On the connected components of fractal cubes
OOn the connected components of fractal cubes.
Dmitry Drozdov, Andrei TetenovGorno-Altaisk State UniversityFebruary 10, 2020
Abstract
We show that a fractal cube F in R may have an uncountable set Q of connected components K α neither of which is contained in anyplane, whereas the set Q is a totally disconnected self-similar subset ofthe hyperspace C ( R ) , isomorphic to a Cantor set. . Primary: 28A80. Keywords and phrases. fractal square, fractal cube, self-similar set, hyper-space
Take n ≥ and some subset D⊂{ , , . . . , n − } k , ≤ D < n k whichwe call a digit set. Then there is unique non-empty compact set F ⊂ R k satisfying F = F + D n , which we call fractal k -cube ( or fractal square if k = 2 ).Let H = F + Z k and H c = R k \ H . We also denote I = [0 , n , T ( A ) := A + D n and F = T ( I ) .For the case k = 2 when F is a fractal square, K.S.Lau, J.J.Luo andR.Hui [3] proved the following Theorem 1.
Let F be a fractal square as in (1.1). Then F satisfies either(i) H c has a bounded component, which is also equivalent to: F contains anon-trivial component that is not a line segment; or(ii) H c has an unbounded component, then F is either totally disconnectedor all non-trivial components of F are parallel line segments. a r X i v : . [ m a t h . M G ] F e b e show that in the case of fractal cubes in R , the situation is completelydifferent. In our short note, we prove the following Theorem 2.
There is a fractal cube F ⊂ R such that H c is connected and H is an uncountable union of unbounded components, each being invariantwith respect to Z -translations. We construct F as a fractal cube with n = 5 and a digit set D⊂{ , , ..., } which is a disjoint union of its subsets D and D . These digit sets definedisjoint connected fractal cubes K and K .The Hutchinson operators T i ( A ) := D i + A for D i can be considered ascontraction maps (cid:101) T i : C ( R ) → C ( R ) whose Lipschitz constant is / . Forany finite ¯ α = α . . . α k ∈ { , } k , k ∈ N , we define T ¯ α = T α ◦ . . . ◦ T α k and F ¯ α = T ¯ α ◦ I. Let K ¯ α be the non-empty compact set satisfying K ¯ α = T ¯ α ( K ¯ α ) . For any infinite string α = α α .... ∈ { , } ∞ we write | ¯ α | = k and define K α = ∞ (cid:92) k =1 T α ...α k ( I ) In our case, each K α is a connected component of F and F = (cid:91) α ∈{ , } ∞ K α . Since all components K α of F are not line segments, the equivalence (i)of Theorem1 does not hold. From the other side, the set H c is connectedand unbounded, but all components of F are not line segments, so (ii) doesnot work too.Thus, the set Q of connected components K α of F is uncountable and itis totally disconnected if we consider it as a subset of the hyperspace C ( R ) . Theorem 3.
The set Q ⊂ C ( R ) of connected components K α of F is a self-similar set generated by two contractions ˜ T and ˜ T of the hyperspace C ( R ) .There is a H¨older homeomorphism ϕ : Q → C / of the set Q to the middle-third Cantor set C / which induces the isomorphism of self-similar struc-tures on these sets. Let α = α α . . . ∈ { , } ∞ . For any m ∈ N let λ m be the density ofzeros in the word α . . . α m . Put ¯ λ α = lim sup m →∞ λ m , λ α = lim inf m →∞ λ m and if2 α = ¯ λ α , we write λ α = lim m →∞ λ m . If α is preperiodic with period length p ,the λ α is equal to the density of zeros in any period of α .We prove the following estimates for the dimension of the components K α : Theorem 4.
For any α ∈ { , } ∞ dim B ( K α ) = ¯ λ α log
13 + (1 − ¯ λ α ) log , ¯dim B ( K α ) = λ α log
13 + (1 − λ α ) log . If α is preperiodic, then dim H ( K α ) = dim B ( K α ) = λ α log
13 + (1 − λ α ) log . F Cross(left) and Frame (right).
Let D and D be the sets of the coordinates defining the cubes ¯ x + I forming subsets "Cross" and "Frame" of a cube · I , so that D + I is a"Cross" and D + I is a "Frame".Let T ( A ) := D + A and T ( A ) := D + A be Hutchinson operatorsdefined by D and D . Let D = D (cid:83) D and T ( A ) := D + A . Let K and K be the fractal cubes corresponding to D and D .3 ractal cubes K (left) and K (right). Applying the approach, developed by L.Cristea and B.Steinsky in [1], wesee that since the set D + I has exactly one pair of entrance and exit pointson each pair of opposite faces of the cube I and has empty intersection withits edges and the intersection graph of D + I is a tree, the fractal cube K is a dendrite, all of whose ramification points have order 6.The fractal cube K is a × × version of Menger sponge. The set T ( I ) (left) and the fractal cube F (right). We see from the construction that D = 13 , D = 44 . Moreover, theHausdorff distance between D + I and D + I is √ , while the minimaldistance d ( D + I, D + I ) between the points of those sets is equal to 1.4 he components K and K Lemma 5.
For any ¯ α, the sets F ¯ α and K ¯ α are connected. Proof:
For i = 0 , , the intersections of F i with any two opposite facesof the cube I are congruent with respect to the translation moving one faceto the other.For any two cubes a + I , b + I ⊂ F i with side / in F i there is a pathfrom one cube to the other which intersects transversely the faces of pairs ofadjacent cubes and does not intersect any of their edges.It follows then, that for any i, j the sets T i ( F j ) are connected and possessthe same opposite face property.Proceeding by induction, we get that the sets T α ...α k ( F i ) are connectedand inherit the same property.Since K ¯ α is the intersection of a nested family of compact connected sets ( T ¯ α ) m ( I ) , K ¯ α is connected too. (cid:4) Lemma 6. If ¯ α, ¯ β ∈ { , } k and ¯ α (cid:54) = ¯ β , then F ¯ α (cid:84) F ¯ β = ∅ , d H ( F ¯ α , F ¯ β ) < √ | ¯ α ∧ ¯ β | , and d ( F ¯ α , F ¯ β ) ≥ | ¯ α ∧ ¯ β | +1 Proof: If α (cid:54) = β then F ¯ α ⊂ F α , F ¯ β ⊂ F β , therefore d H ( F ¯ α , F ¯ β ) ≤ d H ( F ¯ α , F α ) + d H ( F α , F β ) + d H ( F ¯ β , F β ) ≤ √
225 + 2 √
25 + 2 √ < √ (1)and d ( F ¯ α , F ¯ β ) ≥ d ( F α , F β ) > / . Notice, that d H ( F σα , F σβ ) ≤ d H ( F α , F β )5 | σ | to get the desired statement. (cid:4) ¯ α = α α α . . . ∈ { , } ∞ . We define K ¯ α = ∞ (cid:84) k =1 F α ...α k . Corollary 7.
For any ¯ α, ¯ β ∈ { , } ∞ | ¯ α ∧ ¯ β | +1 ≤ d H ( F ¯ α , F ¯ β ) < √ | ¯ α ∧ ¯ β | +1 , and d ( F ¯ α , F ¯ β ) ≥ | ¯ α ∧ ¯ β | +1 (cid:4) Consider ¯ α ∈ { , } k , k ∈ N . The number N δ ( F ¯ α ) of δ -mesh cubes for ( F ¯ α = ( T ¯ α ( I ) and δ = 5 − k is equal to m · k − m , where m = { i ≤ k : α i = 0 } . Thus − log N δ log δ = mk log
13 + (1 − mk ) log .For any α ∈ { , } ∞ , the minimal number of δ -mesh cube containing F α is equal to N δ ( F ¯ α ) if δ = 5 − k and ¯ α = α . . . α k is the k -th initial segmentof α . Taking upper and lower limits as k → ∞ we get desired estimates for dim B ( K α ) . References [1] Cristea L. L., Steinsky B., Curves of infinite length in4
13 + (1 − mk ) log .For any α ∈ { , } ∞ , the minimal number of δ -mesh cube containing F α is equal to N δ ( F ¯ α ) if δ = 5 − k and ¯ α = α . . . α k is the k -th initial segmentof α . Taking upper and lower limits as k → ∞ we get desired estimates for dim B ( K α ) . References [1] Cristea L. L., Steinsky B., Curves of infinite length in4 ×4