scholarly journals Entropies and volume growth of unstable manifolds

2021 ◽  
pp. 1-15
Author(s):  
YUNTAO ZANG
Keyword(s):  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Upper Bound ◽  
Compact Manifold ◽  
Volume Growth ◽  
Ergodic Measure ◽  
Metric Entropy ◽  
Covering Numbers ◽  
Unstable Manifolds

Abstract Let f be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu $ . We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu $ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha },\,\alpha>0$ , case.

Dynamical upper bounds for Hausdorff dimension of invariant sets

1997 ◽  
Vol 17 (3) ◽  
pp. 739-756 ◽  
Author(s):  
YINGJIE ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Upper Bounds ◽  
Invariant Sets ◽  
Topological Pressure ◽  
Expanding Maps ◽  
Unstable Manifolds ◽  
Hyperbolic Sets

We study the Hausdorff dimension of invariant sets for expanding maps and that of hyperbolic sets on unstable manifolds. Upper bounds for the Hausdorff dimension are given in terms of topological pressure, or topological entropy and Lyapunov exponents.

Entropy and volume growth

2000 ◽  
Vol 20 (1) ◽  
pp. 77-84 ◽  
Author(s):  
KURT COGSWELL
Keyword(s):  
Growth Rate ◽  
Probability Measure ◽  
Topological Entropy ◽  
Compact Manifold ◽  
Volume Growth ◽  
Maximal Entropy ◽  
Measure Of Maximal Entropy

We consider a $C^{1+1}$ diffeomorphism $f$ of a compact manifold $M$ which preserves an ergodic probability measure $\mu$. We conclude that $\mu$-a.e. $x \in M$ is contained in a disk $D_x \subset W^u(x)$, with $D_x$ open in the $W^u(x)$ topology, which exhibits an exponential volume growth rate greater than or equal to the measure-theoretic entropy of $f$ with respect to $\mu$. Drawing on results of Newhouse and Yomdin, we then find that when $f$ is $C^\infty$ and $\mu$ is a measure of maximal entropy, this exponential volume growth rate equals the topological entropy of $f$ for $\mu$-a.e. $x$.

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 Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

2014 ◽  
Vol 15 (3) ◽  
pp. 712-732
Author(s):  
Eric Bourgain-Chang
Keyword(s):  
Lyapunov Exponents ◽  
Spectral Theory ◽  
Upper Bound ◽  
Shift Operator ◽  
Numerical Study ◽  
Striking Difference ◽  
Spectral Features ◽  
Random Models

AbstractIn this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper’s equation. This study is motivated by various conjectures on the spectral theory of these ‘pseudo-random’ models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS

1996 ◽  
Vol 06 (05) ◽  
pp. 919-948 ◽  
Author(s):  
D. TURAEV
Keyword(s):  
Lyapunov Exponents ◽  
Center Manifold ◽  
Critical Dimension ◽  
Lyapunov Dimension ◽  
Stable And Unstable Manifolds ◽  
Local Bifurcations ◽  
Non Local ◽  
Unstable Manifolds ◽  
The Difference ◽  
Manifold Theory

An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

scholarly journals Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

2019 ◽  
Vol 40 (9) ◽  
pp. 2552-2570
Author(s):  
SCOTT SCHMIEDING
Keyword(s):  
Asymptotic Behavior ◽  
Automorphism Group ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Shift Of Finite Type ◽  
Dimension Group ◽  
Bounded Below

Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and $\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to $\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of $\unicode[STIX]{x1D719}^{n}$. We give lower bounds on $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of $\unicode[STIX]{x1D719}$ on the dimension group associated to $(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy $h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of $\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general, $h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of $\unicode[STIX]{x1D719}$ on the dimension group.

scholarly journals A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 938
Author(s):  
Raymond Addabbo ◽  
Denis Blackmore
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Shannon Entropy ◽  
Shannon Information ◽  
Metric Entropy ◽  
Special Case

A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov–Sinai (metric) and Adler, Konheim & McAndrew (topological). In particular, metric entropy, with the imposition of some additional properties, is proven to be a special case of topological entropy and Shannon entropy is shown to be a particular form of metric entropy. This is the first of two papers aimed at establishing a dynamically grounded hierarchy comprising Clausius, Boltzmann, Gibbs, Shannon, metric and topological entropy in which each element is ideally a special case of its successor or some kind of limit thereof.

Local unstable entropies of partially hyperbolic diffeomorphisms

2019 ◽  
Vol 40 (8) ◽  
pp. 2274-2304
Author(s):  
WEISHENG WU
Keyword(s):  
Topological Entropy ◽  
Small Diameter ◽  
Conditional Entropy ◽  
Open Cover ◽  
Metric Entropy ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Topological Dynamical Systems ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

SOME RESULTS ON ENTROPY AND SEQUENCE ENTROPY

1999 ◽  
Vol 09 (09) ◽  
pp. 1731-1742 ◽  
Author(s):  
F. BALIBREA ◽  
V. JIMÉNEZ LÓPEZ ◽  
J. S. CÁNOVAS PEÑA
Keyword(s):  
Topological Entropy ◽  
Elementary Proof ◽  
Type Inequality ◽  
Topological Invariants ◽  
Metric Entropy ◽  
Sequence Entropy ◽  
Piecewise Monotone ◽  
Monotone Maps ◽  
Topological Sequence Entropy

In this paper we study some formulas involving metric and topological entropy and sequence entropy. We summarize some classical formulas satisfied by metric and topological entropy and ask the question whether the same or similar results hold for sequence entropy. In general the answer is negative; still some questions involving these formulas remain open. We make a special emphasis on the commutativity formula for topological entropy h(f ◦ g)=h(g ◦ f) recently proved by Kolyada and Snoha. We give a new elementary proof and use similar ideas to prove commutativity formulas for metric entropy and other topological invariants. Finally we prove a Misiurewicz–Szlenk type inequality for topological sequence entropy for piecewise monotone maps on the interval I=[0, 1]. For this purpose we introduce the notion of upper entropy.

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