aa r X i v : . [ m a t h . GN ] S e p Mixed labyrinth fractals
Ligia L. Cristea ∗ Karl-Franzens-Universit¨at GrazInstitut f¨ur Mathematik und Wissenschaftliches RechnenHeinrichstrasse 36, 8010 GrazAustria [email protected]
Bertran SteinskyF¨urbergstr. 56, 5020 SalzburgAustria [email protected]
September 28, 2020
Keywords: fractal, dendrite, tree, graph, length of paths, Sierpi´nski carpets
AMS Classification:
Abstract
Labyrinth fractals are self-similar fractals that were introduced and studied in recentwork [2, 3]. In the present paper we define and study more general objects, called mixedlabyrinth fractals , that are in general not self-similar and are constructed by usingsequences of labyrinth patterns . We show that mixed labyrinth fractals are dendritesand study properties of the paths in the graphs associated to prefractals, and of arcs inthe fractal, e.g., the path length and the box counting dimension and length of arcs. Wealso consider more general objects related to mixed labyrinth fractals, formulate twoconjectures about arc lengths, and establish connections to recent results on generalisedSierpi´nski carpets.
Labyrinth fractals are self-similar fractals that were introduced and studied by Cristeaand Steinsky [2, 3]. In the present paper we deal with mixed labyrinth fractals that area generalisation of the labyrinth fractals studied before [2, 3]. Mixed labyrinth fractalsare fractal sets obtained by an iterative construction that uses labyrinth patterns, asdescribed in Sections 2 and 3, and are in general not self-similar. We remind thatgeneralised Sierpi´nski carpets [4, 5] studied some years ago were also defined with thehelp of patterns, and are in general not self-similar. There are recent results [11] onthe topology of a class of self-similar Sierpi´nski carpets called fractal squares . In thecase of the mixed labyrinth fractals, there are special restrictions on the patterns, thatcorrespond to the properties of labyrinth sets [2, 3]. Labyrinth patterns have three ∗ This author is supported by the Austrian Science Fund (FWF), Project P27050-N26 and by the Austrain-French cooperation project FWF I1136-N26. roperties, which we formulate in Section 3, with the help of graphs that we associateto the patterns. An example for the first two steps of the construction of a mixedlabyrinth fractal is illustrated in Figure 1 and 2. In Section 4 we show that mixedlabyrinth fractals are dendrites. Section 5 is dedicated to properties of the paths inthe graphs associated to the prefractals and of paths in the fractal. In Section 6 weconjecture a property of the length of the path between any two points in a mixedlabyrinth fractal generated by a sequence of blocked labyrinth patterns. In the Sections7 and 8 we discuss other extensions of labyrinth sets and labyrinth fractals. Finally,Section 9 is dedicated to a result on the total disconnectedness of generalised Sierpi´nskicarpets generated by complementary patterns of labyrinth patterns.Before presenting our results let us remark that labyrinth fractals are strongly re-lated to other mathematical objects studied by mathematicians and physicists. First,we mention that during the last years related objects called “fractal labyrinths” havebeen in the attention of physicists, either in the context of nanostructures [9], or offractal reconstructions of images, signals and radar backgrounds [16], who used the for-malism for the self-similar case of labyrinth fractals [2] in order to refine their definitionof these objects, or to formulate it with more precision. We note that while our maininterest regards properties of the limit set, the physicists focus on what we would callthe prefractals of a labyrinth fractal, i.e., on the objects obtained after a finite numerof steps in the iterative construction of the fractal.Mixed labyrinth fractals are also related to the objects introduced 1986 by Falconer[7] in the context of random fractals, as net fractals , and random net fractals . Inthat probabilistic framework the focus is on the Hausdorff dimension of these randomfractals. There, trees and random networks are used in order to define these objectsand study their dimension. Due to the very general setting chosen for the mixedlabyrinth fractals in the present paper, e.g., to the fact that there are no restrictionson the width of the patterns or on the number of black/white squares in the patternsthat generate the mixed labyrinth fractal, we could not apply the results on measuresand dimensions obtained for net fractals to the mixed labyrinth fractals. In the samecontext we also mention the work of Mauldin and Williams on random fractals [12] andon graph directed constructions [13]. Moreover, the graph directed Markov systems(GDMS) studied in much detail in the book of Mauldin and Urbanski [14] are alsorelated to the objects that we study here. However, in the very general setting of thepresent paper, regarding the width and the structure of the patterns that generatethe mixed labyrinth fractal, we chose to extend the results obtained for self-similarlabyrinth fractals to the case of mixed labyrinth fractals by reasoning whithin the sameframework. For an overview on random fractals we also refer to M¨orters’ contributedchapter to the volume on new perspectives in stochastic geometry [17].Finally, let us also mention that there is a lot of ongoing research on V -variablefractals, e.g., [8], that also provide a framework that could be used for certain classes ofgeneralised Sierpi´nski carpets, and, in particular, families of mixed labyrinth fractals.For V -variable fractals and superfractals we also refer to Barnsley’s book [1].To conclude this introductive section, let us remark that in the present paper wefocus on geometric and topological properties of mixed labyrinth fractals as a general-isation of the self-similar labyrinth fractals studied before [2, 3] and stick to the samesetting, to the approach with patterns and no probabilistic frame. Of course, in futurework one can use the probabilistic, GDMS, or V -variable fractals approach, in order toachieve further results on classes of mixed labyrinth fractals. In order to construct labyrinth fractals we use patterns . Figures 1 and 2 show examplesof patterns and illustrate the first two steps of the construction described now. Let , y, q ∈ [0 ,
1] such that Q = [ x, x + q ] × [ y, y + q ] ⊆ [0 , × [0 , z x , z y ) ∈ [0 , × [0 ,
1] we define the function P Q ( z x , z y ) = ( qz x + x, qz y + y ) . Let m ≥ S i,j,m = { ( x, y ) | im ≤ x ≤ i +1 m and jm ≤ y ≤ j +1 m } and S m = { S i,j,m | ≤ i ≤ m − ≤ j ≤ m − } .We call any nonempty A ⊆ S m an m - pattern and m its width . Let {A k } ∞ k =1 bea sequence of non-empty patterns and { m k } ∞ k =1 be the corresponding width-sequence ,i.e., for all k ≥ A k ⊆ S m k . We denote m ( n ) = Q nk =1 m k , for all n ≥
1. We let W = A , and call it the set of white squares of level
1. Then we define B = S m \ W as the set of black squares of level
1. For n ≥ set of white squares oflevel n by W n = [ W ∈A n ,W n − ∈W n − { P W n − ( W ) } . (1)We note that W n ⊂ S m ( n ) , and we define the set of black squares of level n by B n = S m ( n ) \ W n . For n ≥
1, we define L n = S W ∈W n W . Therefore, { L n } ∞ n =1 is amonotonically decreasing sequence of compact sets. We write L ∞ = T ∞ n =1 L n , i.e., the limit set defined by the sequence of patterns {A k } ∞ k =1 . A graph G is a pair ( V , E ), where V = V ( G ) is a finite set of vertices, and the set ofedges E = E ( G ) is a subset of {{ u, v } | u, v ∈ V , u = v } . We write u ∼ v if { u, v } ∈ E ( G )and we say u is a neighbour of v . The sequence of vertices { u i } ni =0 is a path between u and u n in a graph G ≡ ( V , E ), if u , u , . . . , u n ∈ V , u i − ∼ u i for 1 ≤ i ≤ n , and u i = u j for 0 ≤ i < j ≤ n . The sequence of vertices { u i } ni =0 is a cycle in G ≡ ( V , E ),if u , u , . . . , u n ∈ V , u i − ∼ u i for 1 ≤ i ≤ n , u i = u j for 1 ≤ i < j ≤ n , and u = u n . A tree is a connected graph that contains no cycle. A connected component is an equivalence class of the relation, where two vertices are related if there is a pathbetween them. For A ⊆ S m , we define G ( A ) ≡ ( V ( G ( A )) , E ( G ( A ))) to be the graph of A , i.e., the graph whose vertices V ( G ( A )) are the white squares in A , and whose edges E ( G ( A )) are the unordered pairs of white squares, that have a common side. The toprow in A is the set of all white squares in { S i,m − ,m | ≤ i ≤ m − } . The bottomrow, left column, and right column in A are defined analogously. A top exit in A is awhite square in the top row, such that there is a white square in the same column inthe bottom row. A bottom exit in A is defined analogously. A left exit in A is a whitesquare in the left column, such that there is a white square in the same row in the rightcolumn. A right exit in A is defined analogously. While a top exit together with thecorresponding bottom exit build a vertical exit pair , a left exit and the correspondingright exit build a horizontal exit pair .If A = W n , for n ≥
1, we call the top row in A the top row of level n . The bottomrow, left column , and right column of level n are defined analogously.A non-empty m -pattern A ⊆ S m , m ≥ m × m - labyrinth pattern (inshort, labyrinth pattern ) if A satisfies Property 1, Property 2, and Property 3. Property 1. G ( A ) is a tree. Property 2. A has exactly one vertical exit pair, and exactly one horizontal exit pair. Property 3.
If there is a white square in A at a corner of A , then there is no whitesquare in A at the diagonally opposite corner of A . Let {A k } ∞ k =1 be a sequence of non-empty patterns, with m k ≥ n ≥ W n the corresponding set of white squares of level n . We call W n an m ( n ) × m ( n )- mixedlabyrinth set (in short, labyrinth set ), if A = W n satisfies Property 1, Property 2, andProperty 3. A (a 4-pattern), A (a 5-pattern), and A (a 4-pattern)Figure 2: The set W , constructed based on the above patterns A and A , that can alsobe viewed as a 20-pattern Remark.
Any labyrinth pattern is a labyrinth set, and any mixed labyrinth set canbe seen as a labyrinth pattern. We use two distinct notions, in order to make it clearerthat in general the construction of a (mixed) labyrinth set is based on a sequence oflabyrinth patterns.
Lemma 1.
Let {A k } ∞ k =1 be a sequence of non-empty patterns, m k ≥ , and n ≥ .If A , . . . A n are labyrinth patterns, then W n is an m ( n ) × m ( n ) -labyrinth set, for all n ≥ , where m ( n ) = Q nk =1 m k .Proof. The proof is almost literally the same as in the case of self-similar labyrinthfractals [2, Lemma 2]. It only differs in two points in the second half of the proof: hereinstead of using the fact that G ( W ) is a tree, we use the argument that G ( A n ) is atree, and instead of using Property 2 for G ( W ), we use Property 2 for G ( A n ).The limit set L ∞ defined by a sequence {A k } ∞ k =1 of labyrinth patterns is called mixed labyrinth fractal . For n ≥
1, we define G ( B n ) ≡ ( V ( G ( B n )) , E ( G ( B n ))) to be thegraph whose vertices V ( G ( B n )) are the black squares in B n , and whose edges E ( G ( B n ))are the unordered pairs of black squares, that have a common side or a common corner.A border square of level n is a square that lies in the right column, the left column, thetop row, or the bottom row, of level n respectively. A , A , A from Figure 1, and the fourth is A Lemma 2.
Let {A k } ∞ k =1 be a sequence of labyrinth patterns, m k ≥ and n ≥ . Fromevery black square in G ( B n ) there is a path in G ( B n ) to a black border square of level n in G ( B n ) .Proof. The proof uses Lemma 1 and is literally the same as in the case of the labyrinthsets occuring in the construction of self-similar labyrinth fractals [2, Lemma 2].A function is a homeomorphism if it is bijective, continuous, and its inverse iscontinuous. A topological space X is an arc if there is a homeomorphism h from [0 , X . We say X is an arc between h (0) and h (1). For the following two results weskip the proof, since the proofs given in the self-similar case [2] work also in the moregeneral case of the mixed labyrinth sets. For more details and definitions we refer tothe mentioned papers [2, 3]. Lemma 3.
Let {A k } ∞ k =1 be a sequence of labyrinth patterns, m k ≥ . If x is a pointin ([0 , × [0 , \ L n , then there is an arc a ⊆ ([0 , × [0 , \ L n +1 between x and apoint in the boundary fr ([0 , × [0 , . Corollary 1.
Let {A k } ∞ k =1 be a sequence of labyrinth patterns, m k ≥ , and n ≥ . If x is a point in ([0 , × [0 , \ L ∞ , then there is an arc a ⊆ ([0 , × [0 , \ L ∞ between x and a point in fr ([0 , × [0 , . We remind that a continuum is a compact connected Hausdorff space, and a dendrite is a locally connected continuum that contains no simple closed curve.
Theorem 1.
Let {A k } ∞ k =1 be a sequence of labyrinth patterns, m k ≥ , for all k ≥ .Then L ∞ is a dendrite.Proof. L ∞ is the intersection of the compact connected sets L n , n ≥
1, and thus itis connected. One can easily check that for any ǫ >
0, there is an n ≥
1, such thatthe diameter of W n ∈ W n is less than ǫ (e.g., by using the facts that m ( n ) > n andthat the diameter of any square in W n is √ m ( n ) ). Thus, for any ǫ > L ∞ is the finiteunion of connected sets of diameter less than ǫ , by the definition of W n (in Equation1). The Hahn-Mazurkiewicz-Sierpi´nski Theorem [10, Theorem 2, p.256] yields that L ∞ is locally connected. As in the self-similar case [2, 3] one can show, by using the JordanCurve Theorem and Corollary 1 that L ∞ does not contain any simple closed curve. emark. Between any pair of points x = y in L ∞ there is a unique arc [10,Corollary 2, p. 301]. In this section all patterns in the sequence ( A k ) k ≥ used in the iterative constructionof W n are labyrinth patterns.We call a path in G ( W n ) a -path if it leads from the top to the bottom exit of W n .The , , , , and -paths lead from left to right, top to right, right to bottom, bottomto left, and left to top exits, respectively. We denote by ( n ) , ( n ) , ( n ) , ( n ) , ( n ),and ( n ) the length of the respective path in G ( W n ), for n ≥ , and by k , k , k , k , k ,and k the length of the respective path in G ( A k ), for k ≥
1. By the length of sucha path we mean the number of squares in the path. Of course, for n = k = 1 the twopath lengths coincide, i.e., (1) = , . . . , (1) = . Proposition 1.
There exist non-negative × -matrices M k , k = 1 , , . . . , such that kkkkkk = M k · , (2) and for M ( n ) = M · M · · · · · M n , for all n ≥ , the element in row x and column y of M ( n ) is the number of y -squares in the x -path in G ( W n ) . Furthermore, ( n )( n )( n )( n )( n )( n ) = M ( n ) · . (3) Proof.
We explain how the path between all possible pairs of exits can be constructed.In order to show the idea of this construction, we start, e.g., with a path betweenthe right and the bottom exit, as shown in Figure 8. We note that the constructiondescribed below works for all mixed labyrinth fractals.First, we find the path between the right and the bottom exit of W , (or, equivalently A ) shown in Figure 5). Then we denote each white square in the path according to itsneighbours within the path: if it has a top and a bottom neighbour it is called - square (with respect to the path), and it is called , , , , and - square if its neighboursare at left-right, top-right, right-bottom, bottom-left, and left-top, respectively. If thewhite square is an exit, it is supposed to have a neighbour outside the side of theexit. A bottom exit, e.g., is supposed to have a neighbour below, outside the bottom,additionally to its inside neighbour. We repeat this procedure for all possible pathsbetween two exits in G ( W ), as shown in Figure 4, 5, and 6. In order to obtain the-path in G ( W ), which is shown in Figure 8, we replace each -square of the pathin G ( W ) with the -path in G ( A ), which is shown in Figure 7. Analogously, we dothis for the other marked white squares, such that the path between the right and thebottom exit of W (shown in Figure 8) arises. In general, for any pair of exits and n ≥ G ( W n ) with its corresponding pathin G ( A n +1 ) and obtain the path of G ( W n +1 ). We define the matrix M k , k ≥
1, thatoccurs in Equation 2 in the following way: the columns of M k from left to right and the A Figure 5: Paths from top to right and from bottom to right exit of A rows of M k from top to bottom correspond to , , , , , and , and the elementin row x and column y of M k is the number of y -squares in the x -path in G ( A k ). Onecan easily check that the matrix multiplication reflects the substitution of paths. TheEquation 3 can then be shown by induction.We note that in the above example, M = , and M = . For the above matrices we obtain M · M =
11 3 4 9 4 97 7 4 11 4 1111 7 6 11 5 1110 5 4 13 4 125 1 2 1 3 18 5 4 7 4 8 , and onecan check (see also Figure 8) that for this matrix the element in row x and column y is the number of y -squares in the x -path in G ( W ).We call the matrix M k in Proposition 1 the path matrix of the labyrinth pattern A k , k = 1 , , . . . , and M ( n ) the the path matrix of the (mixed) labyrinth set W n , for A Figure 7: Paths from bottom to top and from left to right exit of A Figure 8: The set W constructed with the patterns A and A shown in Figure 1, and thepath from the bottom to the right exit of W (in lighter gray) n = 1 , , . . . .For n ≥ W , W ∈ V ( G ( W n )), let p n ( W , W ) be the path in G ( W n ) from W to W . Lemma 4 can be proven, as in the self-similar case [2, Lemma 6] by using a heorem from the book of Kuratowski [10, Theorem 3, par. 47, V, p. 181]. Lemma 4. (Arc Construction) Let a, b ∈ L ∞ , where a = b . For all n ≥ , there are W n ( a ) , W n ( b ) ∈ V ( G ( W n )) such that(a) W ( a ) ⊇ W ( a ) ⊇ . . . ,(b) W ( b ) ⊇ W ( b ) ⊇ . . . ,(c) { a } = T ∞ n =1 W n ( a ) ,(d) { b } = T ∞ n =1 W n ( b ) .(e) The set T ∞ n =1 (cid:16)S W ∈ p n ( W n ( a ) ,W n ( b )) W (cid:17) is an arc between a and b . Let T n ∈ W n be the top exit of W n , for n ≥
1. The top exit of L ∞ is T ∞ n =1 T n .The other exits of L ∞ are defined analogously. We note that Property 2 yields that( x, , ( x, ∈ L ∞ if and only if ( x,
1) is the top exit of L ∞ and ( x,
0) is the bottomexit of L ∞ . For the left and the right exits the analogue statement holds.The proof of the following proposition is analogous to that in the self-similar case[2, Lemma 7], taking into account that in the case of mixed labyrinth fractals theedgelength of a square of level n is m ( n ) . For the definitions of the parametrisation ofa curve and its length we refer, e.g., to the mentioned paper [2]. Proposition 2.
Let n, k ≥ , { W , . . . , W k } be a (shortest) path between the exits W and W k in G ( W n ) , K = W ∩ fr ([0 , × [0 , , K k = W k ∩ fr ([0 , × [0 , , and c bea curve in L n from a point of K to a point of K k . The length of any parametrisationof c is at least ( k − / (2 · m ( n )) . Let n ≥ W ∈ W n , and t be the intersection of L ∞ with the top edge of W . Thenwe call t the top exit of W . Analogously we define the bottom exit , the left exit and the right exit of W . We note that the uniqueness of each of these four exits is provided bythe uniqueness of the four exits of a mixed labyrinth fractal and by the fact that eachsuch set of the form L ∞ ∩ W , where W ∈ W n , is a mixed labyrinth fractal scaled bythe factor m ( n ). We note that we have now defined exits for three different types ofobjects, i.e., for W n (and A k ), for L ∞ , and for squares in W n . Proposition 3.
Let e , e be two exits in L ∞ , and W n ( e ) , W n ( e ) be the exits in G ( W n ) of the same type as e and e , respectively, for some n ≥ . If a is the arcthat connects e and e in L ∞ , p is the path in G ( W n ) from W n ( e ) to W n ( e ) , and W ∈ W n is a -square with respect to p , then W ∩ a is an arc in L ∞ between the topand the bottom exit of W . If W is an other type of square, the corresponding analoguestatement holds.Proof. Analogously to the self-similar case, the statement follows from Lemma 4.By the construction of mixed labyrinth fractals we obtain the following result.
Proposition 4.
Let {A k } ∞ k =1 be a sequence of labyrinth patterns, m k ≥ , and we set m (0) := 1 .(a) Let t , t and b , b be the Cartesian coordinates of the top exit and bottom exit,respectively, in L ∞ , and x k ,t , x k ,t the Cartesian coordinates of the left lower vertexof the square that is the top exit in A k , for all k ≥ . Then t = b = ∞ X k =1 x k ,t m ( k − , t = 1 , b = 0 . (b) Let l , l and r , r be the Cartesian coordinates of the left exit and right exit,respectively, in L ∞ , and x k ,l , x k ,l the Cartesian coordinates of the left lower vertexof the square that is the left exit in A k , for all k ≥ . Then l = r = ∞ X k =1 x k ,l m ( k − , l = 0 , r = 1 . emark. In the self-similar case it was shown [3, Lemma 4] that for all n ≥ e in L ∞ lies in exactly one square W n ( e ) ∈ W n .The following counterexample shows that the above statement does not hold ingeneral in the case of mixed labyrinth fractals. Let, e.g., A and A be as shown inFigure 1 9, and A k = A , for all k ≥
3. One can check (e.g., with Proposition 4) thatthe left exit of L ∞ is the midpoint of the left side of the unit square, i.e., the point(0 , ), and that this exit lies in two squares of W , and in exactly one square of W n , for all n ≥ . Figure 9: Two labyrinth patterns, A (a 4-pattern) and A (a 5-pattern)Figure 10: Three labyrinth patterns, A , A , and A In Figure 10 we show another counterexample. Let, e.g., A , A , and A be asshown in the figure, and A k = A , for all k ≥
4. One can check that the left exit of L ∞ is the point (0 , + ), and that this exit lies in exactly one square of W , in twosquares of W , and in exactly one square of W n , for all n ≥ . For all n ≥
1, and W ∈ W n let L ∞ | W := L ∞ ∩ W and, for e ∈ { t, b, l, r } , let e ( W )denote the top, bottom, left, or right exit of W , respectively. Proposition 5.
With the above notations, we have: L ∞ = ∪ W ∈W n L ∞ | W . For all n ≥ and W , W ∈ W n there exists a translation φ , such that φ ( L ∞ | W ) = L ∞ | W . Moreover, then we also have φ ( e ( W )) = e ( W ) , for e ∈ { t, b, l, r } .Proof. The above statement follows from the construction of labyrinth sets and thedefinition of a mixed labyrinth fractal.
Proposition 6. If a is an arc between the top and the bottom exit in L ∞ then lim inf n →∞ log( ( n )) P nk =1 log( m k ) = dim B ( a ) ≤ dim B ( a ) = lim sup n →∞ log( ( n )) P nk =1 log( m k ) . For the other pairs of exits, the analogue statement holds. roof. The above inequalities follow, e.g., by using an alternative definiton of the boxcounting dimension [6, Definition 1.3.] and making use of the property that one canuse, instead of δ →
0, appropriate sequences ( δ k ) k ≥ for the computation of the boxcounting dimension (see the cited reference). For δ k = m ( k ) , we have δ k ≤ δ k − , andone immediately gets the above formulæ. An m × m -labyrinth pattern A is called horizontally blocked if the row (of squares)from the left to the right exit contains at least one black square. It is called verticallyblocked if the column (of squares) from the top to the bottom exit contains at least oneblack square. Anogously we define for any n ≥ n . We note that there are no horizontally or vertically blocked m × m -labyrinth patterns for m <
4. As an example, the labyrinth patterns shown inFigure 1 and 13 are horizontally and vertically blocked, while those in Figure 9 are notblocked.
Conjecture 1.
Let {A k } ∞ k =1 be a sequence of (both horizontally and vertically) blockedlabyrinth patterns, m k ≥ . For any two points in the limit set L ∞ the length of thearc a ⊂ L ∞ that connects them is infinite and the set of all points, where no tangentto a exists, is dense in a . In order to define rectangular mixed labyrinth sets and fractals we introduce the follow-ing function P R , in analogy with P Q mentioned in Section 2. Let x, y, p, q ∈ [0 ,
1] suchthat R = [ x, x + p ] × [ y, y + q ] ⊆ [0 , × [0 , z x , z y ) ∈ [0 , × [0 , P R ( z x , z y ) = ( pz x + x, qz y + y ) . For any integers m, s ≥ S i,j ; m,s = { ( x, y ) | im ≤ x ≤ i +1 m and js ≤ y ≤ j +1 s } and S m × r = { S i,j ; m,s | ≤ i ≤ m − ≤ j ≤ s − } .We call any nonempty A ⊆ S m × s a rectangular m × s - pattern , and ( m, s ) the widthsvector of A . A rectangular m × s -pattern is called a rectangular m × s -labyrinth pattern,if it satisfies Properties 1, 2, and 3, in Section 3. Let {A k } ∞ k =1 be a sequence of non-empty rectangular patterns and { ( m k , s k ) } ∞ k =1 be the sequence of the correspondingwidths vectors, i.e., for all k ≥ A k ⊆ S m k × s k . We denote m ( n ) = Q nk =1 m k ,and s ( n ) = Q nk =1 s k , for all n ≥
1. In this case the set W n ⊂ S m ( n ) × s ( n ) of whiterectangles of level n , is defined as in Equation (1), Section 2, and the set B n of blackrectangles of level n , correspondingly.The results shown in Sections 4 and 5 also hold (with analogous proofs) in the caseof rectangular mixed labyrinth sets and fractals . Here, in the proof of Theorem 1 wetake into account the fact that the diameter of any rectangle in W n is strictly less than √ n . The results in Proposition 4 and Proposition 6 hold with small modifications,which we skip here.We remark that rectangular mixed labyrinth fractals are related to the generalSierpi´nski carpets studied by McMullen [15]. We mention that, on the one hand,McMullen uses the same pattern at each step of the construction, and, on the otherhand, no restrictions are imposed on the pattern (except that it is not trivial). In the setting of self-similar labyrinth fractals [2, 3], i.e., if A k = A , for all k ≥ , letus weaken the conditions that define labyrinth patterns: instead of asking Property 1and Property 2 to be satisfied, we use here the following two properties. Property 4 (“Wild Property 1”) . G ( A ) is a connected graph. Property 5 (“Wild Property 2”) . A has at least one vertical exit exit pairpair, andat least one horizontal exit pair. We call a pattern that satisfies Property 4, Property 5, and Property 3 a wildlabyrinth pattern . We note that every labyrinth pattern is also a wild labyrinth pattern.Figure 11 shows on the left a wild labyrinth pattern with two left and two right exits,and on the right a wild labyrinth patternd whose graph contains cycles.We call the self-similar limit set generated by a wild labyrinth pattern a wildlabyrinth fractal . We call mixed wild labyrinth fractal a mixed labyrinth fractal gener-ated by a sequence of wild labyrinth patterns.
Proposition 7.
Every (mixed) wild labyrinth fractal is connected.
One way to prove the above proposition is by using recent results on connectedgeneralised Sierpi´nski carpets [4].We call a wild labyrinth pattern horizontally blocked if each of its horizontal exitpairs (i.e., a left exit together with the corresponding right exit) is blocked, i.e. thereis at least one black square in the row that contains it.
Conjecture 2. If A is a horizontally and vertically blocked wild labyrinth pattern, thenthe self-similar wild labyrinth fractal L ∞ generated by A has the property that for anytwo distinct points x, y ∈ L ∞ the length of an arc that connects them in L ∞ is infinite. In the case of wild labyrinth fractals Lemma 4 does not hold in general. Moreover, inthis case the path between two exits in G ( W n ) is in general not unique. The pattern inFigure 12 generates a (self-similar) wild labyrinth fractal with the following property:the squares of level 2 that lie on the shortest path in G ( W ) from the top to thebottom exit in G ( W ) are not contained in the white squares which correspond to theshortest path between the top and bottom exit in G ( W ), as it can be easily checked, bycomparing the paths in G ( W ) and then in G ( W ), that connect the top and bottom exitin each case. This gives an example for the case when the methods used in Proposition G ( W ) does not liewhithin the shortest path from the top exit to the bottom exit in G ( W )Figure 13: A (6 × A and its complementary pattern A Remark.
Conjecture 2 can be also formulated for mixed wild labyrinth fractals,with some restrictions. If A ∈ S m is an m -pattern, we call the complementary pattern of A the patterndenoted by A that is defined by A = S m \ A , i.e., the pattern obtained by recolouringthe squares in A k such that the black squares are recoloured in white squares and thewhite squares are recoloured in black, see, e.g., Figure 13. With methods and results ofa recent paper on totally disconnected Sierpi´nski carpets [5] one can prove the followingresult. Proposition 8. If ( A k ) k ≥ is a sequence of labyrinth patterns, then the limit setgenerated by the sequence of the corresponding complementary patterns ( ¯ A k ) k ≥ is atotally disconnected generalised Sierpi´nski carpet. Remark
We note that the case of generalised Sierpi´nski carpets generated by asequence of complementary patterns of labyrinth patterns is an example of a situa- ion when the main result about totally disconnected Sierpi´nski carpets [5, Theorem1] holds under conditions, that differ partially from the sufficient conditions of totaldisconnectedness given in the cited paper. Acknowledgement.
The authors thank the referee for his/her useful comments.
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