Invariant measures for position dependent random maps with continuous random parameters

2012 ◽  
Vol 208 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Tomoki Inoue
Keyword(s):  
Invariant Measures ◽  
Random Parameters ◽  
Random Maps

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.

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.

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

2009 ◽  
Vol 19 (01) ◽  
pp. 409-417 ◽  
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measures ◽  
Stochastic Matrix ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Higher Dimension ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Higher Dimensional ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

RELATIVE EQUILIBRIUM STATES AND DIMENSIONS OF FIBERWISE INVARIANT MEASURES FOR RANDOM DISTANCE EXPANDING MAPS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350015 ◽  
Author(s):  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Gibbs State ◽  
Relative Equilibrium ◽  
Equilibrium States ◽  
Expanding Maps ◽  
Hölder Continuous ◽  
Random Maps ◽  
Full Dimension ◽  
Relative Metric

We show that the Gibbs states (known from [10] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps, we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.

CONDITIONALLY INVARIANT MEASURES FOR POSITION DEPENDENT RANDOM MAPS OF AN INTERVAL WITH HOLES

2006 ◽  
Vol 16 (02) ◽  
pp. 437-444
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA
Keyword(s):  
Invariant Measures ◽  
Unit Interval ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Piecewise Monotonic ◽  
Laws Of Motion

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

STOCHASTIC PERTURBATIONS OF POSITION DEPENDENT RANDOM MAPS

2003 ◽  
Vol 03 (04) ◽  
pp. 545-557 ◽  
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Dynamical System ◽  
Invariant Measures ◽  
Invariant Density ◽  
Absolutely Continuous ◽  
Stochastic Perturbations ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant

A random map is a dynamical system consisting of a collection of maps which are selected randomly by means of fixed probabilities at each iteration. In this note, we consider absolutely continuous invariant measures of random maps with position dependent probabilities and prove that they are stable under small stochastic perturbations. This result depends on a new lemma which handles arbitrarily small extra partition elements that may arise from the perturbation of the random map. For perturbations satisfying additional conditions, we give precise estimates of the error in the invariant density.

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