scholarly journals Minimality, transitivity, mixing and topological entropy on spaces with a free interval

2012 ◽  
Vol 33 (6) ◽  
pp. 1786-1812 ◽  
Author(s):  
MATÚŠ DIRBÁK ◽  
ĽUBOMÍR SNOHA ◽  
VLADIMÍR ŠPITALSKÝ
Keyword(s):  
Topological Entropy ◽  
Open Subset ◽  
Topological Structure ◽  
Open Interval ◽  
Topological Transitivity ◽  
Topological Mixing ◽  
Minimal Sets ◽  
Free Interval ◽  
Continuous Maps ◽  
Metrizable Spaces

AbstractWe study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.

Minimal Sets on Graphs and Dendrites

2003 ◽  
Vol 13 (07) ◽  
pp. 1721-1725 ◽  
Author(s):  
Francisco Balibrea ◽  
Roman Hric ◽  
L'ubomír Snoha
Keyword(s):  
Topological Structure ◽  
Partial Characterization ◽  
Minimal Set ◽  
Full Characterization ◽  
Minimal Sets ◽  
Continuous Maps

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

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.

scholarly journals Dynamics of homeomorphisms on minimal sets generated by triangular mappings

1999 ◽  
Vol 59 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Gian Luigi Forti ◽  
Luigi Paganoni ◽  
Jaroslav Smítal
Keyword(s):  
Topological Entropy ◽  
Minimal Set ◽  
Minimal Sets ◽  
Continuous Maps ◽  
Triangular Maps ◽  
Parametric Class

The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

scholarly journals Entropy on noncompact sets

2019 ◽  
Vol 7 (1) ◽  
pp. 29-37
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Topological Spaces ◽  
Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

Jℱ-Class Weighted Backward Shifts

2018 ◽  
Vol 28 (06) ◽  
pp. 1850076 ◽  
Author(s):  
Shengnan He ◽  
Yu Huang ◽  
Zongbin Yin
Keyword(s):  
Topological Entropy ◽  
Sequence Space ◽  
Topological Mixing ◽  
Basic Properties ◽  
Shift Operators ◽  
Recurrence Property ◽  
Backward Shifts

In this article [Formula: see text]-class operators are introduced and some basic properties of [Formula: see text]-vectors are given. The [Formula: see text]-class operators include the [Formula: see text]-class operators and [Formula: see text]-class operators introduced by Costakis and Manoussos in 2008. This class also includes the [Formula: see text]-class and [Formula: see text]-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the [Formula: see text]-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of [Formula: see text]-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.

scholarly journals Diagonals of separately continuous maps with values in box products

2018 ◽  
Vol 6 (1) ◽  
pp. 26-33
Author(s):  
Olena Karlova ◽  
Volodymyr Mykhaylyuk
Keyword(s):  
Connected Space ◽  
Continuous Map ◽  
Continuous Maps ◽  
Baire One ◽  
Metrizable Spaces

Abstract We prove that if X is a paracompact connected space and Z = ∏s∈S Zs is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map g : X → Z there exists a separately continuous map f : X2 → Z such that f (x, x) = g(x) for all x ∈ X.

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.

Minimizing topological entropy for continuous maps on graphs

2000 ◽  
Vol 20 (6) ◽  
pp. 1559-1576 ◽  
Author(s):  
LL. ALSEDÀ ◽  
F. MAÑOSAS ◽  
P. MUMBRÚ
Keyword(s):  
Topological Entropy ◽  
Invariant Set ◽  
Continuous Maps

We study the existence of models with minimal topological entropy among the class of all continuous maps from a given graph to itself with a fixed behavior on a given finite invariant set.

scholarly journals On topological entropy of triangular maps of the square

1993 ◽  
Vol 48 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Lluís Alsedà ◽  
Sergiĭ F. Kolyada ◽  
Ľubomír Snoha
Keyword(s):  
Topological Entropy ◽  
Lower Bounds ◽  
Lower Semicontinuous ◽  
Continuous Maps ◽  
Triangular Maps

We study the topological entropy of triangular maps of the square. We show that such maps differ from the continuous maps of the interval because there exist triangular maps of the square of “type 2∞” with infinite topological entropy. The set of such maps is dense in the space of triangular maps of “type at most 2∞” and the topological entropy as a function of the triangular maps of the square is not lower semicontinuous. However, we show that for these maps the characterisation of the lower bounds of the topological entropy depending on the set of periods is the same as for the continuous maps of the interval.

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