A NOTE ON SPECIFICATION FOR ITERATED FUNCTION SYSTEMS

. We introduce several notions of speciﬁcation for iterated function systems and exhibit some of their dynamical properties. In particular, we show that topological entropy and algebraic pressure [4] of systems with speciﬁcation are approximable by the corresponding expressions for ﬁnitely generated iterated function systems.


1.
Introduction. Let X be a compact metric space and V a family of homeomorphisms v : D(v) → v(D(v)) ⊂ X with closed domain D(v) ⊂ X. The pair (X, V) is called a function system. It is called an iterated funtion system if there is a set O ⊂ X such that, This notion has been studied in [4] where also some general motivation to investigate iterated function systems of this general type is discussed. Fairly general conditions are given in [4] that a function system is in fact an iterated function system and self-homeomorphic in the sense that X = v∈V D(v). It then proceeds investigating some thermodynamic aspects of iterated function systems when V is countable. The algebraic pressure is defined as where φ v : D(v) → R + is a family of continuous functions and the supremum is taken over all x in the domain of v n • v n−1 • ... • v 1 . The limit exists because the sequence is subadditive. One of the basic problems within the framework of its thermodynamic formalism is constructing conformal measures for a given family of potentials φ v and the transition parameter P alg (φ). This problem has been solved essentially in [4] reducing the problem to the existence of aperiodic points for a 3476 WELINGTON CORDEIRO, MANFRED DENKER AND MICHIKO YURI potential φ of bounded variation (Theorem 3.17 in [4]). Constructing aperiodic points can be established using some type of specification property (see Section 2). This is one of the reasons to study the specification property here in more detail. Another problem arises in calculating the algebraic pressure via finite iterated function systems (X, W) where W ⊂ V is finite. We derive a theorem of this type using specification.
In Section 2 we define the notion of specification and show that it implies topological mixing. We introduce a notion of topological pressure in Section 3 (in analogy to the classical definition (see [3,7])) and show that topological entropy is strictly positive for an IFS with specification. The main result for IFS with weak specification appears in Section 5, that its algebraic pressure equals the supremum over the algebraic pressure for finite subsystems. The same type of result holds for the toplogical entropy. These results are in the same spirit as those for countable Markov chains in [5], [6], [2]. We finally add a result on the existence of iterated function systems for a function system with specification (see Section 6). This complements results in [4] where this type of problem was treated in general.

2.
Specification. In this section we let (X, V) denote an iterated function system (IFS) as defined in [4]. This means that X is a metric space with metric d and V is a family of partially defined maps v : D(v) → X, where D(v) is non-empty and closed. Moreover, there exists a subset O ⊂ X such that (1) holds. We begin with a series of definitions.
According to the definition of iterated functions systems (the condition (1)), there exists σ ∈ Σ and x ∈ X such that whenever it is well defined and the orbit of (x, σ) as 2. An iterated function system (X, V) has the specification property if for every > 0 there is M ∈ N such that every M -spaced specification isshadowed by some x ∈ X.
2. An iterated function system (X, V) is called topologically mixing if for all nonempty, open sets V, U ⊂ X there is N > 0 such that for all n > N there is 3. An iterated function system (X, V) is called topologically transitive if for all nonempty, open sets V, U ⊂ X there is σ ∈ Σ and n ∈ N such that v σ(n) (U )∩V = ∅.
2. If (X, V) has the specification property then In particular, (X, V) is locally self-homeomorphic (see [4], Definition 2.2). 3. If (X, V) has the specification property then it is topologically transitive. 4. If (X, V) has the specification property then it is topologically mixing. 5. If v : X → X has the specification property then v is topologically mixing. 6. If (X, V) is topologically transitive and satisfies v∈V intD(v) = X, then there is x ∈ X such that for all y ∈ X and > 0 there is n ∈ N and 2. This follows from 1.
be relatively open. Choose points y and z in these sets and > 0 such that Define P (0) = z and P (n) = y. For any σ we have (τ, P, σ) is a M -spaced specification and by specification property we can find 3. and 4. This has been proved in 2. 5. Is a special case of 4. 6. Let V 1 , V 2 , ... be a basis for the topology. Let x 1 , x 2 , ... be a dense set of points belonging to v∈V intD(v). By transitivity, for i, j ∈ N there is σ i,j ∈ Σ and n ∈ N such that v σi, j is open and dense for each j. By the Baire Category Theorem Corollary 2.6. Let (X, V) have the specification property and let intD(v) = ∅ for all v ∈ V. Then there exists x ∈ X such that for every Proof. We may assume that v∈V intD(v) = X. Let x be chosen as in 6. of Then the (X, V) is an IFS with the specification property, but the corollary is not true for any x ∈ T 2 . 3. Topological entropy. Let (X, V) be an iterated function system. We begin with a modification of toplogical entropy for the case of an IFS (see [3,7]). The basic idea is to count the minimal number of orbits from (X, V) which are needed to shadow all orbits. This leads to the following definitions.
For a compact set We denote the minimum cardinality of an (n, )-spanning set for K by r(n, , K). Since we are not restricting the notion to the case of a finite family V, r(n, , K) may be infinite.
We call a set S ⊂ Ψ(K) an (n, )-separated set for K, if for all pairs (x, σ), (y, ϕ) ∈ S there is some i ∈ {1, ..., n − 1} satisfying We denote the maximum cardinality of an (n, )-separated set for K by s(n, , K).
4. Weak specification. Let (X, V) be an iterated function system. We first extend the definitions in Section 2. A specification S = (τ, P, σ) in V (0) ⊂ V consists of a finite collection τ of finite disjoint intervals J = [a J , b J ] ⊂ N, a map P : I = J∈τ J → X and σ J = (σ J i ) 1≤i<|J| ∈ (V (0) ) |J|−1 such that for each J ∈ τ and t ∈ J . A specification in V agrees with the notion of a specification as introduced in Section 2. This means that a specification is a finite collection of orbit segments where the maps σ · belong to V (0) . A weak specification is a specification where the intervals J ∈ τ are adjacent beginning at 1. Each specification S = (τ, P, σ) defines a weak specification S = ( τ , P , σ) by deleting the spacings, called the reduced specification: If the intervals in τ are denoted by J k = [t k , t k + α k ] (1 ≤ k ≤ q) in increasing order the new intervals are defined by α 0 = 0, k = 1, ..., q and P and σ by for t ∈ J k , k = 1, ..., q. Given a weak specification S, any specification S whose reduced specification is S is called an associated specification.
Definition 4.2. An iterated function system is said to have the finite weak specification property if for all η > 0, and all finite sets V (0) ⊂ V there is a finite set V (1) ⊂ V and M ≥ 0 such that any weak specification in V (0) has an associated specification which is at most M -spaced and is η-shadowed within V (1) by some point in X.

Lemma 4.3.
If an iterated function system (X, V) has the finite specification property then (X, V) has the weak finite specification property.
Proof. Let V (0) ⊂ V be finite and η > 0. Take V (1) and M > 0 as in the finite specification property. If S = ( τ , P , σ) is a weak specification, for each Then, by definition, (τ, P, σ) is an (exactly) M -spaced specification. By finite specification, it is ηshadowed within V (1) by some point in X.

5.
Specification and algebraic pressure. We begin extending the notion of algebraic pressure from [4] slightly. Let φ ∈ Φ, the space of families {φ v : v ∈ V} of uniformly continuous functions φ v : D(v) → R + , and V (0) ⊂ V. Define for each n ≥ 1 the n-th partition function for (X, V) by n denotes the set of n-fold compositions of maps from V (0) . We allow here Z n (V (0) ) = ∞ as well.
If V ( 0) = V we also write Z n (φ) and P alg (φ). Proof. Note that log Z n (φ, V (0) ) is subadditive, since We are now in the position to proof the main result of this note. Let Φ b denote the family of potentials {φ v : v ∈ V} so that {log φ v : v ∈ V} is uniformly continuous.
Theorem 5.2. If (X, V) has the finite weak specification property, then for a potential φ ∈ Φ b we have Proof. Since the right hand side of (3) is clearly bounded by its left hand side it is left to show the converse inequality. Let > 0. Since log φ is a family of uniformly continuous functions, there exists η > 0 such that for φ v ∈ φ and x, y ∈ D(v) with d(x, y) < η we have Since V is finite, there is a finite family Given η > 0 and V (0) , choose a subsystem V (1) ⊂ V and a spacing constant L = L(η) ≥ 0 as in the finite weak specification property. Let k defines a weak specification setting By the finite weak specification property there is an orbit segment z 0 , ..., z M such that It follows that there is a map ρ : M sending a weak specification to the associated w as defined above. This map is clearly injective.
We now use the fact that for the pair (z, w), z = (z 0 , ..., z M ), w = (w 1 , ..., w M −1 ), as above It follows that Let γ = inf v∈V (1) inf y∈D(v) φ v (y) (we may assume that γ < 1). Taking the maximum on the right hand side we arrive at sL max sk+1≤M ≤s(k+L)

WELINGTON CORDEIRO, MANFRED DENKER AND MICHIKO YURI
Since δ > 0 is independent of all other quantities sL max sk+1≤M ≤s(k+L) Finally, if the maximum is attained for M = sk + a 1 sk + a log Z sk+a (φ, V (1) ) + 1 sk + a log(sL) Taking the limit as s → ∞ we obtain using Lemma 5.1 Note that L only depends on η and not on k, hence letting k → ∞ and → 0 ends the proof.
Theorem 5.3. If (X, V) has the finite weak specification property, then The proof of this theorem is similar to the previous proof and therefore omitted.
6. Existence of iterated function systems. Let (X, V) be a family of partially defined hoemeomorphisms v : D(v) → X where D(v) ⊂ X is a closed subset; it is called a function system in [4]. The notions of specification can be defined for such systems in analogy with those for iterated function systems.
Lemma 6.1. Let (X, V) be a function system which has the finite weak specification property and X be compact. Then there is a subfamily V (1) ⊂ V and ∅ = E ⊂ X such that each x ∈ E has an infinite orbit in V (1) .
Then for each n ∈ N we have that A n is nonempty and closed. In fact, if x ∈ D(v * ) we can define the following weak specification: τ = {I 1 , I 2 , ..., I n }, where I i = {i}, i ∈ {1, ..., n}, P : {1, 2, ..., n} → X P (t) = x and σ = (v * , v * , v * , ...). Then there is a specification at most L-spaced and x 0 ∈ X such that x 0 shadows this specification within V (1) . Then, x 0 ∈ A n . If x m → x 0 with x m ∈ A n , then there is w m ∈ V (1) n , such that x m ∈ D(w m ). As V (1) is finite, there are a subsequence m k and w ∈ V (1) n such that x m k ∈ D(w). Therefore, x 0 ∈ D(w) and A n is closed.
Then, we have a nested sequence {A n } of compact and nonempty sets. Therefore, E = n∈N A n = ∅. If x ∈ v∈V (1) v(D(v) ∩ E) there exists a ∈ E and v ∈ V (1) such that v(a) = x. Then a ∈ A n for all n ∈ N, therefore, v(a) ∈ A n+1 for all n ∈ N. As A 1 ⊂ A 2 v(a) ∈ E ⊂ v∈V (1) . As x is arbitrary, v∈V (1) Theorem 6.2. Let (X, V) be a function system which has the specification property and let X be compact. Then there is a a nonempty closed subset X 0 ⊂ X and a subfamily V (0) ⊂ V such that (X 0 , V (0) ) is an iterated function system. Proof. We use Theorem 2.5 in [4]. There exists a pair ( K, K) such that ∅ = K, K ⊂ K, the pair is minimal in the sense that there is no smaller K and no larger K ⊂ K, and if K = ∅ then the claim holds. Hence we need to show that we can produce a pair with nonempty K.
Fix x 0 ∈ X and η > 0. By specification there are sequences w n ∈ V n and x n ∈ X such that w n+1 = vw n with v ∈ V and x n ∈ D(w 1 ) such that d(w n (x n+1 ), w n (x n )) < η.
In particular, w n is a well defined map defined on ∅ = D(w n ) ∩ D(w n+1 ). Therefore Now we can repeat the proof of Theorem 2.5 in [4] to finish the proof.