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

Authors
• 1 Jagiellonian University Cracow (Kraków), Poland , (Poland)
• 2 Ignacy Łukasiewicz Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959 , (Poland)
Type
Published Article
Journal
Publisher
De Gruyter
Publication Date
Jan 16, 2021
Volume
10
Issue
1
Pages
972–981
Identifiers
DOI: 10.1515/anona-2020-0163
Source
De Gruyter
Keywords
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.