Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝ d

Abstract

Abstract Existence of fixed point of a Frobenius-Perron type operator P : L 1 ⟶ L 1 generated by a family {φy } y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L 1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L 1) of the sequence { P j g } j = 1 ∞ $\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g 0. The general result is applied to a family of C 1+α -smooth Markov maps in ℝ d .