Stationary Random Iterations of Lipschitz Maps: A Generalized Displacement Condition for the Existence of Invariant Structures
Abstract
We study random iterations generated by infinitely many Lipschitz maps on non-compact complete separable metric spaces. We generalize classical displacement conditions ensuring the existence and uniqueness of invariant structures. We prove two main results. The first uses the graph transform operator to extend Elton's condition [A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34(1) (1990)] for the existence and uniqueness of invariant graphs. The second uses the Markov operator to generalize a displacement condition for stationary measures, introduced by Diaconis and Freedman [Iterated random functions. SIAM Rev. 41(1) (1999)] and by Kloeckner [Optimal transportation and stationary measures for iterated function systems. Math. Proc. Cambridge Philos. Soc. 173(1) (2022)]. We present several examples illustrating the applicability of our results.
Related
- LiD-GLM: Lipschitz-constrained Deep Generalized Linear Models
- Stationary Structures of the Boussinesq equations
- Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
- Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies
- Deformations of generalized complex structures on 𝒢-flat transitive Courant algebroids
BibTeX
@misc{figliaggi2026stationary,
title = {{Stationary Random Iterations of Lipschitz Maps: A Generalized Displacement Condition for the Existence of Invariant Structures}},
author = {Bella Figliaggi and Edgar Matias and Tiago Pereira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33829},
eprint = {2610.33829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stationary Random Iterations of Lipschitz Maps: A Generalized Displacement Condition for the Existence of Invariant Structures
AU - Bella Figliaggi
AU - Edgar Matias
AU - Tiago Pereira
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33829
UR - https://arxiv.org/abs/2610.33829
AB - We study random iterations generated by infinitely many Lipschitz maps on non-compact complete separable metric spaces. We generalize classical displacement conditions ensuring the existence and uniqueness of invariant structures. We prove two main results. The first uses the graph transform operator to extend Elton's condition [A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34(1) (1990)] for the existence and uniqueness of invariant graphs. The second uses the Markov operator to generalize a displacement condition for stationary measures, introduced by Diaconis and Freedman [Iterated random functions. SIAM Rev. 41(1) (1999)] and by Kloeckner [Optimal transportation and stationary measures for iterated function systems. Math. Proc. Cambridge Philos. Soc. 173(1) (2022)]. We present several examples illustrating the applicability of our results.
ER -