ON A CERTAIN SEPARATION CONDITION FOR ATTRACTORS OF FINITE SYSTEMS OF CONTRACTIVE HOMEOMORPHISMS

Let X be a complete metric space. Denote by M(X) the set of all contracting homeomorphisms w: X → X of the whole space X onto itself. Given a system w1,..., wN ∈ M(X), N ≥ 2, let A = A(w1,..., wN) ⊂ X be the attractor of this system. Denote Σ = {1,..., N} and for every vector i = {i1,..., im} ∈ Σm, let wi(A) = wi1 ◦... ◦ wim(A). Our work studies the following separation condition. For the purposes of this paper, we say that a system of mappings (w1,..., wN) ∈ (M(X))N satisfies the strong Markov property (SMP) if for every m ∈ N, there is an open set Om ⊂ X such that 1. Om ∩ A = A; 2. wi(Om) ∩ wj(Om) = ∅, for every i 6 = j ∈ Σm. This separation condition is weaker than the classical strong open set condition, and not equivalent to the weak separation condition studied in [4] and [7]. We will call a sequence {wk}k∈N fromM(X) strongly poinwise convergent to a mapping w ∈ M(X) and wrtie wk s.p.→w, k → ∞, if it converges to w at every point x ∈ X, and the contraction coefficients of the mappings wk are bounded by a number strictly less than one. If {wk}k∈N ⊂ M(X) is a sequence of similitudes, then the strong pointwise convergence is equivalent to the ”usual ” pointwise convergence. We define topology BN on the space (M(X))N by defining a subset C ⊂ (M(X))N to be closed if C = ∅ or if for every sequence {(wk1,..., wkN)}k∈N ⊂ C, such that {wki} s.p.→wi ∈ M(X), i = 1,..., N, we have (w1,..., wN) ∈ C. Theorem 1. Let X be a complete metric space. The set of systems of mappings (w1,..., wN) ∈ (M(X))N, which satisfy the SMP is a G-delta set in the topology BN. For everym ∈ N, denote by Vm the set of all orderedN-tuples (w1,..., wN) ∈ (M(X))N such that for every i ∈ Σm, there holds wi(A) * j∈Σm, j6=i wj(A). To prove Theorem 1 we show that every set Vm is open and prove the following statement. Theorem 2. Let X be a complete metric space. The system (w1,..., wN) of contracting homeomorphisms of X onto itself satisfies the SMP if and only if (w1,..., wN) ∈ m=