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 Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

2021 ◽  
pp. 1-40
Author(s):  
ALENA ERCHENKO
Keyword(s):  
Lyapunov Exponent ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Maximal Entropy ◽  
Anosov Diffeomorphism ◽  
Dynamical Invariants ◽  
Area Preserving ◽  
Measure Of Maximal Entropy

Abstract We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

scholarly journals Volume growth and closed geodesics on Riemannian manifolds of hyperbolic type

2005 ◽  
Vol 2005 (6) ◽  
pp. 875-893
Author(s):  
Jean-Pierre Ezin ◽  
Carlos Ogouyandjou
Keyword(s):  
Growth Rate ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Volume Growth ◽  
Growth Function ◽  
Hyperbolic Type ◽  
Compact Manifolds ◽  
Closed Geodesics ◽  
Geodesic Spheres

We study the volume growth function of geodesic spheres in the universal Riemannian covering of a compact manifold of hyperbolic type. Furthermore, we investigate the growth rate of closed geodesics in compact manifolds of hyperbolic type.

scholarly journals Intrinsic ergodicity via obstruction entropies

2013 ◽  
Vol 34 (6) ◽  
pp. 1816-1831 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
DANIEL J. THOMPSON
Keyword(s):  
Topological Entropy ◽  
Maximal Entropy ◽  
Unique Measure ◽  
Measure Of Maximal Entropy

AbstractBowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

scholarly journals A point is normal for almost all maps βx+α mod 1 or generalized β-transformations

2009 ◽  
Vol 29 (5) ◽  
pp. 1529-1547 ◽  
Author(s):  
B. FALLER ◽  
C.-E. PFISTER
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Critical Orbit ◽  
Maximal Entropy ◽  
Tent Map ◽  
Almost All ◽  
Analytic Curves ◽  
Measure Of Maximal Entropy

AbstractWe consider the map Tα,β(x):=βx+α mod 1, which admits a unique probability measure μα,β of maximal entropy. For x∈[0,1], we show that the orbit of x is μα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit of x=0 is μα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalized β-transformations (in particular, the tent map). We show that the critical orbit x=1 is normal with respect to the measure of maximal entropy for almost all β.

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.

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

Numerical Analysis of the Effect of Diffusion and Creep Flow on Cavity Growth

2007 ◽  
Author(s):  
Frederick W. Brust ◽  
Joonyoung Oh
Keyword(s):  
Growth Rate ◽  
Numerical Method ◽  
Cavity Growth ◽  
Volume Growth ◽  
Numerical Results ◽  
Shape Evolution ◽  
Cavity Volume ◽  
Cavity Shape ◽  
Cavity Growth Rate ◽  
Fully Coupled

In this paper, intergranular cavity growth in regimes, where both surface diffusion and deformation enhanced grain boundary diffusion are important, is studied. In order to continuously simulate the cavity shape evolution and cavity growth rate, a fully-coupled numerical method is proposed. Based on the fully-coupled numerical method, a gradual cavity shape change is predicted and this leads to an adverse effect on the cavity growth rates. As the portion of the cavity volume growth due to jacking and viscoplastic deformation in the total cavity volume growth increases, the initially spherical cavity evolves to V-shaped cavity. The numerical results are physically more realistic compared to results in the previous studies. The present numerical results suggest that the cavity shape evolution and cavity growth rate based on an assumed cavity shape, whether spherical or crack-like, cannot be used in this regime due to transitional coupled growth mechanisms.

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