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.

CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

2012 ◽  
Vol 22 (10) ◽  
pp. 1250259 ◽  
Author(s):  
ZDENĚK KOČAN
Keyword(s):  
Topological Entropy ◽  
Metric Spaces ◽  
Chaotic Behavior ◽  
Homoclinic Trajectory ◽  
One Dimensional ◽  
Distributional Chaos ◽  
Compact Metric Spaces ◽  
Continuous Maps

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability

2014 ◽  
Vol 24 (02) ◽  
pp. 1450016 ◽  
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Topological Entropy ◽  
Lyapunov Stability ◽  
Metric Spaces ◽  
Periodic Points ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Continuous Map ◽  
Peano Continuum ◽  
Continuous Maps ◽  
Lyapunov Instability

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

scholarly journals Connecting ergodicity and dimension in dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 451-462 ◽  
Author(s):  
C. D. Cutler
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Pointwise Dimension ◽  
Continuous Maps ◽  
Borel Probability ◽  
The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites

2010 ◽  
Vol 31 (1) ◽  
pp. 165-175 ◽  
Author(s):  
ZDENĚK KOČAN ◽  
VERONIKA KORNECKÁ-KURKOVÁ ◽  
MICHAL MÁLEK
Keyword(s):  
Topological Entropy ◽  
Open Problem ◽  
Homoclinic Trajectory ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Continuous Maps

AbstractIt is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

scholarly journals Топологическая унифицированная (r,s)-энтропия непрерывных отображений в квазиметрических пространствах

2021 ◽  
pp. 56-67
Author(s):  
R. Kazemi ◽  
M.R. Miri ◽  
G.R.M. Borzadaran
Keyword(s):  
Topological Entropy ◽  
Shannon Entropy ◽  
Metric Space ◽  
Metric Spaces ◽  
Recent Decade ◽  
Continuous Map ◽  
Continuous Maps ◽  
Separated Sets ◽  
Symmetric Property ◽  
Classical Entropy

The category of metric spaces is a subcategory of quasi-metric spaces. It is shown that the entropy of a map when symmetric properties is included is greater or equal to the entropy in the case that the symmetric property of the space is not considered. The topological entropy and Shannon entropy have similar properties such as nonnegativity, subadditivity and conditioning reduces entropy. In other words, topological entropy is supposed as the extension of classical entropy in dynamical systems. In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified $(r,s)$-entropy. In this paper, we extend the notion of unified $(r, s)$-entropy for the continuous maps of a quasi-metric space via spanning and separated sets. Moreover, we survey unified $(r, s)$-entropy of a map for two metric spaces that are associated with a given quasi-metric space and compare unified $(r, s)$-entropy of a map of a given quasi-metric space and the maps of its associated metric spaces. Finally we define Tsallis topological entropy for the continuous map on a quasi-metric space via Bowen's definition and analyze some properties such as chain rule.

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

1999 ◽  
Vol 09 (09) ◽  
pp. 1719-1729 ◽  
Author(s):  
LLUÍS ALSEDÀ ◽  
MOIRA CHAS ◽  
JAROSLAV SMÍTAL
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Limit Sets ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Countable Ordinal ◽  
Continuous Maps

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

Further discussion about transitivity and mixing of continuous maps on compact metric spaces

2020 ◽  
Vol 61 (11) ◽  
pp. 112701
Author(s):  
Guo Liu ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Anwar Waseem
Keyword(s):  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Continuous Maps

scholarly journals Topological entropy and periodic points of a factor of a subshift of finite type

1986 ◽  
Vol 104 ◽  
pp. 117-127 ◽  
Author(s):  
Takashi Shimomura
Keyword(s):  
Finite Type ◽  
Topological Entropy ◽  
Compact Space ◽  
Metric Spaces ◽  
Periodic Points ◽  
Subshift Of Finite Type ◽  
Continuous Map ◽  
Continuous Maps ◽  
Definition Of ◽  
Uniformly Continuous

Let X be a compact space and f be a continuous map from X into itself. The topological entropy of f, h(f), was defined by Adler, Konheim and McAndrew [1]. After that Bowen [4] defined the topological entropy for uniformly continuous maps of metric spaces, and proved that the two entropies coincide when the spaces are compact. The definition of Bowen is useful in calculating entropy of continuous maps.

CHAOTIC PROPERTIES OF SUBSHIFTS GENERATED BY A NONPERIODIC RECURRENT ORBIT

2000 ◽  
Vol 10 (05) ◽  
pp. 1067-1073 ◽  
Author(s):  
XIN-CHU FU ◽  
YIBIN FU ◽  
JINQIAO DUAN ◽  
ROBERT S. MACKAY
Keyword(s):  
Periodic Point ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Initial Conditions ◽  
Open Problems ◽  
Recurrent Point ◽  
Orbit Closures ◽  
Nonwandering Point ◽  
Sensitive Dependence ◽  
The One

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

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