INVARIANT DENSITIES FOR POSITION-DEPENDENT RANDOM MAPS ON THE REAL LINE: EXISTENCE, APPROXIMATION AND ERROR BOUNDS

2006 ◽  
Vol 06 (02) ◽  
pp. 155-172
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA
Keyword(s):  
Probability Function ◽  
Approximation Error ◽  
Probability Measures ◽  
Countable Number ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Discrete Time Dynamical System ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant ◽  
Number Of Branches

A random map is a discrete-time dynamical system in which a transformation is randomly selected from a collection of transformations according to a probability function and applied to the process. In this note, we study random maps with position-dependent probabilities on ℝ. This means that the random map under consideration consists of transformations which are piecewise monotonic with countable number of branches from ℝ into itself and a probability function which is position dependent. We prove existence of absolutely continuous invariant probability measures and construct a method for approximating their densities. Explicit quantitative bound on the approximation error is given.

Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

2018 ◽  
Vol 39 (10) ◽  
pp. 2769-2792
Author(s):  
GARY FROYLAND ◽  
CECILIA GONZÁLEZ-TOKMAN ◽  
RUA MURRAY
Keyword(s):  
Invariant Measures ◽  
Single Step ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Driving Mechanisms ◽  
The Family ◽  
Ulam’S Method ◽  
Ulam's Method ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

Uniqueness of Invariant Densities for Certain Random Maps of The Interval

1987 ◽  
Vol 30 (3) ◽  
pp. 301-308 ◽  
Author(s):  
Abraham Boyarsky
Keyword(s):  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Time Process ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

AbstractA random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

scholarly journals Approximating invariant densities of metastable systems

2010 ◽  
Vol 31 (5) ◽  
pp. 1345-1361 ◽  
Author(s):  
CECILIA GONZÁLEZ-TOKMAN ◽  
BRIAN R. HUNT ◽  
PAUL WRIGHT
Keyword(s):  
Lebesgue Measure ◽  
Convex Combination ◽  
Invariant Sets ◽  
Positive Lebesgue Measure ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant ◽  
Metastable Systems

AbstractWe consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs. The result is generalized to the case of finitely many invariant components.

scholarly journals A generalization of Straube's theorem: existence of absolutely continuous invariant measures for random maps

2005 ◽  
Vol 2005 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Pawel Góra ◽  
Abraham Boyarsky
Keyword(s):  
Invariant Measures ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Discrete Time Dynamical System ◽  
Absolutely Continuous Invariant Measure ◽  
Necessary And Sufficient ◽  
Absolutely Continuous Invariant

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. In this paper, we study random maps. The main result provides a necessary and sufficient condition for the existence of absolutely continuous invariant measure for a random map with constant probabilities and position-dependent probabilities.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

A Piecewise Linear Maximum Entropy Method for Invariant Measures of Random Maps with Position-Dependent Probabilities

2018 ◽  
Vol 28 (12) ◽  
pp. 1850154 ◽  
Author(s):  
Congming Jin ◽  
Tulsi Upadhyay ◽  
Jiu Ding
Keyword(s):  
Maximum Entropy ◽  
Invariant Measures ◽  
Piecewise Linear ◽  
Maximum Entropy Principle ◽  
Entropy Method ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Moment Functions ◽  
Entropy Principle ◽  
Absolutely Continuous Invariant

We present a numerical method for the approximation of absolutely continuous invariant measures of one-dimensional random maps, based on the maximum entropy principle and piecewise linear moment functions. Numerical results are also presented to show the convergence of the algorithm.

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