A variation on the variational principle and applications to entropy pairs

1997 ◽  
Vol 17 (1) ◽  
pp. 29-43 ◽  
Author(s):  
F. BLANCHARD ◽  
E. GLASNER ◽  
B. HOST
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Open Cover ◽  
Topological Dynamical System ◽  
Positive Topological Entropy ◽  
Finite Open Cover

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

Zero-entropy invariant measures for skew product diffeomorphisms

2009 ◽  
Vol 30 (3) ◽  
pp. 923-930 ◽  
Author(s):  
PENG SUN
Keyword(s):  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Affirmative Answer ◽  
Skew Product ◽  
Hyperbolic Structure ◽  
Combinatorial Properties ◽  
Positive Topological Entropy ◽  
Zero Entropy ◽  
Uniquely Ergodic

AbstractIn this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

CONTROLLING CHAOS AND THE INVERSE FROBENIUS–PERRON PROBLEM: GLOBAL STABILIZATION OF ARBITRARY INVARIANT MEASURES

2000 ◽  
Vol 10 (05) ◽  
pp. 1033-1050 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Stochastic Matrix ◽  
Open Loop ◽  
Global Stabilization ◽  
Invariant Density ◽  
Controlling Chaos

The inverse Frobenius–Perron problem (IFPP) is a global open-loop strategy to control chaos. The goal of our IFPP is to design a dynamical system in ℜn which is: (1) nearby the original dynamical system, and (2) has a desired invariant density. We reduce the question of stabilizing an arbitrary invariant measure, to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. We present a generalization of Baker maps with an arbitrary grammar and whose FP operator is the required stochastic matrix.

scholarly journals Bounded complexity, mean equicontinuity and discrete spectrum

2019 ◽  
Vol 41 (2) ◽  
pp. 494-533 ◽  
Author(s):  
WEN HUANG ◽  
JIAN LI ◽  
JEAN-PAUL THOUVENOT ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Topological Dynamics ◽  
Topological Dynamical System ◽  
The Mean ◽  
Bounded Complexity

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

scholarly journals Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

2020 ◽  
pp. 1-34
Author(s):  
M. KESSEBÖHMER ◽  
J. D. M. RADEMACHER ◽  
D. ULBRICH
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Travelling Waves ◽  
Excitable Media ◽  
Skew Product ◽  
Invariant Subset ◽  
One Dimensional ◽  
Markov System ◽  
Positive Topological Entropy ◽  
Shift Dynamics

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$ .

scholarly journals Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems

2020 ◽  
pp. 2150021
Author(s):  
Xinsheng Wang ◽  
Weisheng Wu ◽  
Yujun Zhu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Topological Entropy ◽  
Equilibrium States ◽  
Metric Entropy ◽  
Unstable Foliation ◽  
Hyperbolic Dynamical Systems ◽  
Partially Hyperbolic

Let [Formula: see text] be a [Formula: see text] random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of [Formula: see text] on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs [Formula: see text]-states are investigated.

scholarly journals Generic points of shift-invariant measures in the countable symbolic space

2018 ◽  
Vol 166 (2) ◽  
pp. 381-413
Author(s):  
AI–HUA FAN ◽  
MING–TIAN LI ◽  
JI–HUA MA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Invariant Measure ◽  
Gibbs Measure ◽  
Relative Entropy ◽  
Invariant Measures ◽  
Gibbs Measures ◽  
Symbolic Space ◽  
Entropy Dimension ◽  
Generic Points

AbstractWe are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

CONSTRUCTING INVARIANT MEASURES FROM DATA

1995 ◽  
Vol 05 (04) ◽  
pp. 1181-1192 ◽  
Author(s):  
GARY FROYLAND ◽  
KEVIN JUDD ◽  
ALISTAIR I. MEES ◽  
DAVID WATSON ◽  
KENJI MURAO
Keyword(s):  
Dynamical System ◽  
Experimental Data ◽  
Phase Space ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Reconstruction Technique ◽  
Continuous Approximation ◽  
Absolutely Continuous ◽  
Finite Set

We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may make our approximation to an invariant measure of the reconstructed system as accurate as we wish. Our method provides us with both a singular and an absolutely continuous approximation, so that the most suitable representation may be chosen for a particular problem.

Entropy pairs for a measure

1995 ◽  
Vol 15 (4) ◽  
pp. 621-632 ◽  
Author(s):  
F. Blanchard ◽  
B. Host ◽  
A. Maass ◽  
S. Martinez ◽  
D. J. Rudolph
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Topological Dynamical System ◽  
Entropy Pair ◽  
Uniquely Ergodic

AbstractWe define entropy pairs for an invariant measure µ on a topological dynamical system (X, T), and show they allow one to construct the maximal topological factorwith entropy 0 for µ. Then we prove that for any µ, a µ-entropy pair is always topologically so, and the reverse is true when (X, T) is uniquely ergodic.

scholarly journals Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators

2010 ◽  
Vol 31 (4) ◽  
pp. 995-1042 ◽  
Author(s):  
A. B. ANTONEVICH ◽  
V. I. BAKHTIN ◽  
A. V. LEBEDEV
Keyword(s):  
Dynamical System ◽  
Markov Chain ◽  
Variational Principle ◽  
Invariant Measures ◽  
Variational Principles ◽  
Weighted Shift ◽  
Explicit Description ◽  
Shift Operators ◽  
Topological Markov Chain ◽  
Dual Object

AbstractThe paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for anarbitrarydynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—thet-entropy.

scholarly journals A LOCAL VARIATIONAL PRINCIPLE FOR RANDOM BUNDLE TRANSFORMATIONS

2013 ◽  
Vol 13 (03) ◽  
pp. 1250023 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical System ◽  
Variational Inequality ◽  
Variational Principle ◽  
Topological Entropy ◽  
Random Dynamical System ◽  
Local Entropy ◽  
Local Measure ◽  
Entropy Formula

We introduce local topological entropy [Formula: see text] and two kinds of local measure-theoretic entropy [Formula: see text] and [Formula: see text] for random bundle transformations. We derive a variational inequality of random local entropy for [Formula: see text]. As an application of such relation we prove a local variational principle in random dynamical system.

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