scholarly journals A Note on the Relation Between the Metric Entropy and the Generalized Fractal Dimensions of Invariant Measures

2021 ◽  
Author(s):  
Alexander Condori ◽  
Silas L. Carvalho
Keyword(s):  
Invariant Measures ◽  
Fractal Dimensions ◽  
Metric Entropy

The topological entropy of non-dense orbits and generalized Schmidt games

2017 ◽  
Vol 39 (2) ◽  
pp. 500-530
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Metric Entropy ◽  
Toral Automorphism ◽  
Partially Hyperbolic ◽  
Dense Orbits ◽  
Bounded Below ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

scholarly journals Invariant measures for higher-rank hyperbolic abelian actions

1996 ◽  
Vol 16 (4) ◽  
pp. 751-778 ◽  
Author(s):  
A. Katok ◽  
R. J. Spatzier
Keyword(s):  
Invariant Measures ◽  
Metric Entropy ◽  
Ergodic Measures ◽  
Anosov Actions ◽  
Higher Rank ◽  
Partially Hyperbolic

AbstractWe investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤkandWe show that they are either Haar measures or that every element of the action has zero metric entropy.

scholarly journals The Pesin entropy formula for diffeomorphisms with dominated splitting

2014 ◽  
Vol 35 (3) ◽  
pp. 737-761 ◽  
Author(s):  
ELEONORA CATSIGERAS ◽  
MARCELO CERMINARA ◽  
HEBER ENRICH
Keyword(s):  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Metric Entropy ◽  
Dominated Splitting ◽  
Dirac Delta ◽  
Asymptotic Statistics ◽  
Entropy Formula ◽  
Almost All

AbstractFor any$C^1$diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.

scholarly journals Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Silas L. Carvalho ◽  
Alexander Condori
Keyword(s):  
Fractal Dimension ◽  
Invariant Measure ◽  
Correlation Dimension ◽  
Invariant Measures ◽  
Fractal Dimensions ◽  
Full Shift ◽  
Generalized Dimensions ◽  
Topological Dynamical Systems ◽  
Dense Set ◽  
Exponential Metric

Abstract In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q > 0 {q>0} , zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system ( X , T ) {(X,T)} (where X = M ℤ {X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q > 1 {q>1} , infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s ∈ ( 0 , 1 ) {s\in(0,1)} and each q > 1 {q>1} , zero lower s-generalized and infinite upper q-generalized dimensions.

On preimage entropy, folding entropy and stable entropy

2020 ◽  
pp. 1-33 ◽  
Author(s):  
WEISHENG WU ◽  
YUJUN ZHU
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Invariant Measures ◽  
Single Point ◽  
Nonequilibrium Statistical Mechanics ◽  
Stat Phys ◽  
Metric Entropy ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Partially Hyperbolic

For non-invertible dynamical systems, we investigate how ‘non-invertible’ a system is and how the ‘non-invertibility’ contributes to the entropy from different viewpoints. For a continuous map on a compact metric space, we propose a notion of pointwise metric preimage entropy for invariant measures. For systems with uniform separation of preimages, we establish a variational principle between this version of pointwise metric preimage entropy and pointwise topological entropies introduced by Hurley [On topological entropy of maps. Ergod. Th. & Dynam. Sys.15 (1995), 557–568], which answers a question considered by Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113]. Under the same condition, the notion coincides with folding entropy introduced by Ruelle [Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys.85(1–2) (1996), 1–23]. For a $C^{1}$ -partially hyperbolic (non-invertible and non-degenerate) endomorphism on a closed manifold, we introduce notions of stable topological and metric entropies, and establish a variational principle relating them. For $C^{2}$ systems, the stable metric entropy is expressed in terms of folding entropy (namely, pointwise metric preimage entropy) and negative Lyapunov exponents. Preimage entropy could be regarded as a special type of stable entropy when each stable manifold consists of a single point. Moreover, we also consider the upper semi-continuity for both of pointwise metric preimage entropy and stable entropy and give a version of the Shannon–McMillan–Breiman theorem for them.

Fractal dimensions of dynamically triangulated random surfaces

1992 ◽  
Vol 2 (12) ◽  
pp. 2181-2190 ◽  
Author(s):  
Christian Münkel ◽  
Dieter W. Heermann
Keyword(s):  
Fractal Dimensions ◽  
Random Surfaces

Structure characterization of ground calcium carbonate flocs by fractal analysis and their effects on handsheet properties

TAPPI Journal ◽  
2013 ◽  
Vol 12 (3) ◽  
pp. 17-23 ◽  
Author(s):  
WANHEE IM ◽  
HAK LAE LEE ◽  
HYE JUNG YOUN ◽  
DONGIL SEO
Keyword(s):  
Fractal Analysis ◽  
Structural Characteristics ◽  
Fractal Dimensions ◽  
Strength Loss ◽  
Structure Characterization ◽  
Uniform Size ◽  
Cross Sectional ◽  
Floc Size ◽  
Mass Fractal ◽  
Handsheet Properties

Preflocculation of filler particles before their addition to pulp stock provides the most viable and practical solution to increase filler content while minimizing strength loss. The characteristics of filler flocs, such as floc size and structure, have a strong influence on preflocculation efficiency. The influence of flocculant systems on the structural characteristics of filler flocs was examined using a mass fractal analysis method. Mass fractal dimensions of filler flocs under high shear conditions were obtained using light diffraction spectroscopy for three different flocculants. A single polymer (C-PAM), a dual cationic polymer (p-DADMAC/C-PAM) and a C-PAM/micropolymer system were used as flocculants, and their effects on handsheet properties were investigated. The C-PAM/micropolymer system gave the greatest improvement in tensile index. The mass fractal analysis showed that this can be attributed to the formation of highly dense and spherical flocs by this flocculant. A cross-sectional analysis of the handsheets showed that filler flocs with more uniform size were formed when a C-PAM/micropolymer was used. The results suggest that a better understanding of the characteristics of preflocculated fillers and their influence on the properties of paper can be gained based on a fractal analysis.

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