scholarly journals Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds

2016 ◽  
Vol 10 (02) ◽  
pp. 497-509 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Topological Entropy ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Positive Topological Entropy ◽  
Reeb Flows

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.

scholarly journals Reeb orbits that force topological entropy

2021 ◽  
pp. 1-44
Author(s):  
MARCELO R. R. ALVES ◽  
ABROR PIRNAPASOV
Keyword(s):  
Topological Entropy ◽  
Contact Homology ◽  
Positive Topological Entropy ◽  
Transverse Knots ◽  
Reeb Flows ◽  
Closed Contact ◽  
Forcing Theory ◽  
Phd Thesis ◽  
Legendrian Contact Homology

Abstract We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1602
Author(s):  
Jan Andres ◽  
Jerzy Jezierski
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Lower Estimate ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Admissible Pairs ◽  
Impulsive Dynamics ◽  
Differential Equations And Inclusions

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

Minimal self-joinings and positive topological entropy

1995 ◽  
Vol 120 (3-4) ◽  
pp. 205-222 ◽  
Author(s):  
F. Blanchard ◽  
E. Glasner ◽  
J. Kwiatkowski
Keyword(s):  
Topological Entropy ◽  
Minimal Self ◽  
Positive Topological Entropy

Stability of the maximal measure for piecewise monotonic interval maps

1997 ◽  
Vol 17 (6) ◽  
pp. 1419-1436 ◽  
Author(s):  
PETER RAITH
Keyword(s):  
Topological Entropy ◽  
Stability Condition ◽  
Finite Union ◽  
Small Perturbations ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Maximal Measure ◽  
Closed Intervals ◽  
Piecewise Monotonic

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.

scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

scholarly journals Two closed orbits for non-degenerate Reeb flows

2020 ◽  
pp. 1-36
Author(s):  
MIGUEL ABREU ◽  
JEAN GUTT ◽  
JUNGSOO KANG ◽  
LEONARDO MACARINI
Keyword(s):  
Contact Manifold ◽  
Contact Manifolds ◽  
Symplectic Homology ◽  
Connected Sums ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Finite Quotient ◽  
Closed Contact ◽  
First Chern Class ◽  
Toric Contact Manifolds

Abstract We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

On curves in 3-dimensional normal almost contact metric manifolds

2020 ◽  
Vol 18 (01) ◽  
pp. 2150004
Author(s):  
Abdullah Yıldırım
Keyword(s):  
Contact Manifolds ◽  
3 Dimensional ◽  
And Topology ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds ◽  
Frenet Curves

The characterization of curves plays an important role in both geometry and topology of almost contact manifolds. Olszak found the equation [Formula: see text] on normal almost contact manifolds. The pair [Formula: see text] denotes the type of these manifolds. In this study, we obtained the curvatures of non-geodesic Frenet curves on [Formula: see text]-dimensional normal almost contact manifolds without neglecting [Formula: see text] and [Formula: see text], and provided the results of their characterization. We exemplified these results with examples.

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