Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

1992 ◽  
Vol 12 (1) ◽  
pp. 13-37 ◽  
Author(s):  
Michael Benedicks ◽  
Lai-Sang Young
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Small Random Perturbations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

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.

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

2015 ◽  
Vol 36 (6) ◽  
pp. 1865-1891 ◽  
Author(s):  
STEFANO GALATOLO ◽  
ISAIA NISOLI
Keyword(s):  
Invariant Measures ◽  
Wasserstein Distance ◽  
Continuous Invariant Measure ◽  
Local Dimension ◽  
Physical Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Rigorous Computation ◽  
Absolutely Continuous Invariant

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

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 RETURN TIME STATISTICS OF INVARIANT MEASURES FOR INTERVAL MAPS WITH POSITIVE LYAPUNOV EXPONENT

2009 ◽  
Vol 09 (01) ◽  
pp. 81-100 ◽  
Author(s):  
HENK BRUIN ◽  
MIKE TODD
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measures ◽  
Return Time ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Normal Fluctuations ◽  
Return Time Statistics ◽  
Absolutely Continuous Invariant

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a "polynomial Gibbs property" for these systems, and that the convergence to the entropy in the Ornstein–Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.

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.

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 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.

scholarly journals Another proof of Jakobson's Theorem and related results

1988 ◽  
Vol 8 (1) ◽  
pp. 93-109 ◽  
Author(s):  
Marek Ryszard Rychlik
Keyword(s):  
Invariant Measure ◽  
Positive Measure ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractThe author shows that any family C2-close to fα(x) = 1 − αx2(2 − ε ≤ α ≤ 2) satisfies Jakobson's theorem: For a positive measure set of α the transformation fα has an absolutely continuous invariant measure. He also indicates some generalizations.

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 ∈ .

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