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].

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 On the origin and development of some notions of entropy

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Francisco Balibrea
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Point Of View ◽  
Topological Dynamics ◽  
Theoretical Point ◽  
Compact Metric Space ◽  
Continuous Maps ◽  
And Control

AbstractDiscrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper we are dealing mainly with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn such notions have to evolve to other recent situations where it is necessary to give some extended versions of them adapted to the new problems.

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 Twist orbits for non continuous maps of degree one

1993 ◽  
Vol 47 (3) ◽  
pp. 415-426
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Discrete Dynamical Systems ◽  
Continuous Maps ◽  
Rotation Set

The existence of twist orbits and twist cycles with a given rotation number is considered for discrete dynamical systems generated by iteration of liftings of maps of the circle into itself. The class of maps for which such orbits exist for every number in the interior of the rotation set is extended to contain an important subclass of non-continuous maps.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS

2005 ◽  
Vol 15 (02) ◽  
pp. 547-555 ◽  
Author(s):  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Relevant Literature ◽  
Discrete Dynamical Systems ◽  
One Dimensional ◽  
Control Techniques ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Continuous Maps

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

On the Dynamics of the q-Deformed Gaussian Map

2020 ◽  
Vol 30 (08) ◽  
pp. 2030021
Author(s):  
J. Cánovas ◽  
M. Muñoz-Guillermo
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Chaotic Attractor ◽  
Schwarzian Derivative ◽  
Discrete Dynamical Systems ◽  
Coexistence Of Attractors ◽  
Wide Range ◽  
Gaussian Map ◽  
Parametric Region

Following the scheme inspired by Tsallis [Jagannathan & Sudeshna, 2005; Patidar & Sud, 2009], we study the Gaussian map and its [Formula: see text]-deformed version. We compute the topological entropies of the discrete dynamical systems which are obtained for both maps, the original Gaussian map and its [Formula: see text]-modification. In particular, we are able to obtain the parametric region in which the topological entropy is positive. The analysis of the sign of Schwarzian derivative and the topological entropy allow us a deeper analysis of the dynamics. We also highlight the coexistence of attractors, even if it is possible to determine a wide range of parameters in which one of them is a chaotic attractor.

ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1437-1438 ◽  
Author(s):  
SERGIĬ KOLYADA ◽  
LUBOMÍR SNOHA
Keyword(s):  
Dynamical System ◽  
Topological Space ◽  
Topological Entropy ◽  
Discrete Dynamical System ◽  
Topological Dynamics ◽  
Limit Sets ◽  
Compact Topological Space ◽  
Continuous Maps ◽  
Uniformly Convergent

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence [Formula: see text] of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence [Formula: see text] either forms an equicontinuous family of maps or is uniformly convergent. We also show that for any continuous maps f and g from a compact topological space into itself the topological entropies h(f ◦ g) and h(g ◦ f) are equal.

scholarly journals Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Qiuju Xing ◽  
Yuming Shi
Keyword(s):  
Banach Space ◽  
Topological Entropy ◽  
Homoclinic Orbits ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Bounded Set ◽  
Transversal Homoclinic Orbits ◽  
Continuous Maps ◽  
Dense Set

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

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