Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets

Hausdorff Dimension and Hausdorff Measure for Non-in teger based Can tor-t y p e Sets Qinghe Yin Mathematic al Scienc e Institute The Austr alian National University, A ustr alia Qinghe.Yin@maths.an u.edu.au Abstract W e consider digits-deleted sets or Cantor-t y p e sets with β - expansions. W e calculate th e Hausdorff dimension d of th ese sets and show th at d is conti nuous with resp ect to β . The d -dimentional Hausdorff measure of these sets is fi n ite and positive. 1 In tro d u ction The Hausdor ff dimension and Hausdorff measur e o f expansions with deleted dig- its are of in terest to mathematicians. The Cantor middle-third set is a clas s ical example. In 1993 , M. Keane p osed the following question,see [22]: Is t he H ausdorff dimension d ( λ ) of t he one p ar ameter family of Cantor-t yp e sets Λ( λ ) = ( ∞ X k =1 i k λ k : i k = 0 , 1 , 3 ) c ontinuous for λ ∈ [ 1 4 , 1 3 ] ? This question is answered b y Pollicott and Simon [22] in 1995 . They show that d ( λ ) = log 3 − log λ for almost a ll λ ∈ [ 1 4 , 1 3 ] and there exists a dense s et with d ( λ ) < log 3 − log λ . So lomy ak [26] in 1 9 98 shows that the d ( λ )-dimensional Hausdor ff measure of Λ ( λ ) is zero for almos t all λ ∈ [ 1 4 , 1 3 ]. Keane, Smoro dinsky a nd Solomy ak study the size of Λ ( λ ) when λ > 1 3 ([15], 19 95). Notice the definition of Λ ( λ ) implies no restrictio n on the digits 0, 1 and 3. When λ > 1 4 , differen t sequences ma y express the same n umber . In this pap er w e study Hausdorff dimension in a mo re restricted case, as so ciated with 0 2000 Mathematics Subje ct Classific ation: primary 28A80; Secondary 37B10, 28A78 . 1 β -e x pansions. W e o btain very spec ific new r esults for digits- deleted sets o r Cantor-t yp e sets with β -expansio ns. β -tr ansformations and β -ex pansions are first intro duced b y R´ enyi [23] in 1957 and further explored by P a rry [20] in 19 60. F or fixed β > 1, T β : [0 , 1) → [0 , 1) is defined by T β x = β x − ⌊ β x ⌋ , (1) where ⌊·⌋ is the floo r function. By (1) we can define the β -expa nsion for an y x ∈ [0 , 1). Let a 1 = ⌊ β x ⌋ and a n = ⌊ β T n − 1 β x ⌋ . Then x = a 1 β + a 2 β 2 + · · · (2) If w e denote x = (0 .a 1 a 2 · · · ) β , then T β x = (0 .a 2 · · · ) β . R ´ enyi [23] and P a r ry [20] study the erg o dic theory for T β . Given β > 2, for 0 ≤ θ 0 < θ 1 < · · · < θ q − 1 ≤ ⌊ β ⌋ , we define C β ; θ 0 ··· θ q − 1 = { x = (0 .a 1 a 2 · · · ) β : a k ∈ { θ 0 , θ 1 , · · · , θ q − 1 }} . Corresp onding to Λ( λ ) defined ab ov e, we hav e C β ;013 where β = 1 λ ∈ (3 , 4]. It is e a sy to see that Λ( λ ) ⊇ C β ;013 with equa lit y only when λ = 1 4 . The cla s sical Cantor middle-third set is C 3;02 . It is natura l to a s k the following questions: What is is the Hausdorff dimension and H ausdorff me asur e of C β ; θ 0 ··· θ q − 1 ? Is dim H ( C β ; θ 0 ··· θ q − 1 ) c ontinuous with r esp e ct to β ? Comparing Λ( λ ) and C β ;013 , we see that the former is an a ttractor of the iterated function system (IFS) ( X ; f 0 , f 1 , f 2 ) wher e X = [0 , 3 λ 1 − λ ] a nd f 0 ( x ) = λx, f 1 ( x ) = λ ( x + 1) and f 2 ( x ) = λ ( x + 3) , but when β is not a n int eger it is imp os s ible to co nsider C 4;013 as an attractor of an IFS without introducing elab orative a dditional machin eries. Given an IFS ( X ; f 0 , · · · , f n − 1 ) and β < n , w e define the β -attracto r as a certain compact subset of the attractor determined by the β - shift. W e show that when all the maps are similarities with the sa me con tr action ratio 0 < r < 1 and with a separa tion co ndition, the Hausdorff dimension of the β -attracto r is given b y s = log β − log r and the s -dimensiona l Hausdorff measur e is p os itiv e and finite. F or fixe d β > 2 and the set of dig its 0 ≤ θ 0 < · · · < θ q − 1 ≤ ⌊ β ⌋ , we show tha t there exists a n um ber α > 1 s uc h that C β ,θ 0 ··· θ q − 1 is an α -attra ctor of an IFS. The Hausdorff dimension of C β ,θ 0 ··· θ q − 1 is s = log α log β and the s -dimensional Hausdorff measure of C β ,θ 0 ··· θ q − 1 is positive and finite. If th e separa tion condition holds we obtain this immedia tely from the genera l r esult for β -a ttractors. How ever the separation co ndition do es no t hold in genera l. W e need a direct pro of. W e also show that dim H ( C β ,θ 0 ··· θ q − 1 ) is con tinuous with respec t to β for β > θ q − 1 and it has a neg ative deriv ative with resp ect to β fo r a lmost a ll β > θ q − 1 . There 2 exists a nowhere dense subset of ( θ q − 1 , θ q − 1 + 1] with Leb esg ue measure 0 suc h that dim H ( C β ,θ 0 ··· θ q − 1 ) has infinite der iv ativ e or infinite one- sided deriv ative. β -a ttractors fit int o the more general symb olic c onstruction of P esin and W eiss [21]. A β -attra ctor is the limit set of a symbolic construction using the β -s hift. If a symbo lic construction is r e gular , Pesin and W eiss in [21] g ive a low er bo und to the Hausdorff dimensio n of the limit set us ing top o logical pressur e. If β is simple then the symbolic construction using the β -shift is regular under a sepa ration condition. It is not kno w n this is true for all β > 1. Bar reira [4] gives a sufficient condition for a symbolic constr uctio n is regula r. How ever the construction of C β ,θ 0 ··· θ q − 1 do es not obey this condition for some v alue o f β . If the symbolic co ns truction is regular and the e quilibrium measure asso cia ted with an l -es timating v ector is a Gibbs meas ur e then [21] shows that the s - dimensional Hausdo rff measure of the limit set of a symbolic construction is po sitive, where s is a low er b ound of the Hausdorff dimension of the limit set related to the l -es timating vector. It is known that if a s ym bo lic system has the specificatio n pr o pe r ty then any equilibr ium measur e is Gibbs ([24]). A β - shift has the specification pro pe r t y if and only if the length of strings o f 0’s in the β -ex pansion of 1 is bo unded ([5]). Obviously this set do es not contain any int e rv al. Sc hmeling [2 5] show ed that this set has Hausdorff dimension 1. Again, it is not known if ev er y β -shift p oss esses a Gibbs measure as its equilibrium measure ass o c iated with the l -estimating vector of [21]. Although the symbolic construction for β -attrac to rs may not b e regular in general, the sym b olic co ns truction of C β ,θ 0 ··· θ q − 1 is reg ular. It follows that the Hausdorff dimensio n of C β ,θ 0 ··· θ q − 1 can be obtained from the results o f [21] by consider ing the top olog ical pressure of the corresp onding α -shift. How ever, as men tioned ab ov e, in this wa y we can not get the information o f Hausdorff measure of C β ,θ 0 ··· θ q − 1 in all cases. T o ov ercome this difficult y we give a new pro of. It is well k nown that the classical Cant or set can b e defined to b e the digit- deleted set, C = C 3;02 . It can a lso b e defined geo metrically as the attracto r of the IFS F = { [0 , 1]; f 0 , f 2 } , where f i ( x ) = x + i 3 for i = 0 , 1 , 2 are the inv e rse branches of T 3 . But fo r 2 < β < 3 the digit- de le ted set C β ;02 is no t the attractor of an IFS of in verse br anches of T β bec ause of the piecewis e definition of T − 1 β , namely T − 1 β x = ( x β , x +1 β , x +2 β , f or x < β − 2 x β , x +1 β , f or x ≥ β − 2 . How ever we will show that there is an asso cia ted lo c al IFS (see [6], p1 77) who se inv ariant sets are rela ted to C β ;02 in an interesting wa y . A lo cal IFS has its domain of at least one of its functions not eq ual to the whole of the underlying space, [0 , 1] in the pres en t case. W e define F β = { [0 , 1]; f 0 : [0 , 1] 7→ [0 , 1 ] , f 2 : [0 , β − 2 ] 7→ [0 , 1] } where no w f 0 ( x ) = x β , and f 2 ( x ) = x +2 β . An inv ar iant set A of the lo cal IFS F β 3 is a nonempt y compac t subset o f [0 , 1] such that A = f 0 ( A ) ∪ f 2 ( A ∩ [0 , β − 2]) . In general, a loca l IFS may hav e no or many inv a riant sets ([6]). W e sho w that, with the exception o f countably ma n y v alues of β , C β ;02 is the unique inv ariant se t of the loca l IFS F β . Otherwise, F β po ssesses another inv ariant set B β ;02 which can b e constructed by an interv a l remov al pro cess. W e hav e C β ;02 ⊂ B β ;02 and B β ;02 \ C β ;02 consists o f countably many isolated p oints. β -tr ansformations a nd β - expansions ar e of interest to mathematicians in a br oad range. After the pioneer w orks of R´ enyi [23] and P a rry [20], many resear ch works r elated to β -trans formations and β -expansions hav e been pub- lished. Among these works, [1 ], [2], [8], [1 1], [13], [14], [17] and [19], for example, hav e studied fractals or fra ctal sets related to β -expansions. Barns ley [7] used β -tr ansformation to study fra ctal tops. In sectio n 2, w e rec a ll concepts and basic results for Hausdorff dimensio n, it- erated function systems, β -expa nsions and sym b olic dynamical s ystems. In sec- tion 3, we study the Hausdorff dimensio n and Hausdorff measure for β -attractor s under a separ ation co ndition. In section 4 , w e study the Hausdorff dimension and Hausdo rff measure for C β ; θ 0 ··· θ q − 1 , the Can tor -type sets constructed by β - expansions. In section 5 , we study the in v a r iant sets of the lo cal IFS F β . In section 6, we point out some interesting topics for further resea rch. 2 Hausdorff Dimensio n, Iterated F unction Sys- tems, β -expansion and Sy mb olic Dynamical Systems In this section w e in tr o duce concepts and definitions for Hausdorff dimension, iterated function systems, β -expa nsion and sym bo lic dynamical systems. Readers can find co ncepts of Hausdor ff measure, Ha usdorff dimension and it- erated function systems in [9] or [10]. F o r the origina l source o f iterated function systems see [12]. Let E b e a subset of a metric spac e X . A δ -c over of E is a countable or finite collectio n { U i } of subsets of X with | U i | ≤ δ s uc h that E ⊂ ∪ ∞ i =1 U i , where | · | is the diameter o f the given set. F or a ny δ > 0 and s ≥ 0, define H s δ ( E ) = inf ( ∞ X i =1 | U i | s : { U i } is a δ -cover of E ) . Clearly , H s δ ( E ) is decreasing with resp ect to either s or δ . Let H s ( E ) = lim δ → 0 H s δ ( E ) . 4 W e call H s ( E ) the s -dimensional Hausdorff out measure of E . The Hausdorff dimension of E is defined by dim H ( E ) = inf { s ≥ 0 : H s ( E ) = 0 } = sup { s ≥ 0 : H s ( E ) = ∞} . An iterated function system (IFS) F = ( X ; f 0 , f 1 , · · · , f n − 1 ) on a compact metric space X consists of a num b er of contractions f i : X → X , where n ≥ 2 . There exis ts a non-e mpt y compact subset E of X such that E = n − 1 [ i =0 f i ( E ) . W e say E is the a ttractor of the IFS. F or any sequence i 1 , i 2 , · · · with 0 ≤ i k ≤ n − 1, and a ny x ∈ X , we hav e lim k →∞ f i 1 ◦ · · · ◦ f i k ( x ) ∈ E . The ab ove limit ex ists and is indepe nden t of x ∈ X . On the other ha nd, for any y ∈ E there exists a sequence ( i 1 , i 2 , · · · ) such that y = lim k →∞ f i 1 ◦ · · · ◦ f i k ( x ) for any x ∈ X . W e say ( i 1 , i 2 , · · · ) is an address of y . One p oint in E may ha ve more than one addre s ses. W e say the Op en Set Condition holds for an IFS, if there exists a non-empt y op en set O such that f i ( O ) ⊂ O and f i ( O ) ∩ f j ( O ) = ∅ for all 0 ≤ i , j ≤ n − 1 and i 6 = j . Assume that X is a compa c t subset of R m . If f i is a similarit y for all i , i.e., for all x, y ∈ X , d ( f i ( x ) , f i ( y )) = r i d ( x, y ) for some r i < 1, and the Op en Set Condition holds, then the Hausdo r ff dimension of the attractor E is given b y dim H ( E ) = s where s satisfies r s 0 + r s 1 + · · · + r s n − 1 = 1. Let Σ n = { 0 , 1 , · · · , n − 1 } N . Then Σ n is compact with r esp ect to the pro duct top ology . Define a shift map σ : Σ n → Σ n by σ ( i 1 , i 2 , · · · ) = ( i 2 , i 3 , · · · ) . W e call Σ n a full shift. A compact s ubs et Σ of Σ n is said to b e a subs hift if σ (Σ) = Σ. A blo ck ( i 1 , · · · , i k ) is a forbidden w or d of Σ if it do es not appea r in any element of Σ. Σ is a subshift of finite type if it is determined b y a finite set of forbidden words (see [1 6]). Given a n n × n 0-1 matrix M , le t Σ M contain all the elements ( i 1 , i 2 , · · · ) ∈ Σ n such that m i k +1 ,i k +1 +1 = 1 for a ll k ≥ 1 . W e say Σ M is a Markov shift. Clearly , a Ma rko v shift is a subshift o f finite type. 5 Fix β > 1. Assume that the β - expansion o f β − ⌊ β ⌋ is β − ⌊ β ⌋ = ǫ 2 β + ǫ 3 β 2 + · · · Let ǫ 1 = ⌊ β ⌋ . Then 1 = ǫ 1 β + ǫ 2 β 2 + ǫ 3 β 3 + · · · (3) W e say that (3) is the β -expansio n of 1 and denote by 1 = (0 .ǫ 1 ǫ 2 · · · ) β . W e s ay β is simple if the β -expa nsion of 1 has finite ma ny non-zer o terms. Let e i = ǫ i if β is non-simple; or let e kn + i = ǫ i and e ( k +1) n = ǫ n − 1, for all k ≥ 0 a nd i = 1 , 2 , · · · , n − 1, if β is simple with ǫ n > 0 and ǫ j = 0 when j > n . Given a sequence ( a 1 , a 2 , · · · ) with a i ∈ { 0 , 1 , · · · , ⌊ β ⌋} , the expr ession a 1 β + a 2 β 2 + · · · (4) is the β -ex pansion of some x ∈ [0 , 1 ) if a nd only if for any n ≥ 1 o ne has ( a n , a n +1 , · · · ) < ( e 1 , e 2 , · · · ) , where “ < ” is according to the lexico graphical or der. When (4) is the β -expa nsion for so me x ∈ [0 , 1), we say the finite sequence ( a 1 , a 2 , · · · , a n ) is β -admissible. Let 1 = ǫ 1 β + ǫ 2 β 2 + ǫ 3 β 3 + · · · and 1 = ǫ ′ 1 α + ǫ ′ 2 α 2 + ǫ ′ 3 α 3 + · · · be the β and α -expansions of 1. If β < α then ( ǫ 1 , ǫ 2 , · · · ) < ( ǫ ′ 1 , ǫ ′ 2 , · · · ) . Use Σ β to denote the closure of all β - admissible se quences under pro duct top ology . Then Σ β is a subshift whic h w e call the β -shift. IF β = n is an int e ger, then Σ β is the full shift. If β is simple, then Σ β a subshift of finite t yp e . F or example, when β = √ 5+1 2 , the golden mean, Σ β is a Mar ko v shift with M =  1 1 1 0  . 3 β - attractors and Th eir Hausdorff Dimension. Let ( X ; f 0 , f 1 , · · · , f n − 1 ) b e an IFS with attr a ctor E . Define φ : Σ n 7→ E by φ ( i 1 , i 2 , · · · ) = lim k →∞ f i 1 ◦ · · · ◦ f i k ( x ) . 6 φ is well defined since the limit in the right hand side exists, is indep endent of x , and is in E . Definition. Given an IFS ( X ; f 0 , f 1 , · · · , f n − 1 ) with attrac to r E . F or a subshift Σ of Σ n let E Σ = φ (Σ) . W e ca ll E Σ the Σ-a ttractor of the g iven IFS. In particular , when Σ is a β -shift for so me β ≤ n , we use E β to denote E Σ β and c a ll it the β -attracto r. Remark. The Σ-a ttractor defined abov e is a special case of the more general symbolic co nstruction o f [21]. Many differen t se ttings of fractals with iterated function sy s tems can b e viewed as Σ-attractor s for some s ubshifts. F o r example, the fractals defined by Ma rko v shifts (see [3] o r [27]), the graph-directed fra ctals first pro po s ed by Mauldin and Williams [18] can fit int o this setting. How ever, when β is non-simple, it is difficult to fit E β int o known settings other than the symbolic co nstruction of [21]. Recall that when all f i ’s are similarities with scale r i < 1 and with the op en set condition, the Hausdorff dimension of the attra ctor E is determined by P n − 1 i =0 r s i = 1. When all the r i ’s are equal (= r , say), w e hav e s = log n − log r . F or β -attra ctors we hav e the following result. Theorem 1. L et ( X ; f 0 , f 1 , · · · , f n − 1 ) b e an IFS such that d ( f i ( x ) , f i ( y )) = r d ( x, y ) for a l l x, y ∈ X and 0 ≤ i ≤ n − 1 , wher e 0 < r < 1 . L et 1 < β < n . Assu me that the β -attr actor E β has the fol lowing sep ar ation c ondition: f i ( E β ) ∩ f j ( E β ) ∩ E β = ∅ , for i 6 = j. Then the Hausdorff dimension of E β is given by dim( E β ) = log β − log r . The s -dimensional Hausdorff me asur e of E β is p ositive and finite, wher e s = dim H ( E β ) . Obviously , theo rem 1 holds when β is an integer. Comparing the integer case, we may in terpret the β -attractor as it is constructed b y β man y similarities. F or some v alue of β , dim H ( E β ) can b e computed by existing metho d. F or example, if β = √ 5+1 2 , then Σ β is a Mar ko v shift with M =  1 1 1 0  (the golden-mean shift). With the separatio n conditio n, the Hausdorff dimension can b e c alculated by k M R s k = 1, where M R s =  r s r s r s 0  and k · k is the Perron-F rob enus eigenv alue of the given matrix. Then we have 1 − r s − r 2 s = 0. This gives r s = β − 1 and s = log β − log r . If β is given by 1 = 1 β + 1 β 3 , then the 7 β -e x pansion of rea l n umber s induce a s hift of finite type with forbidden words determined by the set { (0 , 1 , 1) , (1 , 0 , 1) , (1 , 1 , 0 ) , (1 , 1 , 1) } . Define a n IFS fo r med by the co mpo sed maps, { f 0 ◦ f 0 , f 0 ◦ f 1 , f 1 ◦ f 0 } . Then E β is the Marko v attr a ctor with M =   1 1 1 1 0 0 1 1 0   . Noting that the comp os e d maps are similarities with scales r 2 , then with the separatio n condition, the Hausdo rff dimension of E β is given by k M R 2 s k = 1. This gives us r 6 s + 2 r 4 s + r 2 s − 1 = 0 . But λ 6 + 2 λ 4 + λ 2 − 1 = ( λ 3 + λ − 1)( λ 3 + λ + 1) . Hence we hav e r 3 s + r s = 1 which implies r s = β − 1 , a nd therefore s = log β − log r . Obviously , this discussion can only apply to particular v a lues of β when the set of forbidden words a re “ short” and the transition matrix is “small” o f size whose eigenv alue is “calculatable” . It is hard to apply to general case. Besides, it is not a pplica ble when the re la ted shift is not a subshift of finite t yp e. Use S k β to denote the set of all β - a dmissible sequences of length k . T o prov e Theorem 1 we need to es timate the size of S k β . T he following res ult can be found in [2 3] (equations (4.9), (4.10 )). Lemma 1. We have β k ≤ |S k β | ≤ β k +1 β − 1 . Pr o of of Th e or em 1 . First we sho w that dim H ( E β ) ≤ log β − log r . F or ( i 1 , · · · , i k ) ∈ S k β } , denote ∆ i 1 ··· i k = f i 1 ◦· · ·◦ f i k ( X ). Then the collectio n { ∆ i 1 ··· i k : ( i 1 , · · · , i k ) ∈ S k β } is a cov e r of E β . Since all f i ’s are similar ities with scale r , we hav e | ∆ i 1 ··· i k | ≤ r k | X | . By Lemma 1, w e get X i 1 ··· i k ∈S k β | ∆ i 1 ··· i k | s ≤ β k +1 β − 1 r sk | E β | s = β β − 1 | X | s , where s = log β − log r . This shows that dim( E β ) ≤ s and the s -dimensional Hausdo rff measure of E β is finite. Next we show tha t dim( E β ) ≥ log β − log r . W e will show that for some δ 0 > 0 we hav e H s δ 0 ( E β ) > 0 where s = log β − log r . Then H s ( E β ) = lim δ → 0 H s δ ( E β ) ≥ H s δ 0 ( E β ). T his gives dim H ( E β ) ≥ s and the s - dimensional Haus dorff measure of E β is p os itiv e. 8 By the separ ation condition, δ 0 = min i 6 = j { d ( f i ( E β ) ∩ E β , f j ( E β ) ∩ E β } > 0 . Then there exis ts l > 0 such that r l +1 | X | < δ 0 ≤ r l | X | . Let U = { U i } b e a δ 0 cov er o f E β . Since E β is c o mpact without loss o f generality we may assume that U = { U 1 , · · · , U N } is a finite cov er. W e may also assume that U i ⊂ E β . In fact we can us e U i ∩ E β to replac e U i . Cho os e k i such that r k i +1 | X | < | U i | ≤ r k i | X | . W e claim that U i ⊂ ∆ i 1 ··· i k i − l for some β -admissible ( i 1 , · · · , i k i − l ). T a king y ∈ U i which has an a ddress ( i 1 , i 2 , · · · ) ∈ Σ β , for a n y z ∈ E β with an addre ss ( j 1 , j 2 , · · · ) ∈ Σ β such that ( i 1 , · · · , i k i − l ) 6 = ( j 1 , · · · , j k i − l ). Assume that i 1 = j 1 , · · · , i p = j p but i p +1 6 = j p +1 for so me p < k i − l . Then d ( y , z ) = d ( f i 1 ◦ · · · ◦ f i p ◦ f i p +1 ( y 1 ) , f i 1 ◦ · · · ◦ f i p ◦ f j p +1 ( z 1 )) for so me y 1 , z 1 ∈ E β . It is obvious that f i p +1 ( y 1 ) , f j p +1 ( z 1 ) ∈ E β . Then, d ( y , z ) = r p d ( f i p +1 ( y 1 ) , f j p +1 ( z 1 )) ≥ r p d ( f i p + l ( E β ) ∩ E β , f j p + l ( E β ) ∩ E β ) ≥ r p δ 0 >r p r l +1 | X | ≥ r k i − l − 1 r l +1 | X | = r k i | X | > | U i | . Hence z / ∈ U i and U i ⊂ ∆ i 1 ··· i k i − l . Now we get ano ther cov er of E β : C = { ∆ i 1 ··· i k i − l : U i ⊂ ∆ i 1 ··· i k i − l , r k i +1 | X | < | U i | ≤ r k i | X | , U i ∈ U } . F or s > 0 we ha ve X ∆ i 1 ··· i k i − l ∈C | ∆ i 1 ··· i k i − l | s = X ∆ i 1 ··· i k i − l ∈ ∆ r s ( k i − l ) | X | s ≤ r s ( − l − 1) X U i ∈U | U i | s (5) Let k = max U i ∈U { k i − l : r k i +1 | X | < | U i | ≤ r k i | X |} . Refine C into C ′ = { ∆ i 1 ··· i k : ( i 1 , · · · , i k ) ∈ S k β } . Then | ∆ i 1 ··· i k | s = r s ( k − k i ) | ∆ i 1 ··· i ki − l | s . Set s = log β − log r , then | ∆ i 1 ··· i k | s = β − ( k − k i ) | ∆ i 1 ··· i k i − l | s . By Lemma 1 w e see that ea ch ∆ i 1 ··· i k i − l ∈ C contains a t most β k − k i + l +1 β − 1 many ∆ i 1 ··· i k ∈ C ′ . Then X ( j 1 , ··· ,j k ) ∈S k β j 1 = i 1 , ··· ,j k i − l = i k i − l | ∆ j 1 ··· j k | s ≤ β l +1 β − 1 | ∆ i 1 ··· i k i − l | s . 9 Therefore, X ∆ i 1 ··· i k i − l ∈C | ∆ i 1 ··· i k i − l | s ≥ β − l − 1 ( β − 1) X ∆ i 1 ··· i k ∈C ′ | ∆ i 1 ··· i k | s The r ight hand side contains all the β -admiss ible seq uences in S k β and we have | ∆ i 1 ··· i k | = r k | X | . Hence X ∆ i 1 ··· i k i − l ∈C | ∆ i 1 ··· i k i − l | s ≥ β − l − 1 ( β − 1) X ( i 1 , ··· ,i k ) ∈S k β r sk | X | s ≥ β − l − 1 ( β − 1) β k r sk | X | s = β − l − 1 ( β − 1) | X | s . (6) By (5) and (6) we obtain X U i ∈U | U i | s ≥ r s ( l +1) X ∆ i 1 ··· i k i − l ∈C | ∆ i 1 ··· i k i − l | s ≥ r s ( l +1) β − l − 1 ( β − 1) | X | s = β − 2( l +1) ( β − 1 ) | X | s for any finite δ 0 -cov er U . Hence dim H ( E β ) ≥ s and H s ( E β ) > 0.  4 Hausdorff Dimension for C β ; θ 0 ··· θ q − 1 Let C β ; θ 0 ··· θ q − 1 be as defined in section 1. Use z β to denote the max im um o f C β ; θ 0 ··· θ q − 1 . W e hav e the following result. Lemma 2. z β c an b e expr esse d a s z β = z 1 β + z 2 β 2 + · · · (7) wher e z i ∈ { θ 0 , · · · , θ q − 1 } , and ( z i , z i +1 , · · · ) ≤ ( z 1 , z 2 , · · · ) . (8) Pr o of. By the definition of C β ; θ 0 ··· θ q − 1 , w e ha ve tw o possibilities: the digits o f the β - e xpansion of z β are in { θ 0 , · · · , θ q − 1 } , o r z β is a limit of num b ers who s e β - expansion consis ts o f digits in { θ 0 , · · · , θ q − 1 } . F o r the firs t case, the β -expansio n of z β satisfies the requirement o f Lemma 2. F or the second ca s e, ther e exits a sequence { y k } with digits in { θ 0 , · · · , θ q − 1 } such that y k ↑ z β . Assume that y n = (0 .y k 1 y k 2 · · · ) β . Then we ha ve ( y k 1 , y k 2 , · · · ) < ( y k +1 1 , y k +1 2 , · · · ) . Since Σ β is compa ct under the pro duct top olog y , the sequence { ( y k 1 , y k 2 , · · · ) } has a limit ( z 1 , z 2 , · · · ). It is o bvious that ( z 1 , z 2 , · · · ) satisfies (7). W e show 10 that it satisfies (8) as well. If it is not true, then ( z k , z k +1 , · · · ) > ( z 1 , z 2 , · · · ). Let ˜ y n = (0 .y n k y n k +1 · · · ) β . Then lim n →∞ ˜ y n = z k β + z k +1 β 2 + · · · > z β , a contradiction.  Using ( z 1 , z 2 , · · · ) we define a new sequence ( ω 1 , ω 2 , · · · ) with ω i = j if z i = θ j . Then ( ω k , ω k +1 , · · · ) ≤ ( ω 1 , ω 2 , · · · ) . Let α β ; θ 0 ··· θ q − 1 be determined by 1 = ω 1 α + ω 2 α 2 + · · · ( 9) Then either (9) is the α β ; θ 0 ··· θ q − 1 -expansion of 1 o r ( ω 1 , ω 2 , · · · ) is p erio dic. Theorem 2. The Hausdo rff dimension of C β ; θ 0 ··· θ q − 1 is given by dim H ( C β ; θ 0 ··· θ q − 1 ) = log α β ; θ 0 ··· θ q − 1 log β . The s -dimensional Hausdorff me asur e of C β ; θ 0 ··· θ q − 1 is p ositive and fin ite wher e s = dim H ( C β ; θ 0 ··· θ q − 1 ) . Pr o of. F or 0 ≤ i ≤ q − 1 define f i : h 0 , β β − 1 i 7→ h 0 , β β − 1 i by f i ( x ) = x + θ i β . W e will show that C β ; θ 0 ··· θ q − 1 is the α -attra ctor of the IFS h 0 , β β − 1 i ; f 0 , · · · , f q − 1  . Here and rest of the pro of w e use α to denote α β ; θ 0 ··· θ q − 1 . Then by the pro of of Theorem 1 we get that dim H ( C β ; θ 0 ··· θ q − 1 ) ≤ log α log β and the s -dimensional Hausdorff mea sure of C β ; θ 0 ··· θ q − 1 is finite fo r s = log α log β . If we further hav e the separ ation condition, then Theorem 2 is prov ed. First w e sho w that C β ; θ 0 ··· θ q − 1 is an α -attractor . Use Σ β ; θ 0 ··· θ q − 1 to denote the closur e of all sequences ( a 1 , a 2 , · · · ) w ith a i ∈ { θ 0 , · · · , θ q − 1 } which forms the β -expans io n of so me x ∈ [0 , 1). Then ( a 1 , a 2 , · · · ) 7→ a 1 β + a 2 β 2 + · · · is a 1-1 co r resp ondence b etw een Σ β ; θ 0 ··· θ q − 1 and C β ; θ 0 ··· θ q − 1 . Given x ∈ C β ; θ 0 ··· θ q − 1 , as s ume that x = a 1 β + a 2 β 2 + · · · for some ( a 1 , a 2 , · · · ) ∈ Σ β ; θ 0 ··· θ q − 1 . Define ( i 1 , i 2 , · · · ) ∈ Σ α according to a k = θ i k . Then x = lim k →∞ x k where x k = a 1 β + a 2 β 2 + · · · + a k β k = f i 1 ◦ f i 2 ◦ · · · ◦ f i k (0) . 11 On the o ther hand, fo r any ( i 1 , i 2 , · · · ) ∈ Σ α and any k > 0 we hav e ( θ i 1 , θ i 2 , · · · , θ i k , θ i 0 , · · · ) ∈ Σ β ; θ 0 ··· θ m − 1 . Denote x ∗ = θ 0 β + θ 0 β 2 + · · · . Then x k = f i 1 ◦ f i 2 ◦ · · · ◦ f i k ( x ∗ ) = θ i 1 β + θ i 2 β 2 + · · · + θ i k β k + θ 0 β k +1 + · · · ∈ C β ; θ 0 ··· θ q − 1 . Hence lim k →∞ f i 1 ◦ f i 2 ◦ · · · ◦ f i k ( x ∗ ) ∈ C β ; θ 0 ··· θ q − 1 . This shows that C β ; θ 0 ··· θ q − 1 is the α -attractor o f the IFS h 0 , β β − 1 i ; f 0 , · · · , f q − 1  . If we ha ve f i ( C β ; θ 0 ··· θ q − 1 ) ∩ f j ( C β ; θ 0 ··· θ q − 1 ) = ∅ for i 6 = j then Theorem 2 can b e obtained by Theor em 1. Unfortunately in some cases we do not hav e the separation condition. F or example, when β = 4, the separation condition do es not hold for C 4;013 . F or Hausdo rff dimension we may use the following argument. F or α ′ < α , we use C α ′ β ; θ 0 ··· θ q − 1 to denote the α ′ -attractor of the IFS in consideratio n. Then C α ′ β ; θ 0 ··· θ q − 1 ⊂ C β ; θ 0 ··· θ q − 1 . Since α ′ < α , we ha ve 1 / ∈ C α ′ β ; θ 0 ··· θ q − 1 . Then the separ ation condition holds for C α ′ β ; θ 0 ··· θ q − 1 . Ther e fore, dim H ( C β ; θ 0 ··· θ q − 1 ) ≥ dim H ( C α ′ β ; θ 0 ··· θ q − 1 ) = log α ′ log β for any α ′ < α This implies that dim H ( C β ; θ 0 ··· θ q − 1 ) ≥ log α log β . How ever, this does not supply an y info r mation ab out the Hausdorff mea s ure of dimension log α log β . W e will use similar discussion a s the pro of of Theo rem 1. Notice that when separ a tion condition fails we must ha ve 0 , 1 ∈ C β ; θ 0 ··· θ q − 1 . F or a n α -admissible sequence ( i 1 , · · · , i k ) let I i 1 ··· i k to deno te the smallest closed int e rv al which contains a ll num b ers whose β -expa nsion starting with ( θ i 1 , · · · , θ i k ). If ( i 1 , · · · , i k ) 6 = ( j 1 , · · · , j k ) then I i 1 ··· i k ∩ I j 1 ··· j k contains at most one po in t. Le t U = { U 1 , · · · , U n } b e a finite co ver o f C β ; θ 0 ··· θ q − 1 . If β − k − 1 < | U i | ≤ β − k then U i int e rsects a t most three I i 1 ··· i k . Then X I i 1 ··· i k ∩ U i 6 = ∅ | I i 1 ··· i k | s ≤ 3 β − ks < 3 β s | U i | s Assume tha t | U i | > β − l for all i . By Lemma 1 ( i 1 , · · · , i k ) ca n b e extend to at most α l − k +1 α − 1 many ( i 1 , · · · , i k , · · · , i l ). Then X I i 1 ··· i l ∩ U i 6 = ∅ | I i 1 ··· i l | s ≤ 3 α l − k +1 α − 1 β − ls = 3 α − k +1 α − 1 < 3 α 2 | U i | s α − 1 12 where s = log α log β . Hence N X i =1 | U i | s > α − 1 3 α X ( i 1 , ··· ,i l ) | I i 1 ··· i l | s (10) where the sum is ov er a ll α -admissible ( i 1 , · · · , i l ). W e claim that if the separation condition do es not hold then we hav e α > 2. In fact if α ≤ 2 we ha ve the following three p oss ibilities : 1. θ 1 < ⌊ β ⌋ , which implies 1 / ∈ C β ; θ 0 ··· θ q − 1 ; 2. θ 0 > 0, which implies 0 / ∈ C β ; θ 0 ··· θ q − 1 ; 3. θ 1 ≥ θ 0 + 1. In a ll these three cases we have f 0 ( C β ; θ 0 ··· θ q − 1 ) ∩ f 1 ( C C ) = ∅ . Now w e estimate the num b er of I i 1 ··· i l that | I i 1 ··· i l = β − l . Let A k = { ( i 1 , · · · , i k ) | ( i 1 , · · · , i k − 1 , i k + 1) is α -admiss ible } . If ( i 1 , · · · , i k ) ∈ A k then ( θ i 1 , · · · , θ i k − 1 , θ i k +1 ) is β - a dmissible. Thu s ( θ i 1 , · · · , θ i k − 1 , θ i k + 1) is β -admiss ible since θ i k + 1 ≤ θ i k +1 . Therefore we hav e | I i 1 ··· i k | = β − k . Let S k α of all α -a dmissible ( i 1 , · · · , i k ). By Lemma 1 we hav e |S k α | ≥ β k . W e show that | A k | ≥ c |S k α | for some co nstant c . Let B k = S k α \ A k . W e estimate | B k | | A k | . No tice that | B k +1 | ≤ |S k α | = | A k | + | B k | and | A k +1 | ≥ ⌊ α ⌋ · | A k | . Then | B k +1 | | A k +1 | ≤ | A k | + | B k | ⌊ α ⌋ · | A k | = 1 ⌊ α ⌋  1 + | B k | | A k |  . Contin ue this discussion w e get that for a ny k we have | B k | | A k | ≤ 1 ⌊ α ⌋ + 1 ⌊ α ⌋ 2 + · · · = 1 ⌊ α ⌋ − 1 . Therefore, | A l | |S l α | ≥ 1 1 + 1 ⌊ α ⌋− 1 = ⌊ α ⌋ − 1 ⌊ α ⌋ . By this and (10) we obtain N X i =1 | U i | s > α − 1 3 α X ( i 1 , ··· ,i l ) ∈ A l | I i 1 ··· i l | s ≥ α − 1 3 α · ⌊ α ⌋ − 1 ⌊ α ⌋ · |S l α | · β − sl ≥ α − 1 3 α · ⌊ α ⌋ − 1 ⌊ α ⌋ · α l β − sl = α − 1 3 α · ⌊ α ⌋ − 1 ⌊ α ⌋ 13 for all finite cov er U , where s = log α log β . This pr ov es that the s -dimensional Hausdorff mea s ure of C β ; θ 0 ··· θ q − 1 is p os itiv e.  Remark. Observe that the Ha usdorff dimension of C β ; θ 0 · θ q is rela ted to “ base change”. F or the classica l Can to r set C , if we change base 3 into base 2 and the digit 2 in to 1, we g et an (almost) 1-1 map from C to [0 , 1 ]. Theorem 2 demonstrates that the “ba se change” pr op erty a lso holds for non-integer bases. Example. Let β 1 , β 2 be given b y 1 = 3 β 1 + 2 β 2 1 and 1 = 3 β 2 + 3 β 2 2 . Then β 1 = 3+ √ 17 2 and β 2 = 3+ √ 21 2 . W e consider C β ;013 for β ∈ [ β 1 , β 2 ]. It is easy to see that Σ β 1 ;013 = Σ β 2 ;013 , and hence Σ β ;013 = Σ β 1 ;013 for any β ∈ [ β 1 , β 2 ]. The maximal seq uence of Σ β 1 ;013 is (3 , 1 , 3 , 1 , · · · ). Thus α β ;013 is determined by 1 = 2 α + 1 α 2 + 2 α 3 + 1 α 4 + · · · which giv es α = 1 + √ 3. therefor e, for a n y β ∈ [ β 1 , β 2 ] dim H ( C β ;013 ) = log(1 + √ 3) log β . By this w e see that dim H ( C β ;013 ) is decr easing with respect to β on the interv al [ β 1 , β 2 ]. Theorem 3 . F or fixe d θ 0 , · · · , θ q − 1 , dim H ( C β ; θ 0 ··· θ q − 1 ) is c ontinuous for β > θ q − 1 . Pr o of. W e need to prove that α β ; θ 0 ··· θ q − 1 is contin uous w ith res pect to β for β > θ q − 1 . Obviously , α β ; θ 0 ··· θ q − 1 is non-decr easing with resp ect to β . It is clear that we ha ve α β ; θ 0 ··· θ q − 1 > q − 1. If w e can show that for any α ∈ ( q − 1 , q ], there exists a β s uc h that α = α β ; θ 0 ··· θ q − 1 by the mo notone prop erty we see that it is contin uous. Given α ∈ ( q − 1 , q ], ass ume that the α -expansion o f 1 is 1 = ǫ 1 α + ǫ 2 α 2 + · · · . Let β be deter mined by 1 = θ ǫ 1 β + θ ǫ 2 β 2 + · · · Then we hav e α = α β ; θ 0 ··· θ q − 1 . Therefore , α β ; θ 1 ··· θ q − 1 is, a nd in turn, C β ; θ 0 ··· θ q − 1 is contin uous with r esp e ct to β .  Now let us hav e a clos er lo ok at C β ;013 for β ∈ (3 , 4]. By Theorem 3 w e k now that dim H ( C β ;013 ) is contin uous with resp ect to β . W e hav e dim H ( C 4;013 ) = log 3 log 4 and lim β ↓ 3 dim H ( C β ;013 ) = log 2 log 3 = dim H ( C 3 , 01 ) . 14 As stated in the a bove, dim H ( C β ;013 ) is decr easing for β ∈  3+ √ 17 2 , 3+ √ 21 2  . In fact, if β l and β r be determined b y 1 = 3 β l + m X i =2 a i β i l + 2 β m +1 l (11) and 1 = 3 β r + m X i =2 a i β i r + 3 β m +1 r (12) where a i ∈ { 0 , 1 , 3 } with ( a i , a i +1 , · · · , a m , 3 , 0 , · · · ) < (3 , a 1 , · · · , a m , 3 , 0 , · · · ) , then for any β ∈ [ β l , β r ] we hav e the same v alue for α β ;013 which is given by 1 = 2 α k + m X i =2 b i α i + 2 α m +1 (13) where b i = a i if a i ∈ { 0 , 1 } or b i = 2 if a i = 3. Let V = [3 , 4] \ ∪ ( β l , β r ) where the union is for all p ossible β l , β r defined in the ab ove. Theorem 4. β ∈ V if and only if the β -exp ansion of 1 is 1 = a 1 β + a 2 β 2 + · · · (14) or 1 = a 1 β + · · · + a k β k + 2 β k +1 (15) wher e a i ∈ { 0 , 1 , 3 } . F or β ∈ (3 , 4) \ V we h ave d dim β ( C β ; 0 13) dβ < 0 . If β ∈ V but β 6 = β l or β r , d dim H ( C β ;013 ) dβ = ∞ . If β = β l or β r , t hen the left side d or right side d deriv ative of dim H ( C β ;013 ) with r esp e ct to β is ∞ . Pr o of. Given β ∈ (3 , 4), if the β -expansio n of 1 co n tains digit 2 , then β ∈ [ β l , β r ) for some pair o f β l , β r defined in the a bove. Hence we ha ve either β / ∈ V or β = β l . Ther e fore, β ∈ V implies (14) or (15) and vise versa. 15 Since for β ∈ [ β 1 , β 2 ] we hav e the same v alue o f α β ;013 , w e hav e d dim β ( C β ;013) dβ < 0 for β ∈ (3 , 4) \ V . T aking β ∈ V , we a s sume that β is non-simple and show that lim β ′ ↓ β α − α ′ β − β ′ = ∞ . F or other cases the dis cussion is similar a nd simpler. Let 1 = a 1 β + a 2 β + · · · where a i ∈ { 0 , 1 , 3 } and a i 6 = 0 for infinitely many i . Given k ≥ 1, choose i k > k such that a i k 6 = 3. Let β k be given by 1 = i k − 1 X i =1 a i β i k + 1 β i k − 1 k  a 1 β k + a 2 β 2 k + · · ·  . Use α and α k to deno te α β ;013 and α β k ;013 . Then 1 = b 1 α + b 2 α 2 + · · · and 1 = i k − 1 X i =1 b i α i k + 1 α k i − 1  b 1 α k + b 2 α 2 k + · · ·  . where b i = a i if a i = 0 o r 1 and b i = 2 if a i = 3. W e estimate β k − β and α k − α . β k − β = i k − 1 X i =2  a i β i k − a i β i  + a 1 β i k k − a i k β i k ! + a 2 β i k +1 k − a i k +1 β i k +1 ! + · · · ≤ a 1 − a i k β i k k + · · · < c β i k for so me constant c . α i k k − α i k = i k − 1 X i =1 b i ( α i k − i k − α i k − i ) + ( b 1 − b i k ) + T α k y − T i k α 1 where y = b 1 β k + b 2 β 2 k + · · · a nd T i k α 1 = b i k +1 β + b i k +2 β 2 + · · · . Noting that b 1 = a 1 − 1 = 2 and b i k ≤ 1, α k − α > 1 + T α k y − T n k α 1 α i k − 1 k + α i k − 1 k α + · · · + α i k − 1 > T α k y i k α i k k . Since α k → α when k → ∞ , we hav e T α k y = b 2 β k + b 3 β 2 k + · · · → b 2 β + b 3 β 2 = T α 1 > 0 . 16 Hence α k − α β k − β > cT α k 1 i k  α k β k  i k → ∞ . This sho ws that lim sup β ′ ↓ β α k − α β k − β = ∞ , whic h implies that lim β ′ ↓ β α k − α β k − β since α β ;013 is a monotone function of β . Now we hav e shown that for β ∈ V if it is non-s imple then the right sided deriv ative of α β ;013 with re s pect to β is ∞ . By this w e see that the rig h t sided deriv ative o f dim H ( C β ;013 ) with resp ect to β is ∞ . By similar discussions we can obtain that d dim H ( C β ;013 ) dβ = ∞ , β ∈ V , but β 6 = β l or β r and d dim H ( C β ;013 ) dβ − = ∞ if β = β l , d dim H ( C β ;013 ) dβ + = ∞ if β = β r , wher e w e use “ − ” and “+” to denote the left and righ t handed deriv atives.  Remark. Although Theo rem 4 is stated and pr oved for C β ;013 , we hav e similar result fo r C β ; θ 0 ··· θ q − 1 in general. W e omit the details of the sta tement and pro of in g e ne r al case. 5 View C β ;0 2 through Differen t Ey es. In this section, we as sume that 2 < β ≤ 3. Let F β = { [0 , 1 ]; f 0 : [0 , 1 ] 7→ [0 , 1] , f 2 : [0 , β − 2] 7→ [0 , 1] } be the lo cal IFS defined in Section 1 , where f 0 ( x ) = x β and f 2 ( x ) = x +2 β . W e sho w th at C β ;02 is the unique inv ariant set o f T β , for all β except a countable set. Theorem 5. F or al l β ∈ (2 , 3 ] , C β ;02 is an invariant set of T β . L et Q b e the set of β ∈ (2 , 3] such that the β -exp ansion of 1 is 1 = ǫ 1 β + · · · + ǫ k β k wher e ǫ i ∈ { 0 , 2 } . F or β ∈ (2 , 3] \ Q , C β ;02 is the only invariant set of T β Pr o of. First we prov e that C β ;02 is an inv ariant set of F β by sho wing that C β ;02 = f 0 ( C β ;02 ) ∪ f 2 ( C β ;02 ∩ [0 , β − 2]) . (16) The compactness of C β ;02 ensures that f 0 ( C β ;02 ) ∪ f 2 ( C β ;02 ∩ [0 , β − 2]) is co m- pact. F o r an y x = (0 .a 1 a 2 · · · ) β with a i ∈ { 0 , 2 } for all i , w e hav e T β x = 17 (0 .a 2 a 3 · · · ) β ∈ C β ;02 . Then x = f 0 ( T β x ) ∈ f 0 ( C β ;02 ) if a 1 = 0, o r T β x ∈ [0 , β − 2) and x = f 2 ( T β x ) ∈ f 2 ( C β ;02 ∩ [0 , β − 2]) if a 1 = 2 . Hence f 0 ( C β ;02 ) ∪ f 2 ( C β ;02 ∩ [0 , β − 2]) contains all x ∈ [0 , 1) whose β -expans ion has only 0 or 2. By the definition of C β ;02 and the compactness of f 0 ( C β ;02 ) ∪ f 2 ( C β ;02 ∩ [0 , β − 2]) we get C β ;02 ⊂ f 0 ( C β ;02 ) ∪ f 2 ( C β ;02 ∩ [0 , β − 2]). On the other hand, it is e asy to see that f 0 ( C β ;02 ) ⊂ C β ;02 and f 2 ( C β ;02 ∩ [0 , β − 2]) ⊂ C β ;02 . Therefore, we hav e (16). Next show that if β ∈ (2 , 3 ] \ Q then C β ;02 is the only inv aria nt set of F β . Let A b e an inv ar iant se t of T β . Then A is compact a nd A = f 0 ( A ) ∪ f 2 ( A ∩ [0 , β − 2]) . (17) It is easy to see that 0 ∈ A . By this and (1 7) we get that A contains all x whose β -e x pansion is finite with all entries equal to 0 or 2 . Then we get C β ;02 ⊂ A . Assume that there ex ists x 0 ∈ A \ C β ;02 . W e claim that w e hav e 1 ∈ A . If x 0 6 = 1 then x 0 = (0 .a 1 a 2 · · · ) β with a k = 1 for some k . Let k b e the s mallest int e ger with a k = 1. By (1 7) we have either x 0 ∈ f 0 ( A ) o r x 0 ∈ f 2 ( A ∩ [0 , β − 2 ]). If x 0 ∈ f 0 ( A ), then y 0 = β x 0 ∈ A . This indicates that a 1 = 0 o r x 0 = 1 β . If x 0 ∈ f 2 ( A ∩ [0 , β − 2]), then y 0 = β x 0 − 2 ∈ A . Hence w e alwa ys hav e T β x 0 ∈ A . Contin ue this discussion, we get T k − 1 β x 0 = (0 .a k a k +1 · · · ) β ∈ A . Be cause a k = 1 we ha ve T k − 1 β x 0 < 2 β . Then T k − 1 β x 0 / ∈ f 2 ( A ∩ [0 , β − 2]. If T k − 1 β x 0 > 1 β , then it is not co ntained in f 0 ( A ). Hence w e must hav e T k − 1 β x = 1 β . By 1 β ∈ A we g et 1 ∈ A . If 1 ∈ A then, by the above discussion, T k 1 ∈ A for any k ≥ 0 . By (17) we see that either β -expansio n o f 1 do es not contain 1 or it ha s an o nly 1 as the last non-zero term. How ever, if β is non-s imple and the β -expansion o f 1 do es not contain 1, o r β is simple with the only digit 1 as the last non-zero term, then we have 1 ∈ C β ;02 which implies x 0 ∈ C β ;02 . Hence only when β is simple and the β -expa nsion of does no t cont ain 1, that is, β ∈ Q , w e ha ve 1 ∈ A \ C β ;02 . Therefore, C β ;02 is the only in v ar iant set of T β for β ∈ (2 , 3] \ Q .  Construct a co mpact set B β ;02 by a int e rv al remov al pro cess s imila r to that for the classical Cant o r s et. Firstly we remo ve the interv a l ( 1 β , 2 β ) from the unit int e rv al [0,1]. Next we remove ( 1 β 2 , 2 β 2 ) from the rema ining in terv al [0 , 1 β ] and remov e ( 2 β + 1 β 2 , 2 β + 2 β 2 ) ∩ [ 2 β , 1 ]. fr om [ 2 β , 1 ]. In general, a t step n if a remaining int e rv al [ u n , v n ] has v n − u n > 1 β n , w e remov e ( u n + 1 β n , u n + 2 β n ) ∩ [ u n , v n ] from it. Theorem 6. B β ;02 is an invariant set of T β for al l β ∈ (2 , 3] . F or β ∈ Q , we have B β ;02 6 = C β ;02 and B β ;02 \ C β ;02 is a c ount able set of isolate d p oints. Pr o of . First we show that B β ;02 = C β ;02 for β ∈ (2 , 3 ] \ Q . It is clear that B β ;02 ⊃ C β ;02 . By the construction pro cess, if the β -expans ion of a num b er has a 1 and at least one non-zero term after the digit 1 then it will b e removed at some sta ge. Hence B β ;02 consists of p oints in C β ;02 and possibly the n umbers 18 whose β -expansion is finite w ith an only digit 1 a s the last non-zero term. As discussed ab ov e, when β ∈ (2 , 3] \ Q these n um be r s are contained in C β ;02 . Hence we ha ve B β , 02 = C β ;02 for β ∈ (2 , 3 ] \ Q . Next we show that for β ∈ Q we hav e B β ;02 6 = C β ;02 and B β ;02 is the only inv ariant set of T β other tha n C β ;02 . Assume that 1 = (0 .ǫ 1 ǫ 2 · · · ǫ k − 1 2) β . By the construction of B β ;02 , the interv al  ǫ 1 β + · · · + ǫ k − 1 β k − 1 + 1 β k , ǫ 1 β + · · · + ǫ k − 1 β k − 1 + 2 β k  is removed and 1 remains in B β ;02 . Hence B β ;02 6 = C β ;02 . I t is easy to see that in this cas e B β ;02 contains a ll x = (0 .a − 1 · · · a k − 1 1) β where a i ∈ { 0 , 2 } . Ther efore B β ;02 is just the inv ar iant set A in the pro of o f (5) when A 6 = C β ;02 . Hence B β ;02 is an in v ariant s e t o f T β and T β do es not hav e inv ar iant set other than B β ;02 and C β ;02 . It is o b vious that B β ;02 \ C β ;02 consists of countablly many isolated po in ts.  W e can a lso define a normal IFS ([0 , 1 ]; f 0 , f 2 ) by extending f 2 to the whole int e rv al with f 2 ( x ) = ( x +2 β x ∈ [0 , β − 2] , 1 x ∈ ( β − 2 , 1 ] . Then B β ;02 is the attractor of this IFS. W e omit the details. 6 F u rther Researc h. 1. In Theorem 1 w e obtained the Hausdor ff dimension for β -attractor s when a ll the maps have the same contract ratio under a separa tion condition f i ( E β ) ∩ f j ( E β ) ∩ E β = ∅ , whenever i 6 = j . W e b elieve that the separation condition can be replaced by an op en s e t co ndition. Compar ing with the β - expansion of real num b e r s in [0 , 1), we may define a β -o pen set condition as follows. β -O p en Se t Condition. Let F = ( X ; f 0 , f 1 , · · · , f n − 1 ) be a n IFS where X is a compact subset of R m and f i is a Lipsc hiz ma p with Lipschiz constant r i < 1 fo r 0 ≤ i ≤ n − 1. Let n − 1 < β ≤ n and E β is the β -attractor . W e say F satisfies a β -op en set condition if there exists an op en set O with E β ⊂ ¯ O such that f i ( O ) ⊂ O and f i ( O ) ∩ f j ( O ) = ∅ for 0 ≤ i < j ≤ n − 1. Note that in the ab ov e definition we do not requir e f n − 1 ( O ) ⊂ O but a ssume E β ⊂ ¯ O which is a utomatically true in the origina l version o f op en set condition. It is easy to see that C β ; θ 0 ··· θ q − 1 satisfies a β -op en set condition if we choose O = (0 , 1). When β = n is a n integer the o riginal version of o pe n set co ndition implies the β -op en set co ndition for the same op en set O . When all f i ’s ar e similarities, a β -op en set condition with β b eing an in teger also implies the original version of o pen set condition p ossibly with a different op en set O ′ . Conjecture. L et ( X ; f 0 , f 1 , · · · , f n − 1 ) b e an IFS su ch that d ( f i ( x ) , f i ( y )) = r d ( x, y ) 19 for al l x, y ∈ X and 0 ≤ i ≤ n − 1 , wher e 0 < r < 1 . L et 1 < β < n and E β b e the β -attr actor. Under the β -op en set c ondition, we have dim( E β ) = log β − log r . The Hausdorff me asur e of E β in its d imension is p ositive a nd finite. 2. Given a n IFS ( X ; f 0 , · · · , f n − 1 ) with attractor E and a pr obability vector ( p 0 , · · · , p n − 1 ) with p i > 0, fo r any Borel measur e µ on X w e define F ( µ )( B ) = n − 1 X i =0 p i µ ( f − 1 i ( B )) for any Bore l subset B . Then there is a unique measure ν such that F ( ν ) = ν . ν has E as its suppor t and fo r a ny µ the sequence { F ( k ) ( µ ) } weakly conv erg es to ν . It is natural to study the inv ariant mea sure for β -a ttractors. Ac kno wledgeme nt. This work was suppo rted by an Australian Research Council disco very gra nt . 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