Shaoyuan Xu，Wangbin Xuand Zuoling Zhou

1School ofM aths and Statistics，Hanshan Normal University，Chaozhou 521001，China

2SchoolofM athsand Statistics，HubeiNormalUniversity，Huangshi435002，China3SchoolofLingnan，Zhongshan University，Guangzhou 510275，China

Some Results on the Upper Convex Densities of the Self-Sim ilar Sets at the Con tracting-Sim ilarity Fixed Poin ts

Shaoyuan Xu1，?，Wangbin Xu2and Zuoling Zhou3

1School ofM aths and Statistics，Hanshan Normal University，Chaozhou 521001，China

2SchoolofM athsand Statistics，HubeiNormalUniversity，Huangshi435002，China
3SchoolofLingnan，Zhongshan University，Guangzhou 510275，China

Received 21 June 2014；Accep ted（in revised version）22March 2015

.In this paper，Some Resultson the upper convex densities of self-sim ilar sets at the contracting-sim ilarity fixed pointsare d iscussed.Firstly，a characterization of the upper convex densities of self-sim ilar sets at the contracting-sim ilarity fixed points is given.Next，under the strong separation open set condition，the existence of the best shape for the upper convex densities of self-sim ilar sets at the contracting-sim ilarity fixed points is proven.As consequences，an open problem and a con jectu re，which were posed by Zhou and Xu，are answered.

Self-sim ilar set，upper convex density，Hausdorffm easu re and Hausdorffd im ension，contracting-sim ilarity fixed point.

AM SSub jectClassifications：28A 78，28A 80

1 Introduction and preliminaries

It iswell know n that the theory of Hausdorffm easu re is the basis of fractal geom etry and Hausdorffm easu re isan im portan tnotion in the study of fractals（see［1，2］）.butunfortunately，it is usually difficult to calcu late the exact value of the Hausdorffm easures of fractal sets.Since Hu tchinson［3］first introduced the self-sim ilar set satisfying the open set cond ition（OSC），m any au thors have stud ied this class of fractals and obtained a num ber ofMeaningfu l Results（see［1-10］）.Am ong them，Zhou and Feng's paper［5］has attracted w idesp read attention since itwas published in 2004.In［5］，Zhou and Feng thought the reason for the difficulty in calcu lating Hausdorffm easu res of fractals is neither com pu tational trickiness nor com pu tational capacity，but a lack of fu ll understanding of the essence of Hausdorffm easu re.Some au thors recently stud ied self-sim ilar setsby Means of upper convex density and best covering，which are very im portan t to the study of Hausdorffm easure（see［5-10］）.In［5］，Zhou and Feng posed eightopen problem s and six con jectu res on Hausdorffm easu re of sim ilar sets.Am ong them，a problem and a con jectu re are as follows.

Let E?Rnbe a self-sim ilar setsatisfying OSC，the Hausdorff d im ension of E be s，i.e.，d imHE=s，and x∈Rn.Denote by DsC（E，x）the upper convex density of E at the point x.

Prob lem 1.1（see［5］）.Underw hatcond itions is therea subset Uxin Rnwith|Ux|＞0 such that

Such a set Uxis called a bestshape for the upper convex density of E at the point x.

Con jectu re 1.1（see［5］）.s=d imHE＞1?there isan x∈E such thatFu rtherm oredenotes the Cartesian p roduct of them idd le third Can tor setwith itself and A denotesany vertex of C×C（see［5，F(xiàn)ig.4］）.

Recently，in order to study Con jectu re 1.1 above，Xu［6］and Xu and Zhou［7］introduced the notion”contracting-sim ilarity fixed point”，and obtained a su fficien t and necessary cond ition for the upper convex density of the self-sim ilar s-set at the sim p lecontracting-sim ilarity fixed point less than 1.In［7］，a conjecturewas posed as follows. Con jectu re 1.2（see［7］）.Let E?Rnbea self-sim ilar s-setsatisfying OSC.Suppose that x is a contracting-sim ilarity fixed pointof E.Then）＜1 if and only if Hs（E∩U）＜|U|sfor each com pact subset U in Rnwith x∈U and|U|＞0.

This is an im portant con jectu re connecting Hausdorffm easu re and upper convex density.In Xu［6］，it was show n that Con jectu re 1.2 w ou ld be true if we only considered the upper convex density at the sim p le-contracting-sim ilarity fixed point of a selfsim ilar s-set satisfying the strong separation set cond ition（SSC），instead of the one at the con tracting-sim ilarity fixed poin tof a self-sim ilar s-setsatisfying OSC.In the subsequent section（i.e.，Section 2），wewill setup a characterization of the upper convex densitiesof self-sim ilar set at the contracting-sim ilarity fixed poin ts.Then，under the strong separation cond ition（SSC），we show that the existence of the best shape for the upper convex densitiesof the self-sim ilar setsat the contracting-sim ilarity fixed points.Asapp lications，we answer an open problem（i.e.，Problem 1.1 above），which was posed by Zhou and Feng in 2004.As consequences，we prove Con jectu re 1.2 does hold true in the case that SSC is satisfied，thus generalizing the correspond ing the resu lt in Xu［6］.Some definitions，notationsand know n Resultsare from references［1-4］.

Let d be the standard d istance function on Rn，where Rnis Euclid ian n-space.Denote d（x，y）by|x-y|，?x，y∈Rn.If U is a nonem p ty subsetof Rn，we define the d iam eter of U as|U|=sup｛|x-y|：x，y∈U｝.Letδbe a positive num ber.If E?SiUiand 0＜|Ui|≤δfor each i，we say that｛Ui｝is aδ-covering of E.

Let E?Rnand s≥0.Forδ＞0，define

Lettingδ→0，we call the lim it

the s-d im ensional Hausdorffm easure of E.Note that the Hausdorff d im ension of E is defined as

An Hs-m easu rable set E?Rnwith 0＜Hs（E）＜∞is term ed an s-set.

Let E be an s-set in Rn，and x∈Rn.The upper convex density of E at x is defined as（see［1］）

Note that theabove sup rem um m ay justbe taken over allsubsets U in Rnwith x∈U and 0＜|U|≤ρ.

Now we review the self-sim ilar s-set satisfying the open set cond ition.Let D?Rnbe closed.A m ap S：D→D is called a con tracting sim ilarity，if there is a num ber c with 0＜c＜1 such that

where c is called the sim ilar ratio.Itwas proved by Hutchinson（see［1］）thatgiven m≥2 and contracting sim ilarities Si：D→D，（i=1，2，···，m）with sim ilarity ratios cithere exists a unique nonem ptycom pactset E?Rnsatisfying

The set E is called the self-sim ilar s-set for the iterated function system（IFS）herewe assum e that there is a bounded nonem p ty open set V such that

and

which is often referred to as the open set cond ition（OSC）.In this case，we know that 0＜Hs（E）＜∞and so E isan s-set.Fu rtherm ore，we call E satisfying the strong separation cond ition（SSC），if E m eetswith OSC and satisfies

Denote by Jkthe set of all k-sequences（i1，···，ik），where 1≤i1，···，ik≤m，k≥1 and pu t Ei1···ik=Si1?···?Sik（E），which is referred to as k-contracting-copy of E.Obviously，?（i1，···，ik）∈Jk，we have

It is nothard to see that for each k≥1，

Definition 1.1（see［6，7］）.Let E?Rnbe self-sim ilar s-set for the IFS｛S1，···，Sm｝satisfying OSC and x∈E.x is called a contracting-sim ilarity fixed pointof E，if there are a positive integer k and（i1，···，ik）∈Jksuch that

Note that here x is alternatively called the contracting-sim ilarity fixed point of E associated with Si1?···?Sik.

In particu lar，x is called a sim p le-contracting-sim ilarity fixed poin t of E，if there is i∈｛1，···，m｝such that Si（x）=x.

Definition 1.2（see［5-7］）.Let E be an s-set in Rn.A sequence｛Ui｝of subsets in Rnis called an Hs-a.e.covering of E，if there is an Hs-m easu rable subset E0in Rnwith E0?E and Hs（E0）=0 such that｛Ui｝isa covering of E-E0（i.e.，E-E0?SiUi）.A sequence｛Ui｝of subsets in Rnis called a best Hs-a.e.covering of E if it is an Hs-a.e.covering of E such that

Note thata best Hs-a.e.coveringm ay be alternatively called a bestalm ost covering or an op tim alalm ost covering（see［10］）.

Note that a sequence｛Ui｝of Hs-m easurable subsets in Rnis an Hs-a.e.covering of E if and only if Hs（E-SiUi）=0.

Let M and N be both nonem p ty subsets in Rn.Then the Euclid ianm etric between M and N is defined as follows：

Let A and B be both nonem p ty com pact sets in Rn.Recall that the definition of Hausdorffm etric is as follows：

where

which is called the openε-neighborhood of A.

2 MainResults

In this section，we first give a characterization of the upper convex densities of the selfsim ilar sets at the con tracting-sim ilarity fixed points.As app lications，we prove that under the strong separation open set cond ition，there exists a best shape for the upper convex density of the self-sim ilar setat any contracting-sim ilarity fixed points.

Proposition 2.1.Let E?Rnbe an self-sim ilar s-set satisfying OSC and x∈Rn，then the upper convex density of E at x is equivalently as

Proof.It is easily show n that（2.1）is true by Means of the definition of DsC（E，x）as（1.2）. So，the p roof is com p leted.

Theorem 2.1.Let E?Rnbe an self-sim ilar s-set satisfying OSC.Suppose that x∈E is any contracting-sim ilarity fixed pointofE，then

Proof.W rite

Then，for any closed subset U in Rnwith x∈U and|U|＞0，thereexistsa sequencedefined by

such that x∈Vj，|Vj|＞0 and

for all j≥1.M oreover，we also see|Vj|→0 as j→∞since

where c=m ax｛ci1，···，cik｝＜1.Thus，by the definition of upper convex density and（2.1）and（2.3），we have

which is the desired resu lt.

The follow ing theorem isobviousaccord ing to the definitionsofupper convex density and contracting-sim ilarity fixed poin t.So we om it its p roof.

Theorem 2.2.Let E?Rnbe an self-sim ilar s-set satisfying OSC.Suppose that x∈E is any contracting-sim ilarity fixed point ofE，then

By using Theorem 2.1，we can easily deduce the follow ing corollary.Weom ititsp roof. Corollary 2.1.Let E?Rnbe an self-sim ilar s-setsatisfying OSC.Suppose that x∈E isany contracting-sim ilarity fixed pointof E，then

for any subset U in Rnwith x∈U and|U|＞0.

Rem ark 2.1.If we take E=C×C and x=A=（0，1）in Corollary 2.1，then we see that Corollary 2.1 is reduced to［13，Lemma 5.1］.

Now we will use them ain Results to deduce the existence of the best shape for the upper convex densitiesof the self-sim ilar setssatisfying SSC at the contracting-sim ilarity fixed points.

Lemma 2.1.Let E?Rnbe a self-sim ilar s-set yielded by contracting sim ilarities S1，···，Sm，satisfying SSC and x∈E.Suppose that x isa contracting-similarity fixed point ofE，then there isa realnumberεwith 0＜ε＜|E|such that

Proof.Since x∈E isa contracting-sim ilarity fixed poin tof E，by Definition 1.1，there are a positive integer k and（i1，···，ik）∈Jksuch that

Set

By Theorem 2.2，itsu ffices to prove l≥L，since l≤L isobviously seen by（2.9b）and（2.9c）. By the p roperty of SSC of E，for any com pactset V in Rnwith V?E，if|V|＞0，|V|＜εand x∈V，then V?Si1?···?Sik（E）.Set

It is easy to see that p≥1.Thus，we get

but the follow ing

doesnothold.Now w rite g=（Si1?···?Sik）p，then we have

where cijis the contracting ratio of Sij（1≤ij≤m，j=1，2，···，k）.Since x∈g-1（V）∩Si1?···?Sik（E），g-1（V）is com pact in Rn，g-1（V）?E，and g-1（V）?Si1?···?Sik（E）does not hold，the inequality|g-1（V）|≥εdoeshold and therefore it follows from（2.9b），（2.9c）and（2.10）that l≥L，as required.

Lemma 2.2（see［7］）.Let｛An｝bea sequence ofnonempty compact subsets in Rnand A?Rn. Suppose that｛An｝converges to A in Hausdorffmetric.Then wehave

Now we begin to show them ain resu lt in this section.

Theorem 2.3.Let E?Rnbea self-sim ilar s-set satisfying SSC.Suppose that x isa contractingsimilarity fixed pointofE.Then there isa compactsetUxin Rnwith x∈Ux，|Ux|＞0 and Ux?E such that

That is，there existsabest shape Uxfor theupper convex density

Proof.By Lemma 2.1，there exist a real positive num berε＞0 and a sequence｛Ui｝of com pactsubsets in Rnwith x∈Ui，Ui?E and|Ui|≥ε＞0（?i≥1）such that

Thus，there is a subsequence｛Uik｝?｛Ui｝such that｛Uik｝converges to a com pact set Uxin Rnin Hausdorffm etric（see［11］）.Without loss of generality，assum e that｛Ui｝converges to the com pact set Uxin Hausdorffm etric.Hence，by Lemma 2.2，we see Ux?E，|Ux|≥ε＞0，x∈Uxand

So，by Corollary 2.1，we obtain the chain of inequalities

where｛Uij｝isSomesubsequenceof｛Ui｝.Thisshow s that Hs（Ux）=|Ux|sas required.

By Theorem 2.3，we immed iately obtain the follow ing two corollaries，so we om it their proofs.

Corollary 2.2.Let E?Rnbe a self-sim ilar s-set satisfying SSC.Suppose thatx is a contracting-sim ilarity fixed pointx of E.Then）=1 ifand only if there isa com pact subset U in Rnwith x∈U，|U|＞0 and U?E such that Hs（U）=|U|s.

Corollary 2.3.Let E?Rnbe a self-sim ilar s-set satisfying SSC.Suppose that x is a contracting-sim ilarity fixed point x of E.Then DsC（E，x）＜1 ifand only if Hs（U）＜|U|sfor each com pact set U in Rnwith x∈U，|U|＞0 and U?E.

Rem ark 2.2.Theorem 2.3 givesan answer to Problem 1.1，which was posed by Zhou and Feng in 2004.

Rem ark 2.3.Corollary 2.2 and Corollary 2.3 generalize Theorem 4.1 and Corollary 4.2 in Xu［6］，respectively.In add ition，Corollary 2.3 show s that Con jectu re 1.2，which was posed by Xu and Zhou in 2005，is true in the case that E?Rnis a self-sim ilar s-set satisfying SSC.

Acknow ledgm en ts

The first au thor is partially supported by the foundation of the research item of Strong Departm entof Engineering Innovation，which is sponsored by the Strong Schoolof Engineering Innovation of Hanshan Norm al University，China，2013.The second au thor is partially supported by NationalNatu ral Science Foundation of China（No.11371379）.

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?Correspond ing author.Emailaddress：xushaoyuan@126.com（S.Y.Xu）