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

Advances in Nonlinear Analysis

article

Peter Bugiel¹  Stanisław Wędrychowicz²  Beata Rzepka² 

[1] Faculty of Mathematics and Computer Science, Jagiellonian University Cracow (Kraków);Department of Nonlinear Analysis, Ignacy Łukasiewicz Rzeszów University of Technology

关键词: Markov operator;    Markov maps;    fixed point;    Radon-Nikodym derivative;    Frobenius-Perron operator;   

DOI  :  10.1515/anona-2020-0163

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

Existence of fixed point of a Frobenius-Perron type operator P : L 1 ⟶ L 1 generated by a family { φ y } 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 {Pjg}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 .

【 授权许可】

CC BY