scholarly journals Circle maps having an infinite $\omega $-limit set which contains a periodic orbit have positive topological entropy

2003 ◽  
Vol 131 (11) ◽  
pp. 3547-3551
Author(s):  
Naotsugu Chinen
Keyword(s):  
Periodic Orbit ◽  
Topological Entropy ◽  
Limit Set ◽  
Circle Maps ◽  
Positive Topological Entropy ◽  
Omega Limit Set

Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity

1991 ◽  
Vol 11 (1) ◽  
pp. 115-128 ◽  
Author(s):  
J. Llibre ◽  
R. S. Mackay
Keyword(s):  
Periodic Orbit ◽  
Convex Hull ◽  
Topological Entropy ◽  
Rational Point ◽  
Rotation Vector ◽  
Connected Subset ◽  
Positive Topological Entropy ◽  
Rotation Vectors ◽  
Compact Connected Subset ◽  
Rotation Set

AbstractWe show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

On the finiteness of attractors for piecewise maps of the interval

2017 ◽  
Vol 39 (7) ◽  
pp. 1784-1804
Author(s):  
P. BRANDÃO ◽  
J. PALIS ◽  
V. PINHEIRO
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Critical Points ◽  
Limit Set ◽  
Periodic Attractors ◽  
Flat Maps ◽  
Omega Limit Set

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

Properties of Dynamical Systems on Dendrites and Graphs

2021 ◽  
Vol 31 (07) ◽  
pp. 2150100
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Limit Set ◽  
Open Problems ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Minimal Sets ◽  
Continuous Maps ◽  
Omega Limit Set

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

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.

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).

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

scholarly journals Chain recurrent points of a tree map

1999 ◽  
Vol 59 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Tao Li ◽  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Limit Set ◽  
Recurrent Point ◽  
Continuous Map ◽  
Tree Map

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if thew-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

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