Maximal entropy measures in dimension zero

2012 ◽  
Vol 127 (1) ◽  
pp. 55-66
Author(s):  
Dawid Huczek
Keyword(s):  
Maximal Entropy ◽  
Dimension Zero ◽  
Entropy Measures

Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

2011 ◽  
Vol 32 (1) ◽  
pp. 63-79 ◽  
Author(s):  
J. BUZZI ◽  
T. FISHER ◽  
M. SAMBARINO ◽  
C. VÁSQUEZ
Keyword(s):  
Hopf Bifurcation ◽  
Maximal Entropy ◽  
Open Set ◽  
Entropy Measures ◽  
Unique Measure ◽  
Anosov Systems ◽  
Anosov System ◽  
Partially Hyperbolic ◽  
Structurally Stable ◽  
Measure Of Maximal Entropy

AbstractWe show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

scholarly journals Discontinuity of topological entropy for Lozi maps

2011 ◽  
Vol 32 (5) ◽  
pp. 1783-1800 ◽  
Author(s):  
IZZET BURAK YILDIZ
Keyword(s):  
Topological Entropy ◽  
Small Neighborhood ◽  
Maximal Entropy ◽  
Affine Surface ◽  
Piecewise Affine ◽  
Lower Semi ◽  
Entropy Measures ◽  
Compact Case ◽  
Surface Homeomorphisms ◽  
Lozi Maps

AbstractRecently, Buzzi [Maximal entropy measures for piecewise affine surface homeomorphisms. Ergod. Th. & Dynam. Sys.29 (2009), 1723–1763] showed in the compact case that the entropy map f→htop(f) is lower semi-continuous for all piecewise affine surface homeomorphisms. We prove that topological entropy for Lozi maps can jump from zero to a value above 0.1203 as one crosses a particular parameter and hence it is not upper semi-continuous in general. Moreover, our results can be extended to a small neighborhood of this parameter showing the jump in the entropy occurs along a line segment in the parameter space.

Maximizing entropy measures for random dynamical systems

2016 ◽  
Vol 17 (05) ◽  
pp. 1750032
Author(s):  
Rafael A. Bilbao ◽  
Krerley Oliveira
Keyword(s):  
Dynamical Systems ◽  
Topological Degree ◽  
Integral Formula ◽  
Ergodic Measure ◽  
Random Dynamical Systems ◽  
Maximal Entropy ◽  
Local Diffeomorphism ◽  
Maximizing Measure ◽  
Entropy Measures ◽  
Uniform Expansion

We prove the existence of relative maximal entropy measures for certain random dynamical systems of the type [Formula: see text], where [Formula: see text] is an invertibe map preserving an ergodic measure [Formula: see text] and [Formula: see text] is a local diffeomorphism of a compact Riemannian manifold exhibiting some non-uniform expansion. As a consequence of our proofs, we obtain an integral formula for the relative topological entropy as the integral of the logarithm of the topological degree of [Formula: see text] with respect to [Formula: see text]. When [Formula: see text] is topologically exact and the supremum of the topological degree of [Formula: see text] is finite, the maximizing measure is unique and positive on open sets.

scholarly journals Maximal entropy measures for piecewise affine surface homeomorphisms

2009 ◽  
Vol 29 (6) ◽  
pp. 1723-1763 ◽  
Author(s):  
JÉRÔME BUZZI
Keyword(s):  
Periodic Points ◽  
Point Of View ◽  
Maximal Entropy ◽  
Probability Measures ◽  
Affine Surface ◽  
Piecewise Affine ◽  
Surface Diffeomorphisms ◽  
Positive Topological Entropy ◽  
Entropy Measures ◽  
Surface Homeomorphisms

AbstractWe study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ‘good’ returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.

D-test: A New Test for Analyzing Scale Invariance Using Symbolic Dynamics and Symbolic Entropy

Methodology ◽  
2011 ◽  
Vol 7 (3) ◽  
pp. 88-95 ◽  
Author(s):  
Jose A. Martínez ◽  
Manuel Ruiz Marín
Keyword(s):  
Symbolic Dynamics ◽  
Scale Invariance ◽  
Rating Scales ◽  
Rating Scale ◽  
Marketing Research ◽  
Response Patterns ◽  
Measurement Scales ◽  
Improve Measurement ◽  
Entropy Measures ◽  
The Difference

The aim of this study is to improve measurement in marketing research by constructing a new, simple, nonparametric, consistent, and powerful test to study scale invariance. The test is called D-test. D-test is constructed using symbolic dynamics and symbolic entropy as a measure of the difference between the response patterns which comes from two measurement scales. We also give a standard asymptotic distribution of our statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We applied D-test to a real marketing research to study if scale invariance holds when measuring service quality in a sports service. We considered a free-scale as a reference scale and then we compared it with three widely used rating scales: Likert-type scale from 1 to 5 and from 1 to 7, and semantic-differential scale from −3 to +3. Scale invariance holds for the two latter scales. This test overcomes the shortcomings of other procedures for analyzing scale invariance; and it provides researchers a tool to decide the appropriate rating scale to study specific marketing problems, and how the results of prior studies can be questioned.

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