space-filling curves and geodesic laminations. ii: symmetries

Monatsh Math (2012) 166:543–558DOI 10.1007/s00605-012-0406-9

Space-filling curves and geodesic laminations. II:Symmetries

Víctor F. Sirvent

Received: 2 November 2010 / Accepted: 14 March 2012 / Published online: 4 April 2012© Springer-Verlag 2012

Abstract In 2008, the author introduced a class of space-filling curves associated tofractals that satisfy the a special property. These structures admit geodesic laminationson the disc, which help to understand the geometrical and the dynamical propertiesof the space-filling curves. In the present article we study the relation between thesymmetries of the laminations and the fractals. In particular we prove that the groupof symmetries of the lamination is isomorphic to a subgroup of the full group ofsymmetries of the fractal. We extend the results to a larger class of fractals using theconcept of sub-IFS.

Keywords Space-filling curves · Geodesic laminations · Fractals · Iterated functionsystems · Symmetries

Mathematics Subject Classification 28A80 · 53C22

1 Introduction

Space-filling curves were introduced by G. Peano in 1890 [11] and later other authorsgave different types of space-filling curves [7,9,10,12,14]. For a history of space-filling curves see Ref. [13]. These curves have been used in different contexts andbranches of mathematics, for some references see Ref. [18]. On the other hand geodesic

Communicated by K. Schmidt.

V. F. Sirvent (B)Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000,Caracas 1086, Venezuelae-mail: [email protected]

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544 V. F. Sirvent

laminations on the disc have been used in complex dynamics [8,19], in surface auto-morphism dynamics [3] and in symbolic dynamics [15,17].

In Ref. [18] the author introduced a class of space-filling curves associated to someconnected fractals. These fractals are the fixed point of linear iterated function sys-tems (IFS) satisfying a special property, known as the common point property, itsdefinition is given in Sect. 2. There it was studied the dynamics associated to the IFS,and some of their geometrical properties. In that article the construction of a geodesiclamination on the disc, plays a very important role. In the present paper we continueour study of the geometrical properties of these structures, in particular we explorethe symmetries that exist between the lamination and the attractor of the IFS. Themain result, Theorem 4.1, shows that the group of symmetries of the lamination is asubgroup of the full group of symmetries of the attractor of the IFS. We show thatthe converse is not true in general. Defined on the lamination there is an expandingdynamical system, we show in Proposition 4.2 that it is constant on the symmetryclasses of the lamination.

In Sect. 2 we give the basic definitions and construction of the space-filling curve.In Sect. 3 we introduce the construction of the geodesic lamination. In Sect. 4 we studythe relation between the symmetries of the lamination and the attractor of the IFS. InSect. 5 we define the concept of a sub-IFS and we associate a space-filling curve anda geodesic lamination to them. The importance of this concept is that it allows togeneralize the constructions of Ref. [18] to a wider class of fractals, that do not satisfythe common point property. In Corollary 5.1, we show that the symmetry group of thesub-lamination is a subgroup of the full group of symmetries of the attractor of thesub-IFS. Finally in Sect. 6 we give a list of examples. In Example 1 we show that someof the constructions are valid in non-self-similar IFS. In Example 3 we show that theconverse of Theorem 4.1 and Corollary 5.1 are not true. In Example 4 the open setcondition of the IFS is not satisfied.

2 Space-filling curve

An iterated function system or IFS consists of a complete metric space X together witha finite set of contraction mappings, {ψ1, . . . , ψk}. Let K (X) be the set of all non-empty compact subsets of X , we give to this set the Hausdorff metric. With this distanceK (X) is a complete metric space [2,4]. We define the map � : K (X) → K (X), as�(U ) = ∪k

i=1ψi (U ). Since the maps ψi are contractions the map � is a contractionon K (X). Its fixed point is called the attractor of the IFS {ψ1, . . . , ψk}.

Let {φ1, . . . , φk} be an IFS on Rd with d ≥ 2 and R its attractor, such that:

1. φi (x) = Ai x + vi , where Ai is a matrix, with spectral radius less than 1 andvi ∈ R

d .2. It satisfies the open set condition, i.e. there exists a non-empty bounded open set

V such that ∪ki=1φi (V ) ⊂ V with disjoint union.

3. ∩ki=1φi (R) is exactly one point, say z and there exists one point y ∈ R such thatφ1(y) = φ2(y) = · · · = φk(y) = z.We shall call this condition, the common point property and the point z the inter-section point of R.

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Space-filling curves and geodesic laminations 545

As an example consider the IFS {φ1, . . . , φ4} on the unit square R = {(x1, x2) ∈R

2 : 0 ≤ x1, x2 ≤ 1}, where

φ1(x1, x2) =(−x1 + 1

2,−x2 + 1

2

), φ2(x1, x2) =

(−x2 + 1

2,

x1 + 1

2

),

φ3(x1, x2) =(

x1 + 1

2,

x2 + 1

2

), φ4(x1, x2) =

(x2 + 1

2,−x1 + 1

2

).

Its attractor is R. The intersection of the four regions φi (R) is the middle point of thesquare. i.e. z = (1/2, 1/2), and y = (0, 0), since φi (y) = z for 1 ≤ i ≤ 4. This is aspecial case of the Example 1 considered in Sect. 6.

By the common point property R is connected [6]. Let s be the Hausdorff dimen-sion of R. Each point of R can be obtained as ∩∞

1 (φa1, . . . , φan (R)) for a suitableinfinite sequence a1a2 . . . in {1, . . . , k}N. This sequence is called an itinerary of thepoint, which may not be unique. We say that the IFS {φ1, . . . , φk} is self-similar if‖φi (x)− φi (x′)‖ = ci‖x − x′‖ for all x, x′ ∈ R

d and 1 ≤ i ≤ k. So ci = | det Ai |1/d .On the half-open interval I = [0, 1), we shall define piece-wise linear maps fi :

I → I , for 1 ≤ i ≤ k, on the following way:

• Let αi = csi = | det Ai |s/d , for i = 1, . . . , k. In the self-similar case observe that∑k

i=1 αi = ∑ki=1 cs

i = 1, since s is the Hausdorff dimension of R [4]. In the non-self-similar case, if s = d let αi = | det Ai |. When s = d, we can assume withoutlost of generality that the Lebesgue measure of R is equal 1, so

∑ki=1 αi = 1.

• We define the partition Ii = [∑i−1j=1 α j ,

∑ij=1 α j ) of intervals of length αi , for

i = 1, . . . , k.• Let Ki : I → Ii be the natural and continuous contractions of coefficient αi , i.e.

Ki (x) := αi x + ∑i−1j=1 α j .

• For a fixed θ ∈ (0, 1) we define Rθ : I → I as Rθ (x) := x + θ (mod 1).• Let fi := Ki ◦ Rθ .

So we can write fi as:

fi (t) = f θi (t) ={

αi t + βi if 0 ≤ t < t∗,αi t + βi − αi if t∗ ≤ t < 1

(1)

where t∗ = 1 − θ and βi = αiθ + ∑i−1j=1 α j .

Let us remark that Ii = fi (I ), for 1 ≤ i ≤ k.Let F : I → I be an expanding map defined by F(t) := f −1

i (t) if t ∈ Ii . Itis well-defined since the intervals Ii are half-open. We can think F as a map on S

1,since fi (0) = limt→1− fi (t). Furthermore it is a continuous map on S

1. By definitionthe map F is of degree k, i.e. every point has k-preimages. We say that the collec-tion of maps { f1, . . . , fk} is the system of branches of the inverse function of F . Letν : I → {1, . . . , k}, where ν(x) = i if x ∈ Ii . We define the itinerary of t as theinfinite sequence a1a2 . . . where a j = ν(F j−1(t)) for all j ≥ 1. By definition it hasthe property t ∈ ∩n≥1 fa1 . . . fan (I ). Let us remark that the itinerary of a given pointis well-defined since the intervals Ii are half-open. However there might be more than

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546 V. F. Sirvent

one point with the same itinerary; since the maps fi -s are not continuous, the sets ofthe form fa1 · · · fan (I ) are a finite union of half-open intervals, for all n ≥ 1. The lackof continuity of the maps fi -s plays an important role in construction of the laminationdescribed in Sect. 3.

The collection of maps { f1, . . . , fk} defined in this way imitates the given IFS{φ1, . . . , φk} on R

d . The contraction factor of the maps fi -s, i.e. the numbers αi -s,are chosen in such way that the Lebesgue measure of fi (I ) = Ii is the same as thenormalized s-dimensional Hausdorff measure of φi (R). In Proposition 2.1 we showthe relation between end points of the intervals Ii and the common point of R, i.e. z.In Sect. 6 we show some examples of { f1, . . . , fk} on I and how are related to the IFS{φ1, . . . , φk} on R

d . Throughout this paper we shall denote Ra1...an = φa1 · · ·φan (R)and Ia1...an = fa1 · · · fan (I ). We shall call these sets cylinders of R and I respectively.By the definition of the maps fi , the cylinders Ia1...an consist of a finite union of half-open intervals, whose total Lebesgue measure is αa1 · · ·αan , which goes to zero as ngrowths.

Let ξθ : I → R be the map defined in the following way: Given t a point in I withitinerary a1a2 . . ., then ξθ (t) is the unique point of ∩∞

1 (Ra1...an ). The map ξθ is welldefined since the itinerary of any point in I is unique.

Proposition 2.1 ([18]) If there exists θ so that ξθ (t∗) = y then ξθ is continuous andonto.

Proof In order to verify that the map ξθ is continuous it suffices to check that neigh-bouring cylinders of I are mapped to neighbouring cylinders of R. Wa say that twocylinders are neighbouring cylinders if the intersection of their closures is not empty.Let Ia1...ak and Ib1...bl be neighbouring cylinders of I such that ai = bi for 1 ≤ i < jand a j �= b j . Let suppose that Ia1...ak is to the left of Ib1...bl so the intersectionpoint of these cylinders is tb1...b j = fb1...b j (t

∗). Then ξ( fb1...b j (t∗)) is in Rb1...bl and

ξ( fa1...a j (t∗)) is in Ra1...ak . Therefore

ξ( fb1...b j (t∗)) = φb1...b j (y) = φb1...b j−1(z)

and

ξ( fa1...a j (t∗)) = φa1...a j (y) = φa1...a j−1(z).

Since b1 . . . b j−1 = a1 . . . a j−1, we have ξ( fa1...a j (t∗)) = ξ( fb1...b j (t

∗)). There-fore Rb1...bl ∩ Ra1...ak �= ∅.

Let us check that the map ξ is onto. Let x be an element in R and a1a2 · · · anitinerary of x . If ∩∞

n=1 Ia1···an is not empty, so ξθ (t) = x for all t ∈ ∩∞n=1 Ia1···an .

However this set it might be empty, since we consider I half-open, so the cylindersare union of half-open intervals, but ∩∞

n=1 Ia1···an is not empty. If ∩∞n=1 Ia1···an = ∅

and t ∈ ∩∞n=1 Ia1···an so t is a right extreme point of one of the intervals that form

the cylinders Ia1···ak for k sufficiently large, and t is a left extreme point of one ofthe closed intervals that form the closure of cylinders Ib1···bm for some word b1 · · · bm

with m large enough. Since neighbouring cylinders in I are mapped to neighbouringcylinders in R, we conclude ξ(t) = x . ��

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It follows from the proof of Proposition 2.1, that we require the common pointproperty of the IFS {φi }k

i=1 for the existence of the continuous map ξθ : I → R.However the open set condition for the IFS is not required.

If there exists θ so that 0 < θ < 1 and ξθ continuous then we say that the IFS{φ1, . . . , φk} is realizable by a regular space-filling curve and the map ξθ is a regularspace-filling curve to the IFS {φ1, . . . , φk}. Throughout the article we will assumethat the IFS {φ1, . . . , φk} is realizable by a regular space-filling curve and we will notshow the dependence of θ of the map ξ , we will use ξ for ξθ , where θ is a value sothat ξθ is continuous. In Ref. [18] it is shown that the map ξ is 1/s-Hölder continuouswhen the IFS is self-similar, where s is the Hausdorff dimension of the attractor R.Moreover if the IFS {φ1, · · · , φk} satisfies the open set condition and is self-similarthen the map ξ is a measure preserving map between the Lebesgue measure of I andthe normalized s-dimensional Hausdorff measure of R. If the IFS is not self-similarbut s = d then ξ is measure preserving between the Lebesgue measure of I and thenormalized Lebesgue measure of R ⊂ R

d [18].

3 Geodesic lamination

Let D2 be the closed unit disk in the plane, and S

1 its boundary. We identify S1 with

I = [0, 1). Since the image of 0 and 1 are the same under the maps fi ’s on the interval.We think of these maps as acting on the boundary of the disk. A geodesic in D

2 is anarc of circle that meets the boundary of D

2 perpendicularly.

Definition 3.1 A geodesic lamination on D2 is a non-empty closed set of geodesics

of the disk and that any two of these geodesics do not intersect except at their endpoints.

The construction of the geodesic lamination is as follows: We consider the extrem-ities of the intervals Ii ’s. And we join pairwise consecutive extremities by geodesicsin D

2.Let a1 . . . an be a word in the alphabet {1, . . . , k}. We join by geodesics the points

fa1 · · · fan (ti ) and fa1 . . . fan (ti+1) for 1 ≤ i ≤ k, where i + 1 is taken mod k and theti is the left boundary point of the interval Ii , i.e. ti = α0 +· · ·+αi−1 = fi (t∗), whereα0 = 0. We do this for all possible finite words in this alphabet and later we take theclosure in the topology given by Hausdorff metric of D

2. The elements obtained inthis way are either geodesics of D

2 or points in S1. In the latter case the points are

called degenerate geodesics.

Proposition 3.1 ([18]) is a geodesic lamination on D2.

Proof Let J = Ia1...an be a cylinder and suppose that it consists of more than oneconnected component. Let x1 and x2 be neighbouring end points of this cylinderbelonging to different connected components, see Fig. 1. Without lost of generalitywe can suppose that x2 = fa1···an (t

∗) and x1 = fa1···an (t∗−) := limt→t∗− fa1···an (t).

So x1 = fa1···an−1(tan+1) and x2 = fa1···an−1(tan ) where the sum an + 1 is takenmod k.

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548 V. F. Sirvent

J’

x1 y1 x2 y2

J J’ J

Fig. 1 Figure relative to the proof of Proposition 3.1

3

4

2

βα

1

Fig. 2 The IFS on R of Example 1

Let us consider the cylinder J ′ = Ia1···an−1an+1 whose left end point is x1. We shallshow that J ′ is contained in the gap of J formed by [x1, x2). Let us suppose that thecylinder J ′ consists of more than one connected component and one of these compo-nents is outside the gap [x1, x2), say to the right of x2. So there exists 1 ≤ j ≤ k suchthat fa1...an−1an+1(t j ) is the discontinuity point for this cylinder, i.e.

x1 < y1 = fa1...an−1an+1(t−j ) < x2 < y2 = fa1...an−1an+1(t j ).

So

fa1...an−1(tan+1) < fa1...an−1an+1(t j+1) < fa1...an−1(tan ) < fa1...an−1an+1(t j )

fa1...an−1an+1(t∗) < fa1...an−1an+1(t j+1) < fa1...an−1an (t

∗) < fa1...an−1an+1(t j ).

Since the maps fi preserve the cyclic order of the images of {t1, . . . , tk}. We havethat the points fan+1(t∗), fan+1(t j+1), fan (t

∗) and fan+1(t j ) are in this cyclic orderin S

1. However, the first two points and the last belong to the interval Ian+1 and thethird point to Ian that contradicts the fact that these are disjoint intervals. ��Proposition 3.2 ([18]) Let λ be an element of with end points b and b′. Thenξ(b) = ξ(b′).

Proof By the construction of there are geodesics λi such that they join the imagesunder the maps fa1 · · · fan , for some finite words a1 · · · an in the alphabet {1, . . . , k},of the points t j ’s. So that the geodesics λi converge to λ.

If the end points of λi are bi and b′i then ξ(bi ) = ξ(b′

i ). So by the continuity of thespace-filling curve: ξ(b) = ξ(b′). ��

123


The converse of Proposition 3.2 might not be true in general. If there is a boundarypoint between Ri and Ri+1, different from the intersection point, then its preimageslie in Ii and Ii+1 then they cannot be joined.

These properties allow us to define a map � : → R as follows: Let λ be ageodesic of with end points b, b′. So �(λ) := ξ(b). By Proposition 3.2 this map iswell defined. The map � is continuous and surjective since ξ has these properties.

On the laminationwe define a dynamical system (, ) as follows: Let F : I →I the expanding map defined in Sect. 2. Let λ ∈ whose end points are s and s′, wedefine (λ) as the geodesic that joins F(s) and F(s′). If λ joins fa1 . . . fam (ti ) withfa1 · · · fam (ti+1), for m ≥ 2 then (λ) joins fa2 . . . fam (ti )with fa2 . . . fam (ti+1). Form = 1 we have ti = fi (t∗). Therefore (λ) ∈ . Since the map F is continuous, so is continuous. In Ref. [18], we show there is a transversal measure to the laminationso that the map is expanding with respect to this transversal measure.

4 Symmetries

Let H be the full group of symmetries of i.e. H is the largest subgroup of Euclideanisometries of D

2 such that hγ ∈ for all h ∈ H and γ ∈ . Let us remark that Hacts continuously on .

Proposition 4.1 The group H induces a continuous action on R.

Proof The group H induces an action on R in the following way: Let h ∈ H andγ ∈ , since H is the group of symmetries of , hγ ∈ . So we have

h∗ : �(γ ) �→ �(hγ ).

This action is well defined, in fact: Let

I =∞⋃

n=1

{Ia1···an : 1 ≤ ai ≤ k, for 1 ≤ i ≤ n}.

By the definition of the lamination it is constructed by joining by geodesicsneighbouring extreme points of the connected components of the elements of I, andlater taking the closure in D

2. Given γ ∈ , such that it joins the neighbouring endpoints of some Ia1...an ∈ I. Due to the fact that H is the symmetry group of , wehave that hγ , joins neighbouring end points of some element of I; otherwise hγ /∈ .Therefore H acts on I, we denote by hIa1...an the element of I such that the geodesicsof that join its neighbouring end points, are of the form hγ , where γ joins theneighbouring end points of Ia1...an .

Let us remark that h is an Euclidean isometry of D2, therefore of S

1. Since h acts onI, it permutes the elements of the partition {I1, . . . , Ik}; moreover h sends isometri-cally Ii into I j , where j = j (i), with 1 ≤ i, j ≤ k, hence Ii , I j have the same length,i.e. αi = α j (i). So h ◦ fi = f j (i) where f1, . . . , fk are the maps that acts on I so thatIi = fi (I ) for 1 ≤ i ≤ k. Therefore hIa1a2...an = Ib1a2...an , where h ◦ fa1 = fb1 .

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550 V. F. Sirvent

Let γ, γ ′ be geodesics in such that �(γ ) = �(γ ′). By the definition of , bythe continuity of the action of H on and by the continuity of the map � : → R,we can assume with lost of generality that γ and γ ′ are geodesics joining neigh-bouring extreme points of Ia1...anan+1 and Ia1...ana′

n+1respectively, for some words

a1 . . . an , a1 . . . anan+1 and a1 . . . ana′n+1. Since hIa1...anan+1 and hIa1...ana′

n+1are ele-

ments of I, we have that hγ and hγ ′ are geodesics joining neighbouring extremepoints of hIa1...anan+1 and hIa1...ana′

n+1, respectively. By the previous remark, we have

hIa1a2...an = Ib1a2...an , so hγ and hγ ′ are geodesics joining neighbouring extremepoints of Ib1a2...anan+1 and Ib1a2...ana′

n+1, respectively. Therefore �(hγ ) = �(hγ ′).

The continuity of the map h∗ : R → R, follows from the continuity of the actionof H on . ��

Let us denote by H∗ the group that defines the action described in the previousproposition.

Theorem 4.1 If the IFS {φ1, . . . , φk} is self-similar then the group H∗ is isomorphicto a subgroup of the full group of symmetries of R.

Proof Let h ∈ H , by Proposition 4.1, we have that the map h∗ : R → R is continu-ous. Since H∗ is a group, the map h∗ is a homeomorphism on R. We shall show thatit is a local isometry on R, so it follows that it is an isometry. Hence H∗ is a subgroupof the full group of isometries of R.

In order to show that h∗ : R → R is an local isometry it is sufficient that sendselements of

∞⋃i=1

{Ra1···an : 1 ≤ ai ≤ k, for 1 ≤ i ≤ n}

into itself and of the same diameter. Let {φ1, . . . , φk} and { f1, . . . , fk} the IFS-s whoseattractor are R and I , respectively. As it was explained in the previous Proposition, hpermutes isometrically the elements of the partition {I1, . . . , Ik} of I , i.e. h◦ fi = f j (i)

so that fi , f j (i) have the same contraction factor: αi = α j (i). Therefore h∗ ◦φi = φ j (i)

and φi and φ j (i) have the same contraction factor ci = c j (i). It follows

h∗Ra1a2...am =h∗ ◦ φa1 ◦ φa2 ◦ · · · ◦ φam (R)=φb1 ◦ φa2 ◦ · · · ◦ φam (R)=Rb1a2···am .

Since the maps φl for 1 ≤ l ≤ k are similarities, h∗ maps isometrically Ra1a2···am intoRb1a2...am . From the continuity of h∗ follows that it is a local isometry. Hence it is anisometry of R. ��

If d = 2 it is not clear if the converse of this theorem is true. In Example 3, weshow that the converse is false.

The following proposition shows that the map is invariant under the symmetryclasses of. So the dynamical system (, ) induces a dynamical system on the thesymmetry classes of , i.e. H : /H → /H .

123


Proposition 4.2 Let γ be an element of the lamination and h an element of thegroup H. Then (γ ) = (hγ ).

Proof Let us suppose that λ joins neighbouring end points of Ia1...an , i.e. fa1 fa2 . . . fam

(ti ) with fa1 fa2 . . . fam (ti+1), for m ≥ 2 then by definition (λ) joins fa2 . . . fam (ti )with fa2 . . . fam (ti+1). On the other hand, by the arguments of the proof of Proposi-tion 4.1 hλ joins fb1 fa2 . . . fam (ti ) with fb1 fa2 . . . fam (ti+1), for some b1. So (hγ )joins fa2 . . . fam (ti )with fa2 . . . fam (ti+1). In the case m = 1, we have that ti = hi (t∗).If γ joins ti with ti+1 then hγ joins t j with t j+1, where j = j (i) so that hIi = I j , asit was explained in the proof of Proposition 4.1. So (γ ) = (hγ ). ��

5 Sub-IFS, sub-laminations and the corresponding symmetries

We say that {ψ1. . . . , ψl} is a sub-IFS of {φ1, . . . , φk}, if ψ j = φσ( j) for 1 ≤ j ≤l < k and σ : {1, . . . , l} → {1, . . . , k} injective. Let F be the attractor of the IFS{ψ1. . . . , ψl}.

If ξ : I → R is a regular space-filling curve to the IFS {φ1, . . . , φk}. Let F bea non-empty connected attractor of a sub-IFS of {φ1, . . . , φk}. We can associate aspace-filling curve to F, ξ : I → F, in the following way: Let C = ξ−1(F) ⊂ I .Obviously the map ξ |C : C → F is continuous. This map can be extend continuouslyto I using the Cantor-Lebesgue construction. Let φ j be a map in the original IFS, butnot in the sub-IFS, so the set I j and its iterates under the IFS { f1, . . . , fk} are not inI . By the definition of the map ξ the image, under ξ , of the end points of I j are thesame. So we can define ξ (I j ) = ξ(t j ), in this way if J is a connected component ofI \ C then we define ξ on any element of J as the image under ξ of the end point ofJ . So ξ defined in this way is continuous and ξ |C = ξC . Let us denote ξF = ξ .

To the attractor F and the space-filling curve ξF we associate a sub-lamination F

of , by F = �−1(F).Let H0, G0 be the symmetry group of F and F, respectively. Obviously H0 and

G0 are subgroups of H and G. From Theorem 4.1 follows the corollary:

Corollary 5.1 H0 is isomorphic to a subgroup of G0.

In Example 3, we show that the converse of this corollary is false.We can define a dynamical systems on (F, F) as the restriction of toF. Due

to the definition of the map , we get (F) ⊂ F. From Proposition 4.2 follows ina straight forward manner:

Corollary 5.2 The map F is constant on the symmetry classes ofF defined by H0.

6 Examples

Example 1 In Ref. [18] was introduced the following non-self-similar example. Letthe IFS {φ1, . . . , φ4} on the unit square R = {(x1, x2) ∈ R

2 : 0 ≤ x1, x2 ≤ 1}, where

φ1(x1, x2) = (−αx1 + α,−αx2 + α), φ2(x1, x2) = (−αx2 + α, βx1 + α),

φ3(x1, x2) = (βx1 + α, βx2 + α), φ4(x1, x2) = (βx2 + α,−αx1 + α)

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552 V. F. Sirvent

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Fig. 3 The graph of the expanding map F of Example 1

with α, β positive numbers such that α + β = 1. Note this IFS is self-similar onlyin the case α = β = 1/2. Using the geometry of the transformations φi -s, it canbe checked, without difficulty, that itineraries for the intersection point, i.e. (α, α)are 113 = 1133 . . ., 213, 313 and 413. The itinerary 13 corresponds to the point(0, 0) which is the point y in the definition of the common point property. We geta regular space-filling curve with the system of maps { f1, . . . , f4} on I , taking θ =α − α3/2:

f1(t) ={α2t + (α2 − α4/2) if 0 ≤ t < α2/2

α2t − α4/2 if α2/2 ≤ t < 1

and f2(t) = (β/α) f1(t) + α2, f3(t) = (β/α)2 f1(t) + α2 + αβ, f4(t) =(β/α) f1(t) + α2 + αβ + β2. For this value of θ the itinerary of t∗ = α2/2 is13. The collection of maps { f1, . . . , f4} is the system of branches of the inversefunction of F , whose graph is shown in Fig. 3 for the value α = (

√5 −

1)/2.

In Ref. [18] was proved, using direct methods, that the associated lamination has a reflection symmetry, which corresponds to the symmetry along the diagonal thatpasses through (0, 0) and (1, 1) of R. This symmetry and the identity correspondsto the full group of symmetries of the partition {R1, . . . ,R4}, which in this case is astrict subgroup of the full group of symmetry of R, i.e. D4. In the self-similar caseα = β = 1/2 the lamination has the full group of symmetries of R [16].

In Fig. 4 can be seen the lamination when α = (√

5−1)/2. The space-filling curveassociated to this value is known as the golden space-filling curve.

We can modify the maps of this IFS, taking the composition of any φi with thereflection along the main diagonal of R. It can be easily checked that the associatedlamination to this new IFS is the same of the lamination to the original IFS.

Example 2 Here we consider sub-IFS-s of the IFS of Example 1 and the correspond-ing laminations. The attractors and laminations of the different sub-IFS, with three

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Fig. 4 Lamination of Example 1 where α = (√

5 − 1)/2

Fig. 5 Attractor and lamination associated to the IFS {φ2, φ3, φ4} of Example 2

maps, can be seen in Figs. 5, 6 and 7. In all cases the attractor and the lamination havethe same group of symmetries, only in the case of Fig. 7 the group is trivial.

Example 3 Here we consider the maps f1, . . . , f4 on I as follows: ai = 1/4 for1 ≤ i ≤ 4, θ = 1/16, i.e.

f1(t) :=

⎧⎪⎨⎪⎩

1

4t + 1

64if 0 ≤ t <

15

16,

1

4t − 15

64if

15

16≤ t < 1

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554 V. F. Sirvent



Fig. 8 Attractor and lamination associated to the IFS {ψ1, ψ2, ψ4} of Example 3

and fi (t) = f1(t) + (i − 1)/4 for i = 2, 3 or 4. And the IFS on R {ψ1, . . . , ψ4}where ψ1 = φ4, ψ2 = φ2, ψ3 = φ3 and ψ4 = φ1; and the maps φi -s are of Exam-ple 1. Here the itinerary of (0, 0), the point y ∈ R in the definition of the commonpoint property, is 43. Can be checked that this itinerary also corresponds to the pointt∗ = 1−θ . So we can associate a space-filling curve to this IFS. This example was firstintroduced in Ref. [16], there it was shown directly that the full group of symmetries

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of the corresponding lamination is the rotation of the square R4. In this example weshow that the converse of Theorem 4.1 is false. We have that R has D4 as group ofsymmetries; however, the group of symmetries of is only R4. The sub-laminationsof have only the trivial symmetries; however, the corresponding attractors of thesub-IFS, which are similar to the Example 2), have a richer group of symmetries.Showing that the converse of Corollary 5.1 is false.

Example 4 Here we consider an example that does not satisfy the open set condition.The sets Rm intersect themselves in a set of positive Lebesgue measure. Let the IFS{φ1, . . . φk} on C which we identify to R

2, where φm(x) = ei2(m−1)π/k(x + 1)/2, for1 ≤ m ≤ k, k even and x ∈ C. Its attractor is the regular k-gon on C, inscribe in theunit circle and having x = 1 as one of its vertices. Clearly the sets Rm and Rm+1intersects in a set of non-empty interior. Let f1, . . . , fk be the maps on I defined asfollows:

f1(t) =

⎧⎪⎨⎪⎩

t

k+ k − 1

2k2 if 0 ≤ t <k + 1

2k,

t

k+ k − 1

2k2 − 1

kif

k + 1

2k≤ t < 1

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556 V. F. Sirvent

Fig. 11 Lamination of Example4 with k = 6

Fig. 12 Lamination and attractor of the sub-IFS {φ2, φ3, φ4, φ5, φ6} of Example 4 with k = 6

and fm(t) = f1(t) + (m − 1)/k for m = 2, . . . , k, i.e. αm = 1/k, θ =(k − 1)/2k in the notation of Sect. 2. The itinerary of t∗ = 1 − θ is l1, withl = k/2 + 1, which is the same itinerary, according to the IFS {φi }k

i=1, of thepoint y in the definition of the common point property. So there is a space-fillingcurve to this IFS and a geodesic lamination. In Ref. [16] it was shown that thegroup of symmetries of this laminations in the same group of symmetries of thek-gon, Dk . The lamination in the case of k = 6 can be seen in Fig. 11. Thereare many sub-IFS of the given that gives attractors and laminations of interest-ing symmetries, see Figs. 12, 13, 14, 15 and 16. The symmetries of some of theattractors of the sub-IFS shown here, have been studied in a different context inRef. [5].

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Fig. 13 Lamination and attractor of the sub-IFS {φ1, φ3, φ4, φ6} of Example 4 with k = 6

Fig. 14 Lamination and attractor of the sub-IFS {φ1, φ3, φ4, φ5, } of Example 4 with k = 6

Fig. 15 Lamination and attractor of the sub-IFS {φ2, φ4, φ6} of Example 4 with k = 6

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558 V. F. Sirvent

Fig. 16 Lamination and attractor of the sub-IFS {φ1, φ3, φ5} of Example 4 with k = 6

Like in Example 3 we can have laminations with only rotational symmetries asso-ciated to the k-gon, with k even.

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sity, Cambridge (1988)4. Falconer, K.: Fractal geometry: mathematical foundations and applications. Wiley, Chichester (1990)5. Falconer, K., O’Connor, J.: Symmetry and enumeration of self-similar fractals. Bull. London Math.

Soc. 39, 272–282 (2007)6. Hata, M.: On the structure of self similar sets. Jpn. J. Appl. Math. 2, 381–414 (1985)7. Hilbert, D.: Über die stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen 38

(1891), 459–460. Gesammelte Abhandlungen Vol III, 1–2, Springer-Verlag, Berlin, (1935)8. Keller, K.: Invariant factors, Julia equivalences and the (abstract) Mandelbrot set, Lecture Notes in

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lars, Paris (1904)11. Peano, G.: Sur une courbe qui remplit toute une aire plane. Math. Ann. 36, 157–160 (1890)12. Peano, G.: La Curva di Peano Nel: Formulario Mathematico, vol 5, Frates Bocca, Turin 239–240

(1908)13. Sagan, H.: Space-Filling curves. Springer-Verlag, New York (1994)14. Sierpinski, W.: Sur une nouvelle courbe continue qui remplit toute une aire plane, Bull. Acad. Sci. de

Cracovie, Série A, 462–478 (1912)15. Sirvent, V.F.: Geodesic laminations as geometric realizations of Pisot substitutions. Ergodic Theory

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Math. Soc. Simon Stevin 10, 221–229 (2003)18. Sirvent, V.F.: Space filling curves and geodesic laminations. Geometriae Dedicata. 135, 1–14 (2008)19. Thurston, W.P.: On the combinatorics and dynamics of iterated rational maps. Preprint, Princeton

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