifractal Decomposition

Letµbe a finite regular Borel measure on ℝ and h:[o,∞)→[o,∞)be a dimension function.We write

If we take h(t)=tα∈D,we have the usual level set

Lemma 3.1Let a={an}be a positive non-increasing sequence withconvergent. Suppose c∗=Then there exists a constant c>o such that ai≥c|Iω|for all k≥1 and all i≤τ(k),where ω=i1i2···ik∈Ωk.

Clearly,it follows that

Hence,for all ω=i1i2···ik∈Ωk,we have

The conclusion holds with c=

Lemma 3.2Let a={an}be a positive non-increasing sequence withconvergent. Supposebe the Moran-type set.Then there exists a positive integer K such that for all integer k≥K and x∈Ea,

in Lemma 3.1 and K be the integer such thatThen

For each j=1,2,···,τ(k−K),we have that

The lengths of the gaps adjacent to the(x)are the elements of the set{aj|j≤τ(k−K)}, i.e.,the lengths of the two gaps are greater than r.Hence the ball Br(x)is contained in the union ofand its two adjacent gaps.Thus

ProofGiven r>o,let k be the integer such that

For each x∈Ea,we have r≥(x)is the interval in(k+1)th step containing x.Then

By Lemma 3.2,there exists an integer K such that for all k≥K and all x∈Ea,

Therefore,we have

Theorem 3.4Let a={an}be a positive non-increasing sequence withconvergent.

Let Eabe the Moran-type set.Suppose

ProofFor all k≥1 and all ω=i1i2···ik∈Ωk,we write CLand CRfor the lengths of gaps adjacent to Iω∗ik+1,respectively.Let r=min{CL,|Iω∗ik+1|+CR}.Since a={an}is non-increasing,we have Iω∗ik+1⊇Br(z)∩Ea,where z is the left endpoint of Iω∗ik+1.By the Lemma 3.1,there exists a constant δ>o such that r≥δ|Iω∗ik|.Hence Iω∗ik⊆Br(1+δ−1)(z). Sinceis doubling,we have

where c is the doubling constant ofWe can similarly find a constant c′such that

Thus for all k and all x∈Ea,themeasure ofis comparable to the measure ofand hence to the measure of(x)for each fixed K.By Lemma 3.2,there exists an integer K such that for all k≥K and all x∈Ea,

For each r>o,there exists an integer Kosuch that

Since h is non-increasing,it gives that

where c is the doubling constant of h.Sincgoes to∞as r tends towe have thatwe apply Lemma 3.2 repeatedly and have that

where c3and c4are constants.Moreover,we have following inequality

Combining(3.1)with(3.2)gives that

Let k tends to∞,we have that

and we complete the proof.

The following two corollaries are the immediate consequences of Theorem 3.4.

Corollary 3.5Let a={an}be a positive non-increasing sequenceanconvergent and Eabe the Moran-type set.Suppose

Recall that,for each ω∈Ωω,we write ωafor the unique element in Easuch that ωa∈for all k≥1.

Corollary 3.6Let a={an}be a positive non-increasing sequence withconvergent and Eabe the Moran-type set.Suppose

and p∗=>o.Then

4　The Classification of Moran-Type Sets

In this Section,we will classify Moran-type sets by considering subsets of dimension function D,namely

Theorem 4.1Let a={an}and b={bn}be positive non-increasing sequences withconvergent.Suppose

where θ=a or b.The following are equivalent:

(i)Ea～Eb;

(ii)there exists a constant c≥1 such that for all α>c,there exists an integer K such that for all k>K,

Proof(i)=⇒(ii)Since Ea～Eb,the dimension functions are equivalent,i.e.,ha～hb, that is there exists a constant c≥1 such that

to the same sequence,without loss of generality,we assumeThus hθis strictly increasing.Then for all k≥K

(ii)=⇒(i)Without loss of generality,we assumefor θ=a or b.There exists a constant c≥1 such that,for all α>c,by(ii)and the monotonicity of{an},there exists an integer K such that for k≥K,

Hence,we have

Similarly,the lower limit is bounded by

Since(4.1)and(4.2)hold for all α>c,we have the lower and upper bounds

Let x>o and x∈Ea.Recall thatis the k-th interval in Eacontaining x,by the fact

Let k tends to∞,we have

where caand cbare the doubling constants of haand hb,and it gives that ha～hb,i.e.,Ea～Eb.

Two Moran-type sets Eaand Ebare said to be logarithmically equivalent,denoted by Ea◇Eb,if their dimension functions are logarithmically equivalent.

Theorem 4.2Let a={an}and b={bn}be positive non-increasing sequences withconvergent.Suppose

where θ=a or b.The following are equivalent:

(i)Ea◇Eb;

(ii)there exists a constant c≥1,such that for each,there exists an integer K=K(β)such that for all k≥K,

Two Moran-type sets Eaand Ebare said to be strong-logarithmically equivalent,denoted by Ea◇◇Eb,if their dimension functions are strong-logarithmically equivalent.

Theorem 4.3Let a={an}and b={bn}be positive non-increasing sequences withbnconvergent.Suppose

where θ=a or b.The following are equivalent:

(i)Ea◇◇Eb;

(ii)for each α>1,there exists an integer K=K(α)such that for all k≥K,

The proofs of Theorems 4.2 and 4.3 are similar to Theorem 4.1.

Theorem 4.4Let a={an}and b={bn}be positive non-increasing sequences withbnconvergent.Suppose

where θ=a or b.If Ea◇Eb,there exists a constant c≥1 such that

ProofBy Theorem 4.2,we have Ea◇Ebif and only if there exists a constant c≥1,forthere exists an integer K=K(β)such that for all k≥K,

On the one hand,

and the conlusion holds.

Corollary 4.5Let a={an}and b={bn}be positive non-increasing sequences withbnconvergent.Suppose

where θ=a or b.If Ea◇◇Eb,we have

By Theorem 4.2,logarithmical equivalence can be described in terms of properties of the tail sequences.It enables us to characterize the logarithmical equivalence of Moran-type sets

where ωa∈Ea.

Theorem 4.6Let a={an}and b={bn}be positive non-increasing sequences withnconvergent.Suppose

where θ=a or b.If Ea◇◇Eb,we have

For each k≥K,there exist k1and k2such that

Taking k tends to∞,we have

Similarly,we have

Hence we have

Theorem 4.7Let a={an}and b={bn}be positive non-increasing sequences withconvergent.Suppose

Given ε>o,there exists an integer K=K(ε)such that for all k≥K,we have

For each k,there exists a positive integer

where p∗=sup

The other side is obtained by the same way.Let c=≥1,so we have for allthere exists integer K such that when k≥K,The conclusion holds by Theorem 4.2.

AcknowledgementsThis work was done during the visit to Morningside Center of Mathematics in 2o13,and the authors would like thank their hospitality.The authors are also grateful to Prof.Wen Zhiying and Prof.Xiong Ying for their helpful suggestion.

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∗September 16,2015;revised March 28,2016.Jun Jie Miao was partially supported by NSFC (11201152),STCSM(13dz2260400)and FDPHEC(20120076120001).The author Min Wu was supported by NSFC(11371148),Fundamental Research Funds for the central Universities,scut(2012zz0073),Fundamental Research Funds for the Central Universities SCUT(D2154240)and Guangdong Natural Science Foundation (2014A030313230).†

Min WU.