Separation properties for infinite iterated function systems
© Liu and Zhu; licensee Springer 2014
Received: 5 February 2014
Accepted: 6 March 2014
Published: 20 March 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.
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 .
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.
which yields the statement. □
Then satisfies the WSC.
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.
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.
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.
We call the root vertex and denote it by . Let be the set of all vertices. For , we use the convenient notation .
is called the neighborhood of v (with respect to Ω and ). Note that by definition.
Next, we define an equivalence relation on v.
For and such that , and for any positive integer , an index satisfies if and only if it satisfies .
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.
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 .
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. □
Then if and only if and , are neighbors if and only if , are.
Similarly we have . Hence if and only if , and so if and only if .
This proves the second part of the proposition. □
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.
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
Thus, follows by taking . Lemma 2.3 implies that the satisfies the WSC. □
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