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 ∈ .

scholarly journals Absolutely continuous invariant measures for non-uniformly expanding maps

2009 ◽  
Vol 29 (4) ◽  
pp. 1185-1215 ◽  
Author(s):  
HUYI HU ◽  
SANDRO VAIENTI
Keyword(s):  
Invariant Measure ◽  
Fixed Points ◽  
Large Class ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractFor a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

scholarly journals Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

2008 ◽  
Vol 28 (1) ◽  
pp. 211-228 ◽  
Author(s):  
TOMAS PERSSON
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Bounded Subset ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractA class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

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.

scholarly journals Estimating invariant measures and Lyapunov exponents

1996 ◽  
Vol 16 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Brian R. Hunt
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Recent Interest ◽  
Absolutely Continuous Invariant Measure ◽  
Time Averages ◽  
Absolutely Continuous Invariant

AbstractThis paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Absolutely continuous invariant measures for piecewise expanding C2 transformations in Rn on domains with cusps on the boundaries

1996 ◽  
Vol 16 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Kourosh Adl-Zarabi
Keyword(s):  
Lower Bound ◽  
Invariant Measure ◽  
Finite Number ◽  
Invariant Measures ◽  
Bounded Region ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Derivative Matrix ◽  
Absolutely Continuous Invariant

AbstractLet Ω be a bounded region in Rn and let be a partition of Ω into a finite number of subsets having piecewise C2 boundaries. The boundaries may contain cusps. Let τ: Ω → Ω be piecewise C2 on and expanding in the sense that there exists α > 1 such that for any i = 1, 2,…,m, where is the derivative matrix of and ‖·‖ is the euclidean matrix norm. The main result provides a lower bound on α which guarantees the existence of an absolutely continuous invariant measure for τ.

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.

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.

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