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.

INVARIANT MEASURES FOR RANDOM MAPS VIA INTERPOLATION

2013 ◽  
Vol 23 (02) ◽  
pp. 1350025
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measures ◽  
Piecewise Linear ◽  
Linear Method ◽  
Piecewise Linear Approximation ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Proof Of Convergence

Let T = {τ1, τ2, …, τK; p1, p2, …, pK} be a position dependent random map on [0, 1], where {τ1, τ2, …, τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, …, pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

scholarly journals Maximal absolutely continuous invariant measures for piecewise linear Markov transformations

1990 ◽  
Vol 10 (4) ◽  
pp. 645-656 ◽  
Author(s):  
W. Byers ◽  
P. Góra ◽  
A. Boyarsky
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Piecewise Linear ◽  
Maximal Entropy ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Necessary And Sufficient ◽  
Absolutely Continuous Invariant

AbstractLet be an irreducible 0–1 matrix such that the non-zero entries in each row are consecutive. Let be the class of piecewise linear Markov transformations τ on [0, 1] into [0, 1] induced by for which the absolutely continuous invariant measure has maximal entropy. The main result presents necessary and sufficient slope conditions on τ which guarantee that τ ∈ .

MAXIMUM ENTROPY METHOD FOR POSITION DEPENDENT RANDOM MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1805-1811 ◽  
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measure ◽  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Proof Of Convergence

Let {τ1, τ2,…,τK} be a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2,…,pK} be a collection of position dependent probabilities on [0, 1]. We consider position dependent random maps T = {τ1,τ2,…,τK;p1,p2,…,pK} such that T preserves an absolutely continuous invariant measure with density f*. A maximum entropy method for approximating f* is developed. We present a proof of convergence of the maximum entropy method for random maps.

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.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

1996 ◽  
Vol 06 (06) ◽  
pp. 1143-1151
Author(s):  
E. A. SATAEV
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Expanding Mapping

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

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.

scholarly journals Optimal transport and dynamics of expanding circle maps acting on measures

2012 ◽  
Vol 33 (2) ◽  
pp. 529-548 ◽  
Author(s):  
BENOÎT KLOECKNER
Keyword(s):  
Invariant Measure ◽  
Optimal Transport ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Expanding Circle ◽  
Rigorous Framework ◽  
Infinite Multiplicity ◽  
Set Up ◽  
Absolutely Continuous Invariant

AbstractIn this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

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