Existence of absolutely continuous invariant probability measures for multimodal maps

2001 ◽  
pp. 432-446
Keyword(s):  
Probability Measures ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant

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.

Non-ergodicity for C1 expanding maps and g-measures

1996 ◽  
Vol 16 (3) ◽  
pp. 531-543 ◽  
Author(s):  
Anthony N. Quasf
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR GENERIC MULTI-DIMENSIONAL PIECEWISE AFFINE EXPANDING MAPS

1999 ◽  
Vol 09 (09) ◽  
pp. 1743-1750 ◽  
Author(s):  
J. BUZZI
Keyword(s):  
Open Problem ◽  
Invariant Measures ◽  
Probability Measures ◽  
Higher Dimension ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Piecewise Affine ◽  
Almost All ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

By a well-known result of Lasota and Yorke, any self-map f of the interval which is piecewise smooth and uniformly expanding, i.e. such that inf |f′|>1, admits absolutely continuous invariant probability measures (or a.c.i.m.'s for short). The generalization of this statement to higher dimension remains an open problem. Currently known results only apply to "sufficiently expanding maps". Here we present a different approach which can deal with almost all piecewise expanding maps. Here, we consider both continuous and discontinuous piecewise affine expanding maps.

Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$-analytic mappings of the plane

2000 ◽  
Vol 20 (3) ◽  
pp. 697-708 ◽  
Author(s):  
JÉRÔME BUZZI
Keyword(s):  
Probability Measures ◽  
Bounded Region ◽  
Absolutely Continuous ◽  
Real Analytic ◽  
Absolutely Continuous Invariant ◽  
Analytic Mappings

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

scholarly journals THE DIFFUSION COEFFICIENT FOR PIECEWISE EXPANDING MAPS OF THE INTERVAL WITH METASTABLE STATES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150005 ◽  
Author(s):  
DMITRY DOLGOPYAT ◽  
PAUL WRIGHT
Keyword(s):  
Diffusion Coefficient ◽  
Markov Chain ◽  
Bounded Variation ◽  
Continuous Time ◽  
Probability Measures ◽  
Continuous Time Markov Chain ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.

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.

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 Stochastic stability for fiber expanding maps via a perturbative spectral approach

2016 ◽  
Vol 16 (04) ◽  
pp. 1650011
Author(s):  
Yushi Nakano
Keyword(s):  
Stochastic Stability ◽  
Noise Level ◽  
Upper Bounds ◽  
Probability Measures ◽  
Spectral Approach ◽  
Skew Product ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Small Perturbations ◽  
Absolutely Continuous Invariant

We consider small perturbations of expanding maps induced by skew-product mappings whose base dynamics need not be mixing or invertible. Adapting a previously developed perturbative spectral approach, we show stability of the densities of the unique absolutely continuous invariant probability measures for expanding maps under these perturbations, and upper bounds on the rate of exponential decay of fiber correlations associated to the measures as the noise level goes to zero.

Sign in / Sign up

Export Citation Format

Share Document