scholarly journals Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies

2006 ◽  
Vol 15 (1) ◽  
pp. 61-71 ◽  
Author(s):  
Yongluo Cao ◽  
◽  
Stefano Luzzatto ◽  
Isabel Rios ◽  
◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Hyperbolic Systems

On the convergence to equilibrium states for certain non-hyperbolic systems

1997 ◽  
Vol 17 (4) ◽  
pp. 977-1000 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Periodic Orbits ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Markov Property ◽  
Standard Technique ◽  
Equilibrium States ◽  
Convergence To Equilibrium ◽  
Invariant Density ◽  
Frobenius Operator ◽  
Uniformly Bounded

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

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.

scholarly journals Ergodic properties of invariant measures for C1+α non-uniformly hyperbolic systems

2012 ◽  
Vol 33 (2) ◽  
pp. 560-584 ◽  
Author(s):  
CHAO LIANG ◽  
WENXIANG SUN ◽  
XUETING TIAN
Keyword(s):  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Hyperbolic Case ◽  
Compact Spaces ◽  
Ergodic Properties ◽  
Fixed Set ◽  
Uniformly Hyperbolic Systems ◽  
Time Averages ◽  
Support Sets ◽  
Dirac Measures

AbstractFor every ergodic hyperbolic measure ω of a C1+α diffeomorphism, there is an ω-full-measure set $\tilde {\Lambda }$ (the union of $\tilde \Lambda _l=\mathrm {supp}( \omega |_{\Lambda _{l}})$, the support sets of ω on each Pesin block Λl, l=1,2,…) such that every non-empty, compact and connected subset $V\subseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ coincides with Vf(x), where $\mathcal {M}_{\mathrm {inv}}(\tilde {\Lambda })$ denotes the space of invariant measures supported on $\tilde {\Lambda }$ and Vf(x) denotes the accumulation set of time averages of Dirac measures supported at one orbit of some point x. For each fixed set V, the points with the above property are dense in the support supp (ω) . In particular, points satisfying $V_f(x)=\mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ are dense in supp (ω) . Moreover, if supp (ω) is isolated, the points satisfying $V_f(x)\supseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ form a residual subset of supp (ω) . These extend results of K. Sigmund [On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285–299] (see also M. Denker, C. Grillenberger and K. Sigmund [Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, Ch. 21]) from the uniformly hyperbolic case to the non-uniformly hyperbolic case. As a corollary, irregular + points form a residual set of supp (ω) .

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.

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