scholarly journals MINIMAL GEODESIC FOLIATION ON T2 IN CASE OF VANISHING TOPOLOGICAL ENTROPY

2011 ◽  
Vol 03 (04) ◽  
pp. 511-520 ◽  
Author(s):  
EVA GLASMACHERS ◽  
GERHARD KNIEPER
Keyword(s):  
Topological Entropy ◽  
Rotation Number ◽  
Geodesic Flow ◽  
Universal Cover ◽  
Low Complexity ◽  
Asymptotic Direction ◽  
Geodesic Foliation ◽  
Minimal Geodesic

On a Riemannian 2-torus (T2, g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper [12] we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all r ∈ ℝ ∪ {∞} the universal cover ℝ2 is foliated by minimal geodesics of rotation number r. For irrational r ∈ ℝ all geodesics are minimal, for rational r ∈ ℝ ∪ {∞} all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

Polynomial Entropy of the Logistic Map

2021 ◽  
Vol 58 (2) ◽  
pp. 206-215
Author(s):  
Milan Perić
Keyword(s):  
Topological Entropy ◽  
Logistic Map ◽  
Low Complexity ◽  
Almost All

We study the polynomial entropy of the logistic map depending on a parameter, and we calculate it for almost all values of the parameter. We show that polynomial entropy distinguishes systems with a low complexity (i.e. for which the topological entropy vanishes).

A variation formula for the topological entropy of convex-cocompact manifolds

2010 ◽  
Vol 31 (6) ◽  
pp. 1849-1864 ◽  
Author(s):  
SAMUEL TAPIE
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Formula ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Kleinian Group ◽  
Limit Set ◽  
Analytic Family ◽  
Variation Formula ◽  
Sectional Curvatures ◽  
Convex Cocompact

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

TOPOLOGICAL ENTROPY VERSUS GEODESIC ENTROPY

1994 ◽  
Vol 05 (02) ◽  
pp. 213-218 ◽  
Author(s):  
GABRIEL P. PATERNAIN ◽  
MIGUEL PATERNAIN
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Exponential Growth ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Betti Numbers ◽  
Exponential Growth Rate ◽  
Sphere Bundle ◽  
Space Of Paths ◽  
Simply Connected

Using Yomdin's Theorem [8], we show that for a compact Riemannian manifold M, the geodesic entropy — defined as the exponential growth rate of the average number of geodesic segments between two points — is ≤ the topological entropy of the geodesic flow of M. We also show that if M is simply connected and N ⊂ M is a compact simply connected submanifold, then the exponential growth rate of the sequence given by the Betti numbers of the space of paths starting in N and ending in a fixed point of M, is bounded above by the topological entropy of the geodesic flow on the normal sphere bundle of N.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

1997 ◽  
Vol 17 (5) ◽  
pp. 1043-1059 ◽  
Author(s):  
KEITH BURNS ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Positive Measure ◽  
Geodesic Flows ◽  
Open Set ◽  
Negative Questions

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].

scholarly journals An example of how the Ricci flow can increase topological entropy

2007 ◽  
Vol 27 (6) ◽  
pp. 1919-1932 ◽  
Author(s):  
DAN JANE
Keyword(s):  
Numerical Integration ◽  
Topological Entropy ◽  
Ricci Flow ◽  
Geodesic Flow ◽  
Rotational Symmetry ◽  
Melnikov Method ◽  
Flow Perturbation ◽  
Surface Of Revolution

AbstractWe give a surface for which the Ricci flow applied to the metric will increase the topological entropy of the geodesic flow. Specifically, we first adapt the Melnikov method to apply to a Ricci flow perturbation and then we construct a surface which is closely related to a surface of revolution, but does not quite have rotational symmetry. This is done by adapting the Liouville metric representation of a surface of revolution. The final steps of the Melnikov method require numerical integration.

Minimal entropy and Mostow's rigidity theorems

1996 ◽  
Vol 16 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gérard Besson ◽  
Gilles Courtois ◽  
Sylvestre Gallot
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Metric Space ◽  
Universal Cover ◽  
Compact Metric Space ◽  
Entropy Functional ◽  
Volume Entropy ◽  
Image Position ◽  
A Minor ◽  
Special Metrics

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

scholarly journals Exceptional minimal sets of C1+α-group actions on the circle

1991 ◽  
Vol 11 (3) ◽  
pp. 455-467 ◽  
Author(s):  
S. Hurder
Keyword(s):  
Growth Rate ◽  
Fixed Point ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Closing Lemma ◽  
Finitely Generated ◽  
Exceptional Set ◽  
Minimal Set ◽  
Finitely Generated Group ◽  
Positive Topological Entropy

AbstractWe prove two extensions of Sacksteder's Theorem for the action A: Γ × S1 → S1 of a finitely-generated group Γ on the circle by C1+α-diffeomorphisms. If the action A has an exceptional minimal set K with a gap endpoint of exponential orbit growth rate, or if the action A on K has positive topological entropy, then the exceptional set K is hyperbolic. That is, A has a linearly contracting fixed-point in K. A key point of the paper is to prove a foliation closing lemma using the foliation geodesic flow technique.

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