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

Abstract

Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 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 ∈ L1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L1) of the sequence {Pjg}j=1∞$\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g0. The general result is applied to a family of C1+α-smooth Markov maps in ℝd.

Keywords

• markov operator
• markov maps
• fixed point
• radon-nikodym derivative
• frobenius-perron operator
• 37a30