Almost totally disconnected minimal systems

Balibrea, Francisco; Downarowicz, Tomasz; Hric, Roman; Snoha, Ľubomír; Špitalský, Vladimír

doi:10.1017/s0143385708000540

Cited by 29 publications

(33 citation statements)

References 28 publications

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“…In this case, besides partial results in [Balibrea et al, 2003], a complete characterization has been given recently in [Balibrea et al, 2009] as a consequence of a more general result based on the new notion of almost totally disconnected spaces. A space X is almost totally disconnected if the set of its degenerate components is dense in X.…”

Section: Autonomous Dynamical Systems On Continuamentioning

confidence: 99%

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Recent Developments in Dynamical Systems: Three Perspectives

Balibrea

Caraballo

Kloeden

et al. 2010

Int. J. Bifurcation Chaos

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The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, nonautonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors.

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Section: Autonomous Dynamical Systems On Continuamentioning

confidence: 99%

“…A cantoroid is a compact metric and almost totally disconnected space without isolated points. With these ingredients we can now state the characterization in [Balibrea et al, 2009]. In addition, the following characterization was given in [Balibrea et al, 2009] [Nadler, 1995]).…”

Section: Autonomous Dynamical Systems On Continuamentioning

confidence: 99%

Recent Developments in Dynamical Systems: Three Perspectives

Balibrea

Caraballo

Kloeden

et al. 2010

Int. J. Bifurcation Chaos

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“…While irrational rotations do not have shadowing, odometers have this property and furthermore they are the only possible examples of infinite minimal systems with shadowing [23]. While there is no complete characterization of minimal systems, some sufficient conditions on the structure of spaces admitting minimal maps are avilable [7]. What is immediately visible is the richness of possible types of minimal systems.…”

Section: Introductionmentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.

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“…In [BDHSS09] this was generalized to local dendrites. In that connection the notion of a cantoroid was introduced.…”

Section: Introductionmentioning

confidence: 99%

“…Already on the circle each of the three cases can occur; here in the case (2) M is a Cantor set. Moreover, if X is a (local) dendrite containing a free interval as well as a non-degenerate nowhere dense subcontinuum then there is a cantoroid different from a Cantor set which intersects that free interval and contains that subcontinuum and so by [BDHSS09] it is a minimal set for some continuous selfmap of X. Hence, in the case (2) we cannot replace "cantoroid" by "Cantor set".…”

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confidence: 99%

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

Dirbák¹,

Snoha²,

Špitalský³

2012

Ergod. Th. Dynam. Sys.

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Abstract. We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A 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. IntroductionOne-dimensional dynamics became an object of wide interest in the middle of 1970's, some 10 years after Sharkovsky's theorem, when chaotic phenomena were discovered by Li and Yorke in dynamics of interval maps. As general references one can recommend the monographs [CE80], [BC92], [ALM00] and [dMvS93] (several motivations for studying one-dimensional dynamical systems are discussed in the introduction of [dMvS93]). The interval and circle dynamics are well understood. To extend/generalize the results to graph maps is sometimes quite easy, sometimes extremely difficult (for instance the characterization of the set of periods for graph maps is known only in very special cases). A good, more than 50 pages long survey of some topics in the dynamics of graph maps can be found in an appendix in [ALM00] (the second edition). Also the paper [Bl84] and the paper [Bl86] with its two continuations under the same title are a must for everybody who wishes to study the dynamics of graph maps.When working in one-dimensional topological dynamics it is natural to try to extend results from the interval to more general spaces. One usually extends them (or slight modifications of them) first to the circle/trees and then to general graphs (then perhaps also to special kinds of dendrites or to other one-dimensional continua; in the case of general dendrites often a counterexample can be found). The present paper suggests that sometimes another approach can be more fruitful -we show that some important facts from the topological dynamics on the interval/circle work on much more general spaces than graphs, namely on spaces containing an open part looking like an interval (we will call it a free interval ). It seems that the first result indicating that the presence of a free interval might have important dynamical consequences for the whole space was obtained as early as 1988 in [Ka88], for two other results see [AKLS99] and [HKO11]. However, they were isolated in a sense and apparently have not attracted much attention; a systematic study of the influence of a free interval on dynamical properties of a space has not been done yet. Based on the main results of the present paper, we believe that namely the class of spaces with a free interval is a natural candidate for possible extension of classical results of one-dimensional topological dynamics. Of course, not all of them can be carried over from the interval/circle to spaces with a free interval (and then trees/graphs naturally enter the scene as candidates for possible extension). However, 2010 Mathematics Subj...

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Almost totally disconnected minimal systems

Cited by 29 publications

References 28 publications

Recent Developments in Dynamical Systems: Three Perspectives

Recent Developments in Dynamical Systems: Three Perspectives

Shadowing, entropy and minimal subsystems

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

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