A Closed Graph Theorem for hyperbolic iterated function systems

In this note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be used to approach the topological conjugacy problem for iterated function systems.


Introduction
Since Hutchinson's seminal paper [Hut81], iterated function systems have remained close to the heart of fractal geometry. Iterated function systems have continued to be studied and generalised in numerous directions.
In this article we focus on the dynamics of hyperbolic iterated function systems from a topological viewpoint, rather than geometric or measure theoretic perspectives. Although we work in the hyperbolic setting, we take a viewpoint similar to that of topological iterated function Definition 2.1. A (hyperbolic) iterated function system (X, Γ) consists of a complete metric space (X, d) together with a finite collection Γ of proper contractions on X. That is for each γ ∈ Γ there exists 0 ≤ c γ < 1 such that d(γ(x), γ(y)) ≤ c γ d(x, y) for all x, y ∈ X.
Denote by K(X) the collection of non-empty compact subsets of X. Recall that the Hausdorff metric d H on K(X) is defined by for all A, B ∈ K(X). It is well-known that if (X, d) is a complete metric space then so is (K(X), d H ) (see for example [Kig01, Proposition 1.1.5]).
Given an iterated function system (X, Γ) we abuse notation and also use Γ to denote the Hutchinson operator Γ : K(X) → K(X) defined, for all K ∈ K(X), by Hutchinson [Hut81] showed that Γ is a contraction on (K(X), d H ), and consequently has a unique fixed point A ∈ K(X) by the Contraction Mapping Principle. The fixed-point A is called the attractor of (X, Γ). In particular, Γ(A) = A and for any K ∈ K(X) we have d H (Γ k (K), A) → 0 as k → ∞. This result is collectively referred to as Hutchinson's Theorem. Hutchinson's original result was stated for X = R n , however the proof of the result for general hyperbolic systems remains nearly identical (cf. [Kig01, Theorem 1.1.7]).
A non-empty compact subset K of X is said to be backward invariant if K ⊆ Γ(K). Backward invariant sets were also called sub-self-similar sets by Falconer [Fal95]. In hyperbolic iterated function systems, backward invariant sets are necessarily contained in the attractor. We require the following lemma in the sequel.
In particular, if (X, Γ) is a hyperbolic iterated function system with attractor A and K ∈ K(X) is backward invariant, then K ⊆ A.
Proof. Suppose for contradiction that there exists i ∈ N and For the second statement note that if K ⊆ Γ(K), then Γ k (K) ⊆ Γ k+1 (K) for all k ∈ N. Hutchinson's Theorem implies that d H (Γ k (K), A) → 0, so the second statement follows from the first.

Morphisms of iterated function systems
Topological conjugacy of iterated function systems is a well-established notion and the problem of determining whether two systems are topologically conjugate can be approached in numerous ways (see for example [Kam00, Corollary 1.27]). When generalising conjugacy there are choices to be made about selecting an appropriate notion of morphism, and several approaches have been taken previously. Kieninger, for instance, describes semiconjugacy of topological iterated function systems [Kie02, Definition 4.6.3], which is analogous to the corresponding notion in symbolic dynamics. Another approach is via the fractal homeomorphisms of Barnsley [Bar09] which use shift invariant sections of a code map.
Expanding on the established definition of conjugacy we use the following-somewhat naïvenotion of morphism, related to Kieninger's semiconjugacies.
(i) (f, α) is an embedding if both f and α are injective.
(ii) (f, α) is a semiconjugacy if both f and α are surjective.
(iii) (f, α) is an isomorphism or conjugacy if f is a homeomorphism and α is bijective. In this case we say that (X, Γ) is isomorphic or conjugate to (Y, Λ). Morphisms may be composed by setting (f, α) In Definition 3.1 we have not made use of the metric space structure of either X or Y , and this has its drawbacks. In particular, if (f, α) : (X, Γ) → (Y, Λ) is a morphism, then it is not typically true that (f (X), α(Γ)) is an iterated function system since f (X) is not necessarily a complete metric space. This could be amended by insisting that f is also a closed map or that X is compact (for example if X is the attractor itself), however we do not require either assumption in what follows.
Morphisms in the sense of Definition 3.1 occur fairly naturally.
In the previous examples α was an injection, but this is not always the case.
Example 3.5. Let (X, Γ) be an iterated function system and consider the product system Morphisms intertwine the induced dynamics on compact sets.
For hyperbolic systems, morphisms always map attractors to compact subsets of attractors so that the image is a subsystem of the codomain.

A closed graph theorem for morphisms
In this section we prove the main result of this article, a Closed Graph Theorem for morphisms of hyperbolic iterated function systems. We show that, when restricted to attractors, the graph of a morphism is itself the attractor of an iterated function system. To this end, we introduce a fibred system.
of f restricted to A. Moreover, suppose that β : Γ → Λ and D is the attractor of (X × Y, Γ × β Λ). Gr The "only if" direction of the second statement follows from the first statement. For the "if" direction suppose that D is the graph of a function g : A → B. Since D is closed in A × B it follows from the Closed Graph Theorem for compact Hausdorff spaces that g is continuous. If (x, g(x)) ∈ D, then invariance under the Hutchinson operator implies that for each γ ∈ Γ, we have (γ(x), β(γ)•g(x)) ∈ D. Since D = Gr(g) it follows that β(γ)•g(x) = g•γ(x). Consequently, (g, β) : (A, Γ) → (B, Λ) is a morphism.
Restricting to attractors yields the following Closed Graph Theorem. Remark 4.4. Set theoretically, a function is its graph. Consequently, Corollary 4.3 may be interpreted as saying that f is the attractor of an iterated function system. Theorem 4.2 also implies that morphisms between hyperbolic iterated function systems are rare, and completely determined by α on the attractor. Proof. Theorem 4.2 implies that Gr(f ) and Gr(g) are both attractors of (X × Y, Γ × α Λ), so uniqueness of the attractor gives the result.
In light of Corollary 4.5 we may use Theorem 4.2 to approach the problem of determining whether there exists a morphism between two hyperbolic iterated function systems. If (f, α) : (X, Γ) → (Y, Λ) is a morphism, then so is (f | A , α) : (A, Γ) → (B, Λ). Since f | A is determined completely by α, it suffices to check whether the attractor of (X × Y, Γ × α Λ) is the graph of a function for each of the |Λ| |Γ| possible choices of α. Moreover, if |Γ| = |Λ| we can determine whether a conjugacy exists by checking each of the |Γ|! possible choices of α.
For concrete iterated function systems, the attractor of (X × Y, Γ × α Λ) may be approximated numerically using the Chaos Game algorithm. This can inform existence or non-existence results about morphisms or conjugacy as seen in Example 4.6 below. ], respectively. It is natural to ask whether these two systems are conjugate.
Example 4.7. Consider an iterated function system (A, Γ) with attractor A. Let L : {1, . . . , N } → Γ be a bijective labelling of the maps in Γ and let β L : Σ N → Γ denote the induced map as in Example 4.7. Then the code map π L : Ω N → A is uniquely determined as the function whose graph is the attractor of the fibred system (Ω N × A, Σ N × β L Γ).
Restricting to attractors, the fibred system associated to a morphism is always conjugate to the domain. Proof. Theorem 4.2 implies that Gr(f ) is the attractor of (Gr(f ), Γ × α Λ). Let p : Gr(f ) → A denote the projection onto the first factor. Since Gr(f ) is the graph of a function, p is a continuous bijection from a compact space to a Hausdorff space, and therefore a homeomorphism. It follows that (p, γ α → γ) is the desired conjugacy.
As a consequence of Corollary 4.8, topological properties of the system (A, Γ) are shared by (Gr(f ), Γ × α Λ). For example, the covering dimension of A is equal to the covering dimension of Gr(f ). The relationship at the level of geometric properties is not so clear, because geometric properties of Gr(f ) depend on the choice of metric on Gr(f ).
We finish by showing that morphisms of of iterated function systems lift uniquely to morphisms between code spaces.   Σ N ), and ( f , α) = (π Λ , β Λ ) = (id Ω N , id Σ N ), then commutativity of (1) would imply that such an f is an inverse for π Γ -regardless of A-which is absurd.
As a final remark, we note that the notion of morphism introduced in Definition 3.1 generalises readily to the topological iterated function systems of Kameyama [Kam93] or Kieninger [Kie02]. Although we do not pursue it here, the author thinks that it would be interesting to see how the collection of invariant sets in the fibred system affects the existence of morphisms for more general topological iterated function systems.