Radom se usmjerava pažnja na ulogu i važnost pluriperspektivizma u misli G. W. F. Hegela. U tu se svrhu iščitava alegorija o istinitom kao bakantskom zanosu, a koja se nalazi u njegovom temeljnom djelu Fenomenologija duha. Također, u tekstu se ukazuje na važnost pluriperspektivizma, odnosno naznačuju i razrađuju se njegove bitne karakteristike da: s jedne strane, omogućuje i potiče na čim neisključiviji i obuhvatniji pristup zbiljnosti, dok, s druge strane, omogućuje i potiče rast i razvoj perspektive, tj. povećanje istinitosti istine.In this work I direct attention to the role and importance of pluriperspectivism in the thought of G. W. F. Hegel. For that purpose, I read (interpret) the allegory about the true as a bacchantic ecstasy, which can be found in his fundamental work Phenomenology of Spirit. Also, in the text I indicate the importance of pluriperspectivism, that is, point out and elaborate its essential characteristics: on the one hand, it allows and encourages the more unexclusive and more embracing approach to the reality while, on the other hand, it allows and encourages the growth and development of perspective, that is, the increase of the truthfulness of truth

Dajani, Karma

Jiang, Kan

Kempton, Tom

Filozofska Istraživanja

Using techniques introduced by C. Gunturk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.Peer reviewe

Kempton, Thomas Michael William

St Andrews Research Repository

Self-Affine Sets with Positive Lebesgue MeasureKarma Dajani, Kan Jiang and Tom Kempton∗April 29, 2014AbstractUsing techniques introduced by C. Gu¨ntu¨rk, we prove that the attractors of afamily of overlapping self-affine iterated function systems contain a neighbourhoodof zero for all parameters in a certain range. This corresponds to giving condi-tions under which a single sequence may serve as a ‘simultaneous β-expansion’ ofdifferent numbers in different bases.1 IntroductionGiven real numbers 1 < β1 < β2, we define contractions Ti : R2 → R2 byTi(x, y) =(x+ iβ1,y + iβ2).A classical result of Hutchinson [4] asserts that there exists a unique non-empty compactset Aβ1,β2 satisfyingAβ1,β2 = T−1(Aβ1,β2) ∪ T1(Aβ1,β2).If β1 6= β2 then the contractions Ti are affine contractions and Aβ1,β2 is termed a self-affineset. Since β1, β2 < 2, the two contracted copies T−1(Aβ1,β2) and T1(Aβ1,β2) overlap. Thereare many fundamental open questions about the structure of overlapping self-affine sets,see for example [5, 6, 7].The family Aβ1,β2 of sets was studied in [7], where Shmerkin proved that there exists anopen set K ⊂ (1, 2)2 such that for almost every pair (β1, β2) ∈ K the corresponding setAβ1,β2 has positive Lebesgue measure. This was done by studying the absolute continuityof a certain measure defined on Aβ1,β2 . In this article we prove that Aβ1,β2 contains aneighbourhood of (0, 0) for all (β1, β2) ∈ (1, 1 +C)2 for some positive constant C whichis explicitly defined later.In fact this problem is closely related to the problem of ‘simultaneous β-expansions’studied by Gu¨ntu¨rk in [3]. Given β ∈ (1, 2) and x ∈ [ −1β−1 ,1β−1 ], a β-expansion of x is a∗Mathematics Department, University of Utrecht, Budapestlaan 6, Utrecht, The Netherlands1sequence a ∈ {−1, 1}N for which∞∑n=1anβ−n = x.This definition can be extended to β > 2 by letting the digits an come from a largerdigit set.For typical x the β-expansion of x is not unique, indeed almost every x ∈ [ −1β−1 ,1β−1 ]has uncountably many β-expansions, see [8]. This allows one, given x, to search forβ-expansions of x with interesting properties, such as a given digit frequency or thatthe sequence is a β-expansion of x for more than one β.In [3], Gu¨ntu¨rk proved that given β1, β2 > 1 and (x1, x2) ∈ R2 there exists a sequence(an) ∈ {−1, 1}N satisfying ∞∑n=1anβ−nk = xkfor each k ∈ {1, 2} whenever a certain algorithm can be implemented, see Proposition2.1. It was claimed without proof1 that there exist constants C, δ > 0 such that thealgorithm can be implemented whenever β1, β2 ∈ (1, 1 + C) and (x1, x2) ∈ (−δ, δ)2.We prove this fact and provide suitable constants C and δ explicitly. We also provea number of related results including results on finding β-expansions with given digitfrequency and finding sequences which serve as multiple expansions for a range of β1, β2.An interesting facet of our work is that the techniques of Gu¨ntu¨rk which we use are quitedistinct from the usual fractal geometry techniques for studying self-affine sets.The following is our main theorem.Theorem 1.1. There exists a constant C ≈ 0.05 such that for any 1 < β1 < β2 < 1+C,there exists δ = δ(β1 , β2) such that for any pair (x1, x2) ∈ [−δ, δ]2, there exists a sequence(an) ∈ {−1, 1}N such that(∞∑n=1anβ−n1 ,∞∑n=1anβ−n2 ) = (x1, x2). (1)In the self-affine setting, this theorem corresponds to saying that the sequence (an) is acoding of the pair (x1, x2) in Aβ1,β2 , and in particular that (x1, x2) ∈ Aβ1,β2 . This leadsimmediately to the following corollary.Corollary 1.1. For all any 1 < β1 < β2 < 1 + C we have that the self-affine fractalAβ1,β2 contains a neighbourhood of (0, 0).The constant δ is explicitly computable. If β1 tends to β2 the constant δ tends to zero.Remark 1.1. An important special case of Corollary 1.1 is the case x1 = x2. Thiswas the main motivation of Gu¨ntu¨rk for his original article because of its relevance to1Gu¨ntu¨rk stated in [3] that details would be provided in a later publication, but has confirmed tous that, due to other commitments, no such publication will be forthcoming. Since the techniques of[3] are rather different from the standard techniques for analysing self-affine sets, and the results areinteresting, we take the liberty of providing a proof of the stated results of Gu¨ntu¨rk in this article.2analogue digital conversion, see[3]. While in general the constant δ depends on β1, β2,in the case that x1 = x2 we can choose δ = 0.16 independently of β1, β2 to give that forall 1 < β1 < β2 < 1 + C and x ∈ [−0.16 , 0.16] there exists a sequence (ai) ∈ {−1, 1}Nsuch thatx =∞∑n=1anβn1=∞∑n=1anβn2.Using the same techniques, we can also find β-expansions of real numbers which havecertain given digit frequencies. It was stated in [3] that the following theorem shouldfollow by suitably adapting the proof of Theorem 1.1, we provide the appropriate adap-tation and prove the result giving explicit constants.Theorem 1.2. Let C1 > 0 satisfy (1+C1)+2(1+C1)3 = 6. Then for all 1 < β < 1+C1,there exists δ = δ(β) such that for any x ∈ [−δ, δ] there exists a sequence (an) ∈ {−1, 1}Nsatisfying∞∑n=1anβ−n = x (2)andx = limn→∞a1 + a2 + · · ·+ ann. (3)One can read off limiting digit frequencies of the sequence (an) from equation 3 by notingthat1− a1+a2+···+ann2=|{k ∈ {1, · · ·n} : ak = −1}|n.Proofs of Theorems 1.1 and 1.2 are given in the next two sections. In the final sectionwe state some further corollaries and remarks.2 Proof of Theorem 1.1As stated in the introduction, we are using many of the ideas of [3]. For clarity, wehave amalgamated these ideas to form the following proposition, which was proved in[3]. The remainder of our proof of Theorem 1.1, which gives conditions under which thealgorithm in Proposition 2.1 can be implemented, is new.Proposition 2.1. Given 1 < β1 < β2 < 2 and (x1, x2) ∈ R2 suppose that one canimplement the following algorithm.1. For L > 2 pick real numbers h1, · · · , hL with hL 6= 0 andhL−1 = hL−2 = 0,such that β1, β2 are roots of the polynomial P (z) = zL −∑Lk=1 hkzL−k.2. Pick real numbers u−L+1, u−L+2 which satisfy the equation(x1x2)= hL(β−11 β−21β−12 β−22)(u−L+1u−L+2)Set u−L+3 = · · · = u0 = 0.33. Find a sequence (an) ∈ {−1, 1}N such thatun :=L∑k=1hkun−k − ansatisfies un ∈ [−1, 1] for each n ∈ N.Then the sequence (an)∞n=1 will satisfy equation (1).In this article we give rigorous conditions under which the algorithm of Gu¨ntu¨rk can beimplemented leading to a proof of Theorem 1.1. For completeness we also give the proofof Proposition 2.1. We begin by introducing the polynomial P , P was chosen because ithas relatively low degree and satisfies the conditions of Proposition 2.1, but it is likelythat better bounds on C and δ can be obtained by choosing a better polynomial P .Definition 2.1. Given β1, β2 > 1, we define the polynomial P byP (x) = x4 − h1x3 − h2x2 − h3x− h4whereh1 =(β1 + β2)(β21 + β22)β21 + β1β2 + β22h2 = 0h3 = 0h4 =−(β1β2)3β21 + β1β2 + β22.We further define the constant C byC :=3√√10− 2 ≈ 0.05.Lemma 2.1. The polynomial P satisfies P (β1) = P (β2) = 0.Proof. Defining,b =β1β2(β1 + β2)β21 + β1β2 + β22,andc =(β1β2)2β21 + β1β2 + β22gives us that(x− β1)(x− β2)(x2 + bx+ c) = x4 − h1x3 − h2x2 − h3x− h4 = P (x).Then β1 and β2 are roots of P .Lemma 2.2. For β1, β2 ∈ (1, 1 + C) we have that4∑n=1|hk| = |h1|+ |h4| ≤ 2.4Proof. Expanding out, we see that4∑n=1|hk| = |h1|+ |h4|=(β1 + β2)(β21 + β22) + β31β32β21 + β1β2 + β22≤ (2 + 2C)2(1 + C)2 + (1 + C)63≤ 2whenever β1, β2 ∈ (1, 1+C), as required. Indeed, C was chosen to be the largest constantsuch that the above inequalities hold.We now prove Theorem 1.1 using Proposition 2.1.Proof. We setu−3 =β21 + β1β2 + β22(β2 − β1)β1β2 (x1β22− x2β21),u−2 =β21 + β1β2 + β22(β2 − β1)β1β2 (x2β1− x1β2),u−1 = u0 = 0.These choices of ui ensure that condition (2) of Proposition 2.1 is satisfied, i.e.(x1x2)= h4(β−11 β−21β−12 β−22)(u−3u−2).Condition (1) has already been shown to hold for our choice of P by Lemma 2.1. Itremains to show that condition (3) holds, i.e. that one can choose some sequence(an) ∈ {−1, 1}N such that defining un for n ∈ N byun :=L∑k=1hkun−k − an (4)gives un ∈ [−1, 1] for each n ∈ N. Since h2 = h3 = 0 the above equation for un becomesun = h1un−1 + h4un−4 − an.We setan ={ −1 h1un−1 + h4un−4 < 0+1 h1un−1 + h4un−4 ≥ 0 . (5)Now we observe that, if for some k ∈ N one has that uk−1, uk−4 ∈ [−1, 1], then it followsfrom Lemma 2.2 thath1uk−1 + h4uk−4 ∈ [−2, 2].Hence it follows thatuk := h1uk−1 + h4uk−4 − ak ∈ [−1, 1],5and hence by induction that un ∈ [−1, 1] for each n ∈ N.Now we defineδ =β21β22(β2 − β1)(β21 + β1β2 + β22)(β1 + β2).We see that δ > 0 whenever β2 > β1, but that δ → 0 as β2 − β1 → 0. From thedefinition of u−3, u−2 we see that for x1, x2 ∈ [−δ, δ]2 and β1, β2 ∈ (1, 1 + C) we havethat u−3, u−2 ∈ [−1, 1]. Since u−1 = u0 = 0 it follows by induction that un ∈ [−1, 1] foreach n ∈ N. Hence conditions (1), (2) and (3) of Proposition 2.1 are satisfied, and sothe sequence (an) satisfies equation 1 and Theorem 1.1 is proved.It remains only to give a formal proof of Proposition 2.1.Proof. We give a proof for the case L = 4, which is the case that we have used. Fromcondition (2), we have thatxi = h4(u−3β−1i + u−2β−2i )for i = 1, 2. Rewriting condition (3) gives us that an =4∑k=0hkun−k. Then summing givesus that∞∑n=1anβ−ni =∞∑n=14∑k=0hkun−kβ−ni (6)where h0 = −1. Since the sequence (un) is bounded, we have by Fubini’s theorem that∞∑n=1anβ−ni =4∑k=0∞∑n=1hkun−kβ−ni =4∑k=0hkβ−ki∞∑n=−k+1unβ−niHere the first equality involved using equation (6) and swapping the order of summationby Fubini. The second equality is just a change of variables. Now, by separating theterms for positive and negative n in the right hand side of the above equation, we havethat∞∑n=1anβ−ni =(4∑k=0hkβ−ki)( ∞∑n=1unβni)+ h1β−1i u0 + h2β−2i (u0 + u−1βi)+ h3β−3i (u0 + u−1βi + u−2β2i ) + h4β−4i (u0 + u−1βi + u−2β2i + u−3β3i ).Since βi is the root of P (x) we have4∑k=0hkβ−ki = 0 and so the first term vanishes. Fromconditions (1) and (2), we have u−1 = u0 = h2 = h3 = 0. Then, removing the zeroterms, the right hand side of the above equation becomes h4(u−3β−1i + u−2β−2i ), whichby condition (2) is equal to xi. We conclude that∞∑n=1anβ−ni = h4(u−3β−1i + u−2β−2i ) = xias required. This completes the proof of Proposition 2.1.6Finally we comment that in the case that x1 = x2 we can give values of δ which areindependent of β1, β2 ∈ (1, 1+C). Our bound on δ was to ensure that u−2, u−3 ∈ [−1, 1].If x1 = x2 then|u−2| ≤ |u−3| =∣∣∣∣x1(β21 + β1β2 + β22(β31β32))(β1 + β2)∣∣∣∣≤ |x1|6(1 + C)3 ≤ 1whenever |x1| ≤ δ = 16(1+C)3 ≈ 0.16.3 β-expansions with a given digit frequency.With some modifications, the algorithm used in the proof of Proposition 2.1 can also beutilized to prove Theorem 1.2. The following is analagous to Proposition 2.1.Proposition 3.1. Given 1 < β < 2 and x ∈ [−δ, δ] for some δ which will be set in theprocess of proof, suppose that one can implement the following algorithm.1. For L > 2 pick real numbers h1, · · · , hL with hL 6= 0 andhL−1 = hL−2 = 0,such that 1, β are roots of the polynomial P (z) = zL −L∑k=1hkzL−k.2. Pick a real number u−L+1, which satisfies the equationx =hL(β − 1)u−L+1β(β − 2) .Set u−L+2 = · · · = u0 = 0.3. Find a sequence (an) ∈ {−1, 1}N such thatun :=(L∑k=1hkun−k)+ x− ansatisfies un ∈ [−1, 1] for each n ∈ N.Then the sequence (an)∞n=1 satisfies equations (2) and (3).Such sequences are known as ‘hybrid encoders’. We begin by proving Proposition 3.1,this is similar to the proof of Proposition 2.1.Proof. We begin by rearranging condition (3) of Proposition 3.1 to givean =(4∑k=1hkun−k)+ x− un =(4∑k=0hkun−k)+ x.7Then we have that∞∑n=1anβ−n =∞∑n=14∑k=0hkun−kβ−n + x∞∑n=11βn(7)where h0 = −1. We now follow the reasoning of the proof of Proposition 2.1 exactly, toyield that∞∑n=1anβ−n = h4(u−3β−1 + u−2β−2) +xβ − 1 .Unlike in Proposition 2.1, we also have that u−2 = 0, so we conclude that∞∑n=1anβ−n = h4(u−3β−1) +xβ − 1 ,and picking u−3 =β(β−2)xh4(β−1) yields thatx =∞∑n=1anβ−n.It remains to prove part two of the theorem, thatx = limn→∞a1 + a2 + · · ·+ ann.Now we have from the condition (3) of Proposition 3.1 that(L∑k=1hkun−k)+ x− an − un =(L∑k=0hkun−k)+ x− an = 0.Then (1NN∑n=1L∑k=0hkun−k)+ x− 1NN∑n=1an = 0.We shall prove thatN∑n=1L∑k=0hkun−k is bounded by some constant independent of N , whichwill givelimN→∞1NN∑n=1L∑k=0hkun−k = 0and hence thatx = limn→∞a1 + a2 + · · ·+ ann.Now we have thatN∑n=1L∑k=0hkun−k = (u1 + u2 + · · ·+ uN−L)(h0 + h1 + · · ·+ hL) + extra terms,8where there are N(L+1)− ((N −L)(L+1)) = L(L+1) extra terms, each of which arebounded in absolute value by( maxk∈0,··· ,L|hk|)(supn∈Nun) ≤ maxk∈0,··· ,L|hk| ≤Mfor some constant M . But h0 + h1 + · · ·+ hL = 0, and so we see that∣∣∣∣∣N∑n=1L∑k=0hkun−k∣∣∣∣∣ ≤ 0 + L(L+ 1)Mwhich is independent of N , and solimN→∞1NN∑n=1L∑k=0hkun−k = 0as required.Now we prove Theorem 1.2.Proof. The proof of∞∑n=1anβ−n = x is almost the same as the proof of Theorem 1.1. Wefollow the construction of Lemma 2.1 replacing β1 by 1 and β2 by β. This givesP (z) = zL −L∑k=1hkzL−k = (z − 1)(z − β)(z2 + az + b),where b =β(1 + β)1 + β + β2and c =β21 + β + β2.Then we have h1 =(1+β)(1+β2)1+β+β2and h4 =−β31+β+β2. We choose C1 such that4∑k=1|hk| = |h1|+ |h4|=(1 + β)(1 + β2) + β31 + β + β2<(1 + C1)((1 + C1)2 + 1) + (1 + C1)33= 2where C1 is the real root of(1+x)((1+x)2+1)+(1+x)33= 2.Since4∑k=1|hk| = |h1| + |h4| < 2, we can choose δ0 > 0 such that4∑k=1|hk| = |h1| + |h4| ≤2− δ0 < 2, thus for any x satisfying |x| ∈ [0, δ0] we have4∑n=1|hk| = |h1|+ |h4| ≤ 2− δ0 ≤ 2− |x| < 29The next step is to prove the boundness of un. Choosing δ1 =h4(β−1)β(β−2) we have that|u−3| =∣∣∣∣β(β − 2)xh4(β − 1)∣∣∣∣ ≤ 1.Finally, if we take δ = min{δ0 , δ1}, then this choice can ensure that4∑n=1|hk| = |h1|+ |h4| ≤ 2− xand |u−3| ≤ 1 hold simultaneously. We also have that u−2 = u−1 = u0 = 0. We let thesequence (an) be chosen as follows:an ={ −1 ∑Lk=1 hkun−k + x < 0+1∑Lk=1 hkun−k + x ≥ 0. (8)Then by induction we have that un ∈ [−1, 1] for all n ∈ N, and hence the conditions ofProposition 3.1 are fulfilled and Theorem 1.2 is proved.4 Further RemarksWe have the following further remarks.(i) We have proved that if β1 and β2 are very close to 1 then Aβ1,β2 has an interior,but it is unlikely that our bounds are optimal, see for example the diagrams in[3]. Our proof was based on choosing an expansion (an)∞n=1 of pairs (x1, x2) usingequation (3). Perhaps by using a more sophisticated algorithm one may hope togain a truer picture of the conditions under which our technique can be made towork.(ii) The IFS which we study is a little different to that studied by Shmerkin in [7],since we use digit set {−1, 1} rather than (− 1γ,− 1λ) and ( 1γ, 1λ). However suchchanges of digit set do not affect whether the attractor of the corresponding IFShas an interior.(iii) We note that if β1β2 > 2 then Aβ1,β2 cannot have an interior. Gu¨ntu¨rk gave avolume covering argument to prove this. In fact one can say more, the sets Aβ1,β2fall into the setting of ‘self-affine sets of Kakeya type’ studied in [5], and so byTheorem 3.3 of that paper we have thatdimB(Aβ1,β2) = 1 +log 2β1log β2< 2whenever β1β2 > 2 and 1 < β1 < β2 < 2.(iv) Our approach to generating sequences a which satisfy the conditions of Theorem1.1 is in some sense dynamical, we have an algorithm which chooses a value of(an) based on the vector (un, un−1, un−2, un−3), and then maps this vector to the10vector (un+1, un, un−1, un−2) and repeats the operation. This system is reminiscentof shift radix systems, see [1], except that we have a displacement by an.Our algorithm is far less simple than corresponding algorithms for generatingexpansions in the one dimensional case, such as the random β-transformation of[2]. It would be nice to have an analogue of the random β-transformation forthe higher dimensional case which produces expansions of pairs (x1, x2) in a moredirect and understandable way.(v) One can use Remark 1.1 to consider when, for a specific sequence (an) and realnumber x, there exist β1, β2 such thatx =∞∑i=1aiβ−i1 =∞∑i=1aiβ−i2 .Given a ∈ {−1, 1}N we define the function fa : (1, 2]→ R byfa(β) =∞∑n=1anβn.The function fa is continuous and differentiable. We call a sequence a = (an)a simultaneous encoder of x if there exists there exist 1 < β1 < β2 < 2 suchthat x = fa(β1) = fa(β2). By Remark 1.1, for any 1 < β1 < β2 < 1 + Cand any x ∈ [−0.16, 0.16] one can find a simultaneous encoder a of x satisfyingx = fa(β1) = fa(β2). By the extreme value theorem, the function fa has globalextrema in [β1, β2]. We let β, β0 ∈ [β1, β2] be the values where the global minimumand global maximum take place. Let y1 = fa(β) and y2 = fa(β0). Then bythe intermediate value theorem, the sequence a is a simultaneous encoder for allz ∈ (y1, y2). Thus, if we define the setEa := {x ∈ R : a is a simultaneous encoder of x}.then the above argument shows that either Ea is empty, or is a single point orcontains an interval.AcknowledgementsThe second author was supported by the China Scholarship Council. The third authorwas supported by the Dutch Organisation for Scientific Research (NWO) grant number613.001.022.References[1] Shigeki Akiyama and Klaus Scheicher. From number systems to shift radix systems.Nihonkai Math. J., 16(2):95–106, 2005.[2] Karma Dajani and Cor Kraaikamp. Random β-expansions. Ergodic Theory Dynam.Systems, 23(2):461–479, 2003.11[3] C.S. Gu¨ntu¨rk. Simultaneous and hybrid beta-encodings. In Information Sciencesand Systems, 2008. CISS 2008. 42nd Annual Conference on, pages 743–748, 2008.[4] John E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J., 30(5):713–747, 1981.[5] Antti Ka¨enma¨ki and Pablo Shmerkin. Overlapping self-affine sets of Kakeya type.Ergodic Theory Dynam. Systems, 29(3):941–965, 2009.[6] Yuval Peres and Boris Solomyak. Problems on self-similar sets and self-affine sets:an update. In Fractal geometry and stochastics, II (Greifswald/Koserow, 1998),volume 46 of Progr. Probab., pages 95–106. Birkha¨user, Basel, 2000.[7] Pablo Shmerkin. Overlapping self-affine sets. Indiana Univ. Math. J., 55(4):1291–1331, 2006.[8] Nikita Sidorov. Almost every number has a continuum of β-expansions. Amer. Math.Monthly, 110(9):838–842, 2003.12

Indagationes Mathematicae

English

Almost every number has a continuum of β-expansions.

Fractals and self-similarity.

From number systems to shift radix systems.

Overlapping self-affine sets of Kakeya type. Ergodic Theory Dynam.

Overlapping self-affine sets.

Problems on self-similar sets and self-affine sets: an update. In Fractal geometry and stochastics,

Random β-expansions.

Simultaneous and hybrid beta-encodings.

Self-affine sets with positive Lebesgue measure

Using techniques introduced by C. G ̈unt ̈urk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different base

Utrecht University Repository

Self-Affine Sets with Positive Lebesgue Measure

Sub Mathematical Modeling

Sub Stochastics and Decision Theory begr

Mathematical Modeling

NARCIS 

Karma Dajani

Kan Jiang

Tom Kempton

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Using techniques introduced by C. Gunturk, we prove that the attractors of afamily of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneousβ-expansion’ of different numbers in different bases

University of St. Andrews - Pure

University of St Andrews Research Portal

Borš, Vanja

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