aa r X i v : . [ m a t h . M G ] M a r FINITENESS PROPERTIES FOR SELF-SIMILAR SETS.
ANDREI TETENOV
Abstract.
We consider self-similar sets possessing finite intersec-tion property and analyze topological structure nearby their localcut points. . Primary: 28A80.
Keywords and phrases. self-similar set, dendrite, OSC, finite intersec-tion property, intersection graph
Introduction
Let S = { S , ..., S m } be a system of injective contraction maps ina complete metric space X . A non-empty compact set K satisfyingthe equation K = m S i =1 S i ( K ) is called the attractor of S and the sets K i = S i ( K ) are called the pieces of the set K . We say the system S isa system with finite intersection property (or call it a FIP system ) iffor any non-equal i, j , the intersection of pieces K i ∩ K j is finite.For a long time it seemed highly likely that finite intersection prop-erty could imply Open Set Condition, at least in case of one-pointintersections. C.Bandt and H.Rao proved in [3] that FIP systems ofsimilarities in R with connected attractor satisfy OSC. From the otherside, the author of this paper proved in [6] that in R this does not holdand in [7] it was also shown that there are one-point intersection sys-tems with totally disconnected attractor in R which violate OSC.Nevertheless, the attractor K of a FIP system S of injective contrac-tions has remarkable topological properties caused by the fact that thebase of its topology consists of the sets ˙ S nk =1 K i k whose boundary isfinite.In this paper we try to collect some structural outcomes of this prop-erty. We consider the relations between the following finiteness propertiesof self-similar sets:
P1.
We say the system S has finite intersection property if for any i, j ∈ I , the set K i ∩ K j is finite. P2.
Denote the boundary of a piece K i by ∂K i = K i ∩ ( K \ K i ) . Wesay the attractor K is ∂ − finite , if there is such M that for any i ∈ I ∗ , ∂K j ≤ M . P3.
The set ∂K = S i ∈ I ∗ S − i ( ∂K i ) is called the self-similar boundaryof the attractor K . We say K has finite self-similar boundary if the set ∂K is finite. P4.
The system S is called post-critically finite if the set P ( S ) = π − ( ∂K ) ⊂ I ∞ is finite. WSP.
The system S satisfies weak separation condition if Id is anisolated point in the set F = { S − j S i , i , j ∈ I ∗ } .Note first that P ⇒ P ⇒ P ⇒ P Some inverse implications also may hold under some additional as-sumptions.We will try to establish how these backward implications depend onWeak Separation Property and connectedness or simply-connectednessof the attractor and how these conditions affect the local structure ofself-similar sets.In Section 1 we consider the intersection graph Γ( S ) of a FIP sys-tem as a bipartite graph whose "white" vertices are the pieces K i and"black" vertices are the intersection points p ∈ K i ∩ K j and prove thefollowing Intersection Graph Criterion for self-similar dendrites: Theorem 1.7.
Let S be a system of injective contraction maps in acomplete metric space X , such that the intersection graph Γ ( S ) is atree. Then the attractor K of the system S is a dendrite. In Section 2 we prove that if FIP system S of contracting similaritiessatisfies WSP, then (a) the conditions P1—P3 are equivalent; (b) thenumber of addresses π − ( x ) of any point x ∈ K has uniform finitebound; and, moreover (c) the degrees at local cut points of K also INITENESS PROPERTIES FOR SELF-SIMILAR SETS. 3 have uniform finite bound. Further, we prove the following Theoremon stability of local structure for FIP systems:
Theorem 2.5.
If a FIP-system S = { S , ..., S m } of contracting similar-ities in R d satisfies WSP, and x ∈ K then there is a family J ≺ π − ( x ) such that:(i) for any non-equal j , k ∈ J , K j ∩ K k = { x } ;(ii) If K is connected, α ∈ π − ( x ) , j ∈ J and j ❁ k ❁ α then thefamilies Q j , Q k connected components of the sets K j \{ x } and K k \{ x } have the same number of components, and for any component Q ∈ Q j , Q ∩ K k ∈ Q k . and the ramification order estimate for self-similar dendrites: Theorem 2.8.
If a system S = { S , ..., S m } of contracting similaritiesin R n satisfies WSP, and its attractor K is a dendrite. Then for any x ∈ K , Ord( x, K ) ≤ M / . In Section 3 we discuss FIP systems of similarities with connectedattractor on the plane. By [3], such systems satisfy OSC. For a cyclicvertex x of the attractor K of such system we prove the existence ofsemi-invariant Jordan arcs in each of the components of K \{ x } anddefine a parameter of the cyclic vertex. then we have the followingParameter matching theorem for points with multiple preperiodic ad-dresses: Theorem 3.6.
Let S = { S , ..., S m } be a FIP-system of contractingsimilarities in R with a connected attractor K . If x = fix( S j ) , x = fix( S j ) are cyclic vertices, y = S i ( x ) = S i ( x ) , and S i ( K ) ∩ S i ( K ) = { y } . Then λ x = λ x The preprint will be further elaborated, filling possible gaps andadding missing references so the readers are welcome to point at all thearising doubts.
Some notation and introductory information.
For a system S = { S , . . . , S m } of contraction maps we denote by I := { , , ..., m } the set of indices, I ∗ := ∞ S n =1 I n is the set of allmultiindices j = j j ...j n , S j = S j S j ...S j n and r i = Lip S i . Also, r min = min { r , ..., r m } is the smallest contraction ratio for S , . . . , S m .For a bounded set A we denote its diameter by | A | .The set of all infinite sequences I ∞ = { α = α α . . . , α i ∈ I } isthe index space ; and π : I ∞ → K is the index map , which sends thesequence α to the point x = ∞ T n =1 K α ...α n and α is called the address of ANDREI TETENOV x . If x ∈ K , π − ( x ) is the set of all addresses of a point x .If for any i, j ∈ I , K i ∩ K j is finite, we set m = max { K i ∩ K j ) , i, j ∈ I, i = j } .The system S is said to satisfy the open set condition (OSC), if thereexists an open set O such that S i ( O ) ⊂ O and S i ( O ) ∩ S j ( O ) = ∅ forall distinct i, j ∈ I = { , . . . , m } .We denote by F = { S j , j ∈ I ∗ } the semigroup, generated by S . Incase when S is a system of similarities in R n then F = F − ◦ F , ora set of all compositions S − j S i , i , j ∈ I ∗ , is the associated family ofsimilarities . The system S has the weak separation property (WSP) iff Id / ∈ F \ Id . Definition of M a . According to Zerner’s Theorem [8], if the system S of contraction similarities satisfies the Weak Separation Condition,then for any a > there is a positive number(1) M a = sup U ⊂ R n { S j : a | U | r min < | K j | ≤ a | U | & K j ∩ U = ∅ } If the system S possesses the finite intersection property it has no exactoverlaps, and S i = S j iff i = j , then (1) becomes(2) M a = sup U ⊂ R n { j : a | U | r min < | K j | ≤ a | U | & K j ∩ U = ∅ } INITENESS PROPERTIES FOR SELF-SIMILAR SETS. 5 FIP set systems and their intersection graphs.
We start with a definition of a FIP set system.
Definition 1.1.
A system of compact subsets A = { A i , i ∈ I } in atopological space X possesses finite (resp. one-point) intersection prop-erty, if for any i = j ∈ I , the intersection P ij = A i ∩ A j is finite (resp.contains at most 1 point). We say then that A is a FIP (resp. FIP1) set system .Let A = S A i ∈ I A i and P = S i = j P ij . We also denote P i = S j = i P ij . Thenin the induced topology on the subspace A ⊂ X , ˙ A i = A i \ P i and the set P i is the boundary of the set A i in A .For each FIP set system we define its intersection graph: Definition 1.2.
The intersection graph Γ( A ) of a FIP set system A isa bipartite graph ( P, A ; E ) , for which { A i , p } ∈ E iff p ∈ A i . We call A i ∈ A white vertices and p ∈ P – black vertices . The set N ( A i ) of the neighbors of a white vertex A i is P i whereas for a blackvertex p , N ( p ) = { A i : p ∈ A i } . Since p is the intersection point of atleast two of the sets A i , deg( p ) ≥ .The main application of these definitions is to the systems of con-traction maps and their attractors. Let S = { S , ..., S m } be a systemof contraction maps on a complete metric space X and let K be itsattractor. Let A ( S ) = { K , ..., K m } and A n ( S ) = { K i : i ∈ I n } . Definition 1.3. S is called FIP (resp. FIP1) system of contractions ifthe system A ( S ) is a FIP (resp. FIP1) set system. The n-th intersec-tion graph Γ n ( S ) of the system S is the intersection graph of the system A n S If the intersection graph Γ( A ) of a FIP set system is a tree, then A is a FIP1-system. Moreover, we can refine such system A to get a newFIP1 system A ′ with a tree graph: Proposition 1.4.
Let A = { A i , i ∈ I } , B = { B i , i ∈ J } , and A k ∈ A . Let the intersection graphs Γ( A ) , Γ( B ) be trees with parts ( A , P A ) , ( B , P B ) . Let f : S B B j → A k be a homeomorphism and B f = { f ( B i ) , i ∈ J } . Then the intersection graph of the set system A ′ = ( A \{ A k } ) ∪ B f is a tree. ANDREI TETENOV
Proof:
Consider the graph Γ( A \{ A k } ) . It is a disjoint union offinite number of connected components Q p each being a non-degeneratetree containing one of the vertices p ∈ P k . Since f : B → A k is ahomeomorphism, there are exactly n k = P k points p ′ = f − ( p ) , p ∈ P k , contained in S i ∈ J B i .These points are of two types. First are the points p ′ ∈ f − ( P k ) ∩ P B which correspond to black vertices of Γ( B ) . The second type are thepoints p ′ ∈ f − ( P k ) \ P B . Each of these points is contained in someunique B i ∈ B . We construct an extension ˜Γ of the graph Γ B , addingthe points of second type to the set of black vertices P B and respectiveedges ( B i , p ′ ) to the edge set E B . Such extension does not producecycles and the graph ˜Γ is again a tree with two parts.Now we identify each of the points f − ( p i ) , p i ∈ P k , with the point p i ∈ Q i . Thus we paste the tree Q i to the graph ˜Γ .The resulting graph Γ( A ′ ) is a tree with two parts – A ′ = A k ∪ B and P A ′ = P A k ∪ ( P B \ f − ( P k )) . The degree of each vertex p ∈ P A ′ is ≥ . (cid:4) The refinement operation can be applied to a FIP system S of con-tractions and its intersection graphs Γ n ( S ) : Corollary 1.5.
Let S = { S i , i ∈ I } be a system of injective contractionmaps, such that the intersection graph Γ ( S ) is a tree. Then for any n ∈ N , the intersection graph Γ n ( S ) is a tree. Proof:
Suppose that the intersection graph Γ n − of the system { K i , i ∈ I n − } is a tree. Applying Proposition1.4 step by step to eachof similarities S i ...i n − : S i ∈ I K i → K i ...i n − , we obtain finally that theintersection graph Γ n is also a tree. (cid:4) If the intersection graph of a FIP1 set system A is a tree, then asimple loop in A cannot cross any of the boundaries between the sets A i : Proposition 1.6.
Let A be a FIP1-system of sets and Γ( A ) be a tree.Let γ be a simple closed curve in A . Then there is unique A k ∈ A suchthat γ ⊂ A k . Proof:
Without loss of generality we suppose that all sets A i areconnected. Let p be some point in P and let Q i , Q j be the componentsof A \{ p } . Suppose γ is a closed curve containing some a ∈ Q i and b ∈ Q j . Since each path connecting a and b passes through p , the point p is a multiple point of γ . Therefore if γ ∩ ˙ A i , then γ ∈ A i . (cid:4) INITENESS PROPERTIES FOR SELF-SIMILAR SETS. 7
Theorem 1.7.
Let S be a system of injective contraction maps in acomplete metric space X , such that the intersection graph Γ ( S ) is atree. Then the attractor K of the system S is a dendrite. Proof:
Let γ ∈ K be a simple closed curve. Since for any n ∈ N thegraph Γ n is a tree, there is unique j ∈ I n such that γ ∈ K j . Therefore | γ | = 0 . (cid:4) Stable neighbourhoods and ramification points.
Proposition 2.1.
Let S = { S , ..., S m } be a FIP-system of injectivecontractions in a complete metric space X , then:(i) for any i , j ∈ I ∗ , K i ∩ K j ) ≤ m ;(ii) the sets ∂K i are finite. Proof: (i) Suppose k = i ∧ j = i ...i l . Then (i) follows from K i ∩ K j ⊂ S k ( K i l +1 ∩ K j l +1 ) (ii) Since ∂K i = S j ∈ I n \{ i } ( K i ∩ K j ) , the set ∂K i is finite. (cid:4) Definition 2.2.
Let P ∗ (resp. P n ) be the set of all intersection pointsof the pieces (resp. pieces of order n ) in K : P ∗ = [ i ∈ I ∗ ∂K i ; P n = [ i ∈ I n ∂K i Proposition 2.3.
If a FIP-system S = { S , ..., S m } of contracting sim-ilarities in R d satisfies WSP then:(i) it is ∂ − finite;(ii) for any point x ∈ K , π − ( x ) < M ;(iii) if K is connected, then for any connected neighborhood W of apoint x ∈ K the number of connected components of W \{ x } is less orequal to M / . Proof: (i) For any x ∈ ∂K i there is such j ∈ I ∗ such that x ∈ K j ,and | K i | r min < | K j | ≤ | K i | The number of such j is at most M , for each j , the number K i ∩ K j ) ≤ m , therefore ∂K i ≤ M · m .(ii) If x / ∈ P ∗ , then π − ( x ) = 1 . Let x ∈ P ∗ . Take some ρ > andconsider the set C ρ = { j ∈ I ∗ : ρr min < | K j | ≤ ρ , K j ∩ B ( x, ρ/ = ∅ } ANDREI TETENOV
By [8, Theorem 1], C ρ ≤ M . Since for any ρ > and any α ∈ π − ( x ) there is j ❁ α such that j ∈ C ρ , π − ( x ) ≤ sup { C ρ , ρ > } ≤ M Therefore Q ≤ M (iii) Let Q = { Q , ..., Q n } be some finite set of connected compo-nents of W \{ x } . Take such ρ , that for any Q k ∈ Q , Q k \ B ( x, ρ ) = ∅ .Each component Q k ∈ Q contains such y k , that d ( x, y k ) = 3 ρ/ . Let j k be such that y k ∈ K j k , and ρr min / < | K j k | ≤ ρ/ . Since x / ∈ K j k , K j k ⊂ Q k . Therefore all j k are incomparable and the number of such j is no greater than M / . (cid:4) Definition 2.4.
Let
J, J ′ ⊂ I ∗ be two sets of multiindices, and let A ⊂ I ∞ be a set of addresses. We write J ≺ J ′ (resp. J ≺ A ), if there is a bi-jection ϕ : J → J ′ (resp. ϕ : J → A ), such that for any j ∈ J , j ❁ ϕ ( j ) . The family V ( x ) := { V J = S j ∈ J K j : J ≺ π − ( x ) } is a neighborhoodbasis in K for the point x . If K is connected, V ( x ) consists entirely ofconnected sets. In the conditions of 2.3, this family has some remark-able properties: Theorem 2.5.
If a FIP-system S = { S , ..., S m } of contracting similar-ities in R d satisfies WSP, and x ∈ K then there is a family J ≺ π − ( x ) such that:(i) for any non-equal j , k ∈ J , K j ∩ K k = { x } ;(ii) If K is connected, α ∈ π − ( x ) , j ∈ J and j ❁ k ❁ α then thefamilies Q j , Q k connected components of the sets K j \{ x } and K k \{ x } have the same number of components, and for any component Q ∈ Q j , Q ∩ K k ∈ Q k . Proof:
Let π − ( x ) = { α , ..., α n } . There is such J ≺ π − ( x ) , that allits elements j ∈ J are incomparable. Therefore the set P J = S j ∈ J ∂K j isfinite. Take such ρ > that B ( x, ρ ) ∩ P J = { x } . If J ′ ≻ J is such that J ≺ J ′ ≺ π − ( x ) and for any j ∈ J ′ , diam( K j ) < ρ , then P J ′ = { x } .This proves (i).Take α ∈ π − ( x ) and let j k ❁ α be the initial substring of length k in α .The sequence { p k := Q j k } is non-decreasing and has the upperbound by the Proposition 2.3, there is k such that if k ≥ k , p k = p k . INITENESS PROPERTIES FOR SELF-SIMILAR SETS. 9
For any l > k ≥ k , Q ∈ Q j k implies Q ∩ K j l ∈ Q j l . Choosing respective j ❁ α for each α ∈ π − ( x ) we get the desired J ≺ π − ( x ) . (cid:4) Definition 2.6.
The set S j ∈ J Q j defined by Proposition2.5(ii), is calledthe stable set of components for the point x . Proposition 2.7.
If a FIP-system S = { S , ..., S m } of contracting sim-ilarities in R d satisfies WSP, then for any i ∈ I ∗ there is a finite subset J ⊂ I ∗ such that for any j ∈ J , K i ∩ K j ) = 1 , and the set K i ∪ S j ∈ J K j is a neighborhood of K i in K , and the sets K j \ ∂K i are disjoint. Proof:
Let A = π − ( ∂K i ) . Since A is finite, we can take some set ofincomparable multiindices J ≺ A . Let ρ = 1 / { d ( x, y ) , x, y ∈ P J } .Take such J ′ ≺ A , that J ≺ J ′ and for any j ∈ J ′ , | K j | < ρ . Thenfor any j ∈ J ′ , the intersection K j ∩ P J is a singleton and therefore iscontained in ∂K i . (cid:4) If S satisfies the assumptions of the Proposition 2.5 and its attractor K is a dendrite, then by Proposition 2.5, for any x ∈ K , Ord( x, K ) ≤ M / .Surprisingly, to prove this last statement, finite intersection propertyis not required: Theorem 2.8.
If a system S = { S , ..., S m } of contracting similaritiesin R n satisfies WSP, and its attractor K is a dendrite. Then for any x ∈ K , Ord( x, K ) ≤ M / . Proof:
Let Q , ..., Q n be some finite set of connected componentsof K \{ x } . Let ρ < min ≤ k ≤ n diam( Q k ) . For each ≤ k ≤ n take some z k ∈ ∂B ( x, ρ ) ∩ Q k . Take such j k ∈ I ∗ that z k ∈ K j k and diam( K j k ) < ρ .Since K is a dendrite and x / ∈ K j k , the sets K j k are disjoint. Thereforeby [8, Theorem 1], n ≤ M / for any x ∈ K . (cid:4) Remark.
There are such systems S , satisfying OSC that their at-tractors are dendrites and for some j ∈ I ∗ the set Q j = { Q i } of thecomponents K \ K j is infinite, and the union of all Q i ∈ Q j is dense in K . 3. Preperiodic points and parameter matching. If S is a FIP system of contracting similarities on a plane whoseattractor K is connected, then it follows from [3], that it satisfies OpenSet Condition, therefore all the statements from previous sections holdunder much simpler assumptions. We begin with the definition of a semi-invariant arc:
Definition 3.1.
Let S be a contracting map on a metric space X and x = fix( S ) . A Jordan arc γ ⊂ K with endpoints x, y is called a semi-invariant arc for S , if for some k ∈ N , S k ( γ ) ⊂ γ . Proposition 3.2.
If a FIP-system S = { S , ..., S m } of contracting sim-ilarities in R d with a connected attractor K satisfies WSP, and for some i = i ...i n , x = fix S i , then:(i) π − ( x ) = { i ...i n } ;(ii) The set Q of connected components of the set K \{ x } is a stable setof components for the point x ;(iii) for any component Q ∈ Q there is a semiinvariant Jordan arc γ ⊂ Q for S i with endpoints x and y , where y ∈ S − i ( ∂K i ) . Proof: (i) If α ∈ π − ( x ) and α = i ...i n , then for any k ∈ N , i k α ∈ π − ( x ) , therefore the set π − ( x ) is infinite.(ii) Let Q = { Q k , k = 1 , ..., s } be a stable set of components for thepoint x . Then for some l the family Q ′ = { Q k ∩ S l i ( K ) , i = 1 , ..., s } is alsoa stable set of components for x . Then, since K \{ x } = s S k =1 ( S − l i ( Q k ∩ K )) , the family { S − l i ( Q k ∩ K ) } is also the stable stable set of compo-nents for x .(iii) Let Q be a component of K \{ x } and let D = S − i ( ∂K i ∩ Q ) . Thenfor any l ∈ N , it follows from (ii) that ∂S l i ( K ) ∩ Q = S l − i ( D ) .We define a map ϕ : D → D the following way:Since Q is arcwise connected, for any z ∈ D there is a Jordan arc δ ∈ Q with endpoints z and x . Let δ ′ ( z ) be the closure of maximal subarc of δ ∩ ( Q \ S i ( Q )) containing z and put ϕ ( z ) = S − i ( z ′ ) .There is such n ≤ D and y ∈ D that ϕ n ( y ) = y . For any non-negative integer l < n , δ ′ ( ϕ l ( y )) is a Jordan arc in Q \ S i ( Q ) connecting ϕ l ( y ) and S i ( ϕ l +1 ( y )) .Therefore γ ′ = n S l =0 S l i ( δ ′ ( ϕ l ( y )) is a Jordan arc connecting y and S n i ( y ) which lies in Q \ S n i ( Q ) and γ = ∞ S k =0 S kn i ( γ ′ ) ∪ { x } be the desired semi-invariant arc in Q with endpoints x and y . (cid:4) Remark.
Since the proof of the Proposition 3.2 is purely combina-torial and is based on FIP and on the assumptions that both π − ( x ) and the set of components of K \{ x } are finite, this Proposition can INITENESS PROPERTIES FOR SELF-SIMILAR SETS. 11 be extended to general case of FIP systems of contractive injections inmetric spaces.
Definition 3.3.
Let S = { S , ..., S m } be a system of contracting simi-larities in R and K be its attractor. A point x ∈ K is called a cyclicvertex , if for some i ∈ I ∗ , x = fix( S i ) and for some unbounded compo-nent A of the set R \ K , x ∈ ¯ A . Parameter of a semiinvariant arc.
Let γ be a Jordan arc C \{ a } with endpoints z , z . We denote by ∆(arg( z − a )) | γ the increment of arg( z − a ) along the arc γ as z travels from z to z .Let γ = γ xy be a semiinvariant arc for the similarity S , x = fix( S ) ,and S k ( γ ) ⊂ γ . Denote by γ ′ the subarc in γ with endpoints y and S k ( y ) . The parameter λ γ of the arc γ is the number λ γ = ∆(arg( z − x )) | γ ′ k log Lip S Proposition 3.4.
Let S = { S , ..., S m } be a FIP-system of contractingsimilarities in R satisfying WSP with a connected attractor K , and x = fix( S i ) be a cyclic vertex. If γ , γ are semiinvariant arcs for S i ,then λ γ = λ γ =: λ x . Proof:
Consider R as a complex plane and let ϕ ( z ) = x + z bea double cover of the set R \{ x } by a complex plane C . Let γ be aJordan arc in the unbounded component A with endpoints at x and ∞ . Let U , U be the components of ϕ − ( R \ γ . Let γ ′ i be ϕ − ( γ i ) ∩ U i .Since γ i ∩ A = ∅ , γ ′ and γ ′ are non-intersecting periodic arcs in C ,so by V.V.Aseev’s Lemma about disjoint periodic arcs [1, Lemma 3.1], λ γ = λ γ . Since the parameter λ γ i is the same for all semiinvariantarcs γ i , we call it a parameter λ x of the cyclic vertex x . (cid:4) The last Proposition allows us to define the parameter of a cyclicvertex.
Definition 3.5.
The number λ x is called the parameter of the cyclicvertex x ∈ K . Thus we arrive to parameter matching theorem for FIP systems ofsimilarities on the plane:
Theorem 3.6.
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