Spectral Criteria for Weighted Wasserstein Contractivity of Markov-Switched Iterated Function Systems

We study one-dimensional iterated function systems in which the choice of map is governed by a finite-state Markov chain rather than by independent sampling. Let X⊂ R be a compact interval, let f₁, ,fₘ:X→ X be C¹ maps, and let (Jₙ)n≥ 0 be a finite-state Markov chain on \1, ,m\ with transition matrix P=(Pᵢⱼ) . The associated random dynamical system is Xₙ₊₁=f_Jₙ₊₁(Xₙ), and its natural Markov state is the lifted process (Xₙ,Jₙ)∈ X× \1, ,m\. We introduce the vector-valued derivative transfer operator ( LΦ )ᵢ(x) = ∑ ⱼ₌₁ᵐ Pᵢⱼ|fⱼ'(x)|\,φ ⱼ(fⱼ(x)), Φ =(φ ₁, ,φ ₘ), where each component φ ᵢ∈ C(X) corresponds to a switching mode. The operator acts on positive continuous vector fields on X . Our main result proves the Collatz–Wielandt type formula ρ ( L) = inf Φ >0 max 1≤ i≤ m x∈ X ( LΦ )ᵢ(x)/φ ᵢ(x). Equivalently, ρ ( L) is the optimal contraction threshold over all continuous mode-dependent weights wᵢ:X→ (0,∞ ) . In particular, ρ ( L)<1 is equivalent to the existence of such weights for which the lifted Markov operator is a strict contraction along fibres with fixed mode marginal, in the associated mode-dependent weighted Wasserstein geometry. When the switching chain is irreducible and aperiodic, this fibrewise contraction combines with finite-state mode mixing to yield a unique invariant probability measure on X× \1, ,m\ and exponential convergence to it in a full weighted Wasserstein metric. Conversely, if ρ ( L)≥ 1 , then no continuous choice of mode-dependent weights can yield fibrewise Wasserstein contractivity of this type with contraction factor strictly smaller than one. The result gives a sharp spectral criterion for contractive geometry in Markov-switched random dynamical systems and shows explicitly how temporal correlations in the switching process affect ergodicity beyond the averaged i.i.d. theory.