stationary measures
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Author(s):  
Charlene Kalle ◽  
Marta Maggioni

In this paper, we employ a random dynamical systems approach to study generalized Lüroth series expansions of numbers in the unit interval. We prove that for each [Formula: see text] with [Formula: see text] Lebesgue almost all numbers in [Formula: see text] have uncountably many universal generalized Lüroth series expansions with digits less than or equal to [Formula: see text], so expansions in which each possible block of digits occurs. In particular this means that Lebesgue almost all [Formula: see text] have uncountably many universal generalized Lüroth series expansions using finitely many digits only. For [Formula: see text] we show that typically the speed of convergence to an irrational number [Formula: see text] of the corresponding sequence of Lüroth approximants is equal to that of the standard Lüroth approximants. For other rational values of [Formula: see text] we use stationary measures to study the typical speed of convergence of the approximants and the digit frequencies.


Author(s):  
BENOÎT R. KLOECKNER

Abstract In this paper we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalised moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a “linear response formula” at almost every parameter of the perturbation.


Author(s):  
Alexandre Baraviera ◽  
Pedro Duarte ◽  
Maria Joana Torres

We extend the theory of isospectral reductions of Bunimovich and Webb to infinite graphs, and describe an application of this extension to the problems of existence and approximation of stationary measures on infinite graphs.


2021 ◽  
pp. 2150039
Author(s):  
José F. Alves ◽  
Carla L. Dias ◽  
Helder Vilarinho

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.


2019 ◽  
Vol 399 ◽  
pp. 73-85
Author(s):  
Min Ji ◽  
Zhongwei Shen ◽  
Yingfei Yi
Keyword(s):  

2019 ◽  
Vol 180 (1-6) ◽  
pp. 366-397
Author(s):  
Gioia Carinci ◽  
Cristian Giardinà ◽  
Errico Presutti

AbstractWe study the Ginzburg–Landau stochastic models in infinite domains with some special geometry and prove that without the help of external forces there are stationary measures with non-zero current in three or more dimensions.


Author(s):  
Krzysztof Barański ◽  
Adam Śpiewak

Abstract We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question by Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.


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