scholarly journals Invariant measures for interval maps without Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
JORGE OLIVARES-VINALES
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Critical Points ◽  
Invariant Measures ◽  
Full Measure ◽  
Interval Maps ◽  
Interval Map

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

UNIFORM LYAPUNOV EXPONENTS ARE REALISED BY ERGODIC INVARIANT MEASURES

2001 ◽  
Vol 01 (01) ◽  
pp. 113-126 ◽  
Author(s):  
HANS CRAUEL
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Differentiable Manifold ◽  
Invariant Set ◽  
Random Dynamical System ◽  
Lyapunov Dimension ◽  
Finite Dimensional ◽  
Random Invariant Set

Given a differentiable random dynamical system on a finite-dimensional differentiable manifold with a compact random invariant set, we show that there exists an ergodic invariant measure, supported by the invariant set, such that the leading Lyapunov exponent associated with this invariant measure equals the uniform Lyapunov exponent with respect to the invariant set. This is extended to sums of Lyapunov exponents and to the Lyapunov dimension of the set.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

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 A relation between Lyapunov exponents, Hausdorff dimension and entropy

1981 ◽  
Vol 1 (4) ◽  
pp. 451-459 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Unstable Manifold ◽  
Positive Lyapunov Exponent ◽  
Axiom A ◽  
Generic Points

AbstractFor an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

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.

DYNAMICAL CHARACTERISTICS OF DISCRETIZED CHAOTIC PERMUTATIONS

2002 ◽  
Vol 12 (10) ◽  
pp. 2087-2103 ◽  
Author(s):  
NAOKI MASUDA ◽  
KAZUYUKI AIHARA
Keyword(s):  
Invariant Measure ◽  
Periodic Orbit ◽  
Lyapunov Exponents ◽  
Chaos Theory ◽  
Chaotic Dynamics ◽  
Invariant Measures ◽  
Random Sequences ◽  
Dynamical Characteristics ◽  
Chaotic Sequences ◽  
Continuous State

Chaos theory has been applied to various fields where appropriate random sequences are required. The randomness of chaotic sequences is characteristic of continuous-state systems. Accordingly, the discrepancy between the characteristics of spatially discretized chaotic dynamics and those of original analog dynamics must be bridged to justify applications of digital orbits generated, for example, from digital computers simulating continuous-state chaos. The present paper deals with the chaotic permutations appearing in a chaotic cryptosystem. By analysis of cycle statistics, the convergence of the invariant measure and periodic orbit skeletonization, we show that the orbits in chaotic permutations are ergodic and chaotic enough for applications. In the consequence, the systematic differences in the invariant measures and in the Lyapunov exponents of two infinitesimally L∞-close maps are also investigated.

Invariant measures for Anosov maps with small holes

2000 ◽  
Vol 20 (4) ◽  
pp. 1007-1044 ◽  
Author(s):  
N. CHERNOV ◽  
R. MARKARIAN ◽  
S. TROUBETZKOY
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Escape Rate ◽  
Positive Lyapunov Exponent ◽  
Smooth Measure ◽  
Anosov Diffeomorphisms ◽  
Small Holes

We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure $\mu_+$. Here we show that the iterations of any initially smooth measure, after renormalization, converge to $\mu_+$. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.

scholarly journals Invariant measures for interval maps with critical points and singularities

2009 ◽  
Vol 221 (5) ◽  
pp. 1428-1444 ◽  
Author(s):  
Vítor Araújo ◽  
Stefano Luzzatto ◽  
Marcelo Viana
Keyword(s):  
Critical Points ◽  
Invariant Measures ◽  
Interval Maps

Backward continued fractions, Hecke groups and invariant measures for transformations of the interval

1996 ◽  
Vol 16 (6) ◽  
pp. 1241-1274 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
Andrew Haas
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Symbolic Dynamics ◽  
Continued Fractions ◽  
Geodesic Flow ◽  
Invariant Measures ◽  
Interval Maps ◽  
Hecke Groups ◽  
Type Group ◽  
New Type

AbstractWe develop a new type of backward continued fractions that can be associated to each Hecke-type group. We study its symbolic dynamics, and the corresponding interval maps and their invariant measures. These measures are infinite if and only if the corresponding groups are discrete. For the discrete Hecke groups the invariant measure is computed explicitly by studying the geodesic flow on the associated Riemann surface.

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