An extension of the ergodic closing lemma

2009 ◽  
Vol 30 (3) ◽  
pp. 773-808 ◽  
Author(s):  
SHUHEI HAYASHI
Keyword(s):  
Lyapunov Exponents ◽  
Small Perturbation ◽  
Approximation Theorem ◽  
Ergodic Measure ◽  
Homoclinic Tangency ◽  
Hyperbolic Structure ◽  
Extended Version ◽  
Closing Lemma ◽  
One Dimensional ◽  
Ergodic Measures

AbstractAn extended version of the ergodic closing lemma of Mañé is proved. As an application, we show that, C1 densely in the complement of the closure of Morse–Smale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure (dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and one-dimensional center direction when zero Lyapunov exponents are involved) over the supports of all non-atomic ergodic measures. As another application, we prove an approximation theorem, which claims that approximating the Lyapunov exponents of any non-atomic ergodic measure by those of an atomic ergodic measure by a C1 small perturbation is possible.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

scholarly journals Measure-theoretic chaos

2012 ◽  
Vol 34 (1) ◽  
pp. 110-131 ◽  
Author(s):  
TOMASZ DOWNAROWICZ ◽  
YVES LACROIX
Keyword(s):  
Ergodic Measure ◽  
Uniform Measure ◽  
Ergodic System ◽  
Topological System ◽  
Ergodic Measures ◽  
Topological Chaos ◽  
Probability Spaces ◽  
Entropy Zero ◽  
Standard Probability

AbstractWe define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

Poincaré Recurrences in a Nonautonomous Chaotic Map

2014 ◽  
Vol 24 (08) ◽  
pp. 1440016 ◽  
Author(s):  
Yaroslav Boev ◽  
Nadezhda Semenova ◽  
Galina Strelkova ◽  
Vadim Anishchenko
Keyword(s):  
Lyapunov Exponents ◽  
Distribution Density ◽  
Chaotic Map ◽  
Nonautonomous System ◽  
Harmonic Force ◽  
One Dimensional ◽  
The Relationship

The statistics of Poincaré recurrences is studied numerically in a one-dimensional cubic map in the presence of harmonic and noisy excitations. It is shown that the distribution density of Poincaré recurrences is periodically modulated by the harmonic force. It is established that the relationship between the Afraimovich–Pesin dimension and Lyapunov exponents is violated in the nonautonomous system.

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.

scholarly journals Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

2009 ◽  
Vol 29 (5) ◽  
pp. 1479-1513 ◽  
Author(s):  
LORENZO J. DÍAZ ◽  
ANTON GORODETSKI
Keyword(s):  
Lyapunov Exponents ◽  
Residual Subset ◽  
Homoclinic Class ◽  
Zero Measure ◽  
Ergodic Measures ◽  
Homoclinic Classes

AbstractWe prove that there is a residual subset 𝒮 in Diff1(M) such that, for everyf∈𝒮, any homoclinic class offcontaining saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure off.

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