Separation properties for infinite iterated function systems
Journal of Inequalities and Applications volume 2014, Article number: 118 (2014)
In this paper, we study the infinite iterated function systems (IFSs) of contractive similitudes with overlaps. We extend the notions of the weak separation condition and the generalized finite type condition for finite IFSs to the infinite case. We show that for an infinite IFS of contractive similitudes the generalized finite type condition implies the weak separation condition.
The separation properties are useful for studying the IFSs. At first, let’s recall the separation property of finite IFSs. Suppose X is a nonempty compact subset of . Let () be a contractive self-map on . We call a finite similar IFS of contractive similitudes on if there exists such that
There exists a nonempty subset K of X such that
We say that satisfies the open set condition (OSC) if there exists a nonempty bounded open set such that
Such a U is called a basic open set for . If, moreover, , the is said to satisfy the strong open set condition (SOSC). It is a classical result (see [1, 4, 5]) that if a similar IFS satisfies the OSC, then it satisfies the SOSC. Fan et al. (see [6–8]) extended the result to finite conformal IFSs.
IFSs that do not satisfy the OSC are said to have overlaps. In this case, it is in general much harder to get acquainted with the structure of the corresponding invariant set K. The weak separation condition (WSC) and the generalized finite type condition are weaker than the OSC but still strong enough to obtain good results (see [9–14]etc.).
However, the circumstances for infinite IFSs are distinct . Szardk and Wedrychowica  showed that for infinite IFSs, the OSC does not imply the SOSC. Moran  also showed that self-similar set generated by a countable system of similitudes may not be s-set even if the open set condition is satisfies. So it is necessary to look for some separation conditions for infinite IFSs that are weaker than the OSC. Moran defined a weak separation property for infinite IFSs . Suppose is an infinite conformal IFS on an open set . Moran  defined the infinite IFS satisfies the finite open set condition if for any integer n, there is a nonempty open set such that for any and for any , . The finite strong open set condition holds if furthermore . It is easy to see that if the IFS satisfies the OSC then it satisfies the finite strong open set condition. Moran uses this separation property to study the Hausdorff dimension of invariant set with respect to the infinite IFS .
Our goal in this paper is to study the infinite iterated function systems (IFSs) of contractive similitudes with overlaps. We define the WSC and the generalized finite type condition for the infinite IFSs. Next, we study the relationship of the two separation conditions. We show that the generalized type condition implies the WSC. Our main result is Theorem 1.1.
Theorem 1.1 Let be an infinite iterated function system of contractive similitude on . If is of generalized finite type condition, then it satisfies the weak separation condition.
This paper is organized as follows. In Section 2, we define the weak separation condition for infinite IFSs and give some examples. In Section 3, we introduce the generalized finite type condition for infinite IFSs and provide examples of IFSs satisfying this condition. Finally, in Section 4, we prove that the generalized finite type condition implies the weak separation condition (i.e. Theorem 1.1).
2 The weak separation condition
Let , and . Let be an IFS of contractions defined on a compact subset with . Let be the contractive ratio of , and , for . We define .
Definition 2.1 We say that an IFS satisfies the weak separation condition (WSC) if there exist and such that, for any , the ball of radius b contains at most γ points of for any . Here we let
Remark 1 For any starting point , it is easy to see that will satisfy the WSC if there exists such that, for any , either
For any and any bounded subsets and , we let
Here denotes the diameter of U. We have two lemmas with respect to the definition of the weak separation condition which are needed to prove our main result.
Lemma 2.2 Let be an infinite IFS of contractive similitudes on , for any and any nonempty subset , . Then satisfies the WSC.
Proof Let . Let be a nonempty subset and let be defined as above. Then for any and any ball of radius b,
which yields the statement. □
Lemma 2.3 Let be an infinite IFS of contractive similitudes on a compact subset . If there exist a constant and a subset with , such that, for any and ,
Then satisfies the WSC.
Proof We denote by the closed ball with radius b and center x. Let ℒ denote the Lebesgue measure on . Let such that , and such that . Then for any ,
Let , , , we have
Suppose . By assumption,
This completes the proof of the lemma. □
Lemma 2.4 Suppose is an infinite IFS on a compact subset , and it satisfies the OSC. Then it satisfies the WSC.
Proof Suppose U is an open set guaranteed by the open set condition. For any , we write and
For any with , the open set condition implies that . Since
So . Then the remark implies that satisfies the WSC. □
Example 2.5 , , (). This IFS satisfies the WSC. It is easy to see that this infinite IFS does not satisfy the OSC. We know that (), (), (, and ). By Lemma 2.4 and the example in  satisfies the WSC.
3 The generalized finite type condition
In this section we promote the generalized finite type condition to infinite IFSs. The generalized finite type condition for infinite IFSs is slightly modified from that for finite IFSs . The definition consists of two parts. The first part concerns the sequence of nested index sets. The second part entails the concept of neighborhood types.
Let be an infinite IFS of contractive similitudes on a compact subset , , and . For any , , we let . For , if I is an initial segment of J or , we write . We denote by if does not hold. Consider a sequence of index sets , where for all , is a finite subset of . Let
Definition 3.1 We say that is a sequence of nested index sets if it satisfies the following conditions:
Both and are nondecreasing, and .
For each , is an antichain in .
For each with , there exists such that .
For each with , there exists such that .
There exists a positive integer n, independent of k, such that, for all with , we have .
By letting for all , we obtain an example of sequence of nested index sets.
To define neighborhood types, we fix a sequence of nested index sets . For each integer , let be the set of k th level vertices (with respect to ) defined as
We call the root vertex and denote it by . Let be the set of all vertices. For , we use the convenient notation .
Fix any nonempty bounded open set which is invariant under . Two k th level (allowing ) are said to be neighbors (with respect to Ω and ) if . The set of vertices
is called the neighborhood of v (with respect to Ω and ). Note that by definition.
Next, we define an equivalence relation on v.
Definition 3.2 Under the above assumptions, two vertices and are equivalent, denoted by if, for , the following conditions hold:
For and such that , and for any positive integer , an index satisfies if and only if it satisfies .
It is easy to see that ∼ is an equivalence relation. We denote the equivalence class containing v by and call it the neighborhood type of v (with respect to Ω and ). We define two important infinite directed graphs and . The graph has vertex set and directed edges defined as follows. Let and . Suppose there exist , , and such that
Then we connect a directed edge L from v to u and denote this by . We call v a parent of u in and u an offspring of v in . We write , where E is the set of all directed edges defined above.
The reduced graph is obtained from by first removing all but the smallest (in the lexicographical order) directed edge going to a vertex. More precisely, let , , be all the directed edges going to the vertex , where are distinct and thus are distinct also. Suppose in the lexicographical order. Then we keep only in the reduced graph and remove all the edges (). Next, we denote the resulting graph by . It is possible that a vertex in does not have any offspring (see the example in ). We remove all vertices that do not have any offspring in , together with all vertices and edges leading only to them. The resulting graph is the reduced graph, denoted by , where is the set of vertices and is the set of all edges.
It follows from the invariance of Ω under that only vertices in can be parents of any offspring of v in . In fact, if is an offspring of v in and if , then for any index ,
Hence w cannot be a parent of u.
Proposition 3.3 Let and u, be their offspring. If v and are not neighbors, then neither are u and .
Proof Let and for some . Since and , we have
This leads to the conclusion. □
Proposition 3.4 Let Ω be a bounded invariant open set for the IFS and let be the corresponding reduced graph. Then there exists a unique path in from the root vertex to any given vertex.
Proof The existence of a path is obvious. Next, we prove the uniqueness. Suppose , if there are two different paths in from to v, then the vertex at which the two paths cross will have two parents in , it is a contradiction. □
Proposition 3.5 Suppose and be two vertices with offspring and () in . Suppose and let
Suppose that in the graph we have edges , such that
Then if and only if and , are neighbors if and only if , are.
Proof Observe that
Similarly we have . Hence if and only if , and so if and only if .
This proves the second part of the proposition. □
Proposition 3.6 Suppose and be two vertices with offspring and () in . Suppose that , and let
Proof The proposition says roughly that two vertices of the same neighborhood type have equivalent offspring. Let W and be the set of offspring of the vertices in and . We define a map as follows: Suppose that u is an offspring of in by an edge k. We let be the offspring of by an edge k. Propositions 3.4 and 3.5 show that χ is a one-to-one correspondence. Furthermore, by (1) we have
By Proposition 3.4 only vertices in can be parents of any offspring of v in . Again by Propositions 3.4 and 3.5, u is an offspring of v in if and only if is an offspring of in . This yields (2). □
Definition 3.7 Let be an self-map IFS on a subset . We say that is of generalized finite type if there exist a sequence of nested index sets and a nonempty invariant open set such that, with respect to Ω and , is a finite set. In this case, we say that Ω is a generalized finite type set.
In the rest of this section, we establish classes of infinite IFSs of generalized finite type condition.
Proposition 3.8 If is of OSC, then it is of the generalized finite type condition.
Proof Let Ω be the open set of OSC. Suppose . For each , the OSC implies that . Let , i.e. , we have . By Proposition 3.4, . So satisfies the generalized finite type condition. □
Example 3.9 , , , (), this IFS satisfies the generalized finite type condition.
Proof Let . For each let . Upon iterating the IFS once, the root vertex generates the following vertices:
Obviously, (). So () with . Upon one more iteration, there is no new neighborhood type generated (see the example 2.8 in ). So and the result follows. □
4 Proof of the main theorem
Proof of Theorem 1.1 Assume that is a finite type similar IFS on X and let and Ω be defined as above. We will show that there exists such that, for all and ,
Let . List all elements of Ψ as . For there exists a unique such that . The choice of the particular does not affect the following proof. We assume that is chosen such that is maximum, i.e., if and for some l, then and . We assume without loss of generality that
For each j, here , let be the initial segment of such that . In particular, . Since for all , it follows that
are neighbors of . The finite type condition implies that the number of vertices in each neighborhood is uniformly bounded by some constant M independent of x, b, and the choice of . That is,
Since each belongs to , we have
Also, by the definition of , there exists a constant n, independent of x and b, such that for all , here . Hence,
It also implies that there exists a constant , independent of x and b, such that
Combining (3) and (4) yields
We write , for any . For each , (5) implies that
If we let , then . The finite type condition implies that
Thus, follows by taking . Lemma 2.3 implies that the satisfies the WSC. □
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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Liu, H., Zhu, X. Separation properties for infinite iterated function systems. J Inequal Appl 2014, 118 (2014). https://doi.org/10.1186/1029-242X-2014-118