Characterizations of $\omega$-limit sets in topologically hyperbolic systems
Andrew D. Barwell Chris Good Piotr Oprocha Brian E. Raines
It is well known that $\omega$-limit sets are internally chain transitive and have weak incompressibility; the converse is not generally true, in either case. However, it has been shown that a set is weakly incompressible if and only if it is an abstract $\omega$-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) $\omega$-limit set. In this paper we generalise these and other results, proving that the characterization for shifts of finite type holds in a variety of topologically hyperbolic systems (defined in terms of expansive and shadowing properties), and also show that the notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces.
keywords: shadowing $\omega$-limit set pseudo-orbit tracing property expansivity topologically hyperbolic. weak incompressibility Omega-limit set internal chain transitivity
Distributional chaos via isolating segments
Piotr Oprocha Pawel Wilczynski
Recently, Srzednicki and Wójcik developed a method based on Wazewski Retract Theorem which allows, via construction of so called isolating segments, a proof of topological chaos (positivity of topological entropy) for periodically forced ordinary differential equations. In this paper we show how to arrange isolating segments to prove that a given system exhibits distributional chaos. As an example, we consider planar differential equation

ż$=(1+e^{i \kappa t}|z|^2)\bar{z}$

for parameter values $0<\kappa \leq 0.5044$.

keywords: distributional chaos isolating segments fixed point index.
Topological mixing, knot points and bounds of topological entropy
Piotr Oprocha Paweł Potorski
In the paper we provide exact lower bounds of topological entropy in the class of transitive and mixing maps preserving the Lebesgue measure which are nowhere monotone (with dense knot points).
keywords: transitive knot point mixing ergodic. Entropy interval map
Dynamics of spacing shifts
John Banks Thi T. D. Nguyen Piotr Oprocha Brett Stanley Belinda Trotta
Spacing subshifts were introduced by Lau and Zame in 1973 to provide accessible examples of maps that are (topologically) weakly mixing but not mixing. Although they show a rich variety of dynamical characteristics, they have received little subsequent attention in the dynamical systems literature. This paper is a systematic study of their dynamical properties and shows that they may be used to provide examples of dynamical systems with a huge range of interesting dynamical behaviors. In a later paper we propose to consider in more detail the case when spacing subshifts are also sofic and transitive.
keywords: scrambled set. transitivity weak mixing chaotic pair Spacing subshift entropy
Chain recurrence in multidimensional time discrete dynamical systems
Piotr Oprocha
The author investigates the behavior of multidimensional time discrete dynamical systems. Problems of expansivity, P.O.T.P. and chain recurrence are considered in particular. The main result of this article is a general version of Spectral Decomposition Theorem.
keywords: chain recurrence hyperbolicity spectral decomposition. expansivity group action shadowing
Shadowing is generic---a continuous map case
Piotr Kościelniak Marcin Mazur Piotr Oprocha Paweł Pilarczyk
We prove that shadowing (the pseudo-orbit tracing property), periodic shadowing (tracing periodic pseudo-orbits with periodic real trajectories), and inverse shadowing with respect to certain families of methods (tracing all real orbits of the system with pseudo-orbits generated by arbitrary methods from these families) are all generic in the class of continuous maps and in the class of continuous onto maps on compact topological manifolds (with or without boundary) that admit a decomposition (including triangulable manifolds and manifolds with handlebody).
keywords: inverse shadowing semidynamical system Shadowing manifold pseudotrajectory periodic shadowing $C^{0}$ topology. continuous surjection genericity continuous map
Preface special issue: Advances and applications in qualitative studies of dynamics
Piotr Oprocha Alfred Peris
The idea of this special issue was to gather a number of articles on various mathematical tools in studies on qualitative aspects of dynamics. Of course not every important topic could be included in this issue due to space limitations. First of all we decided to focus on qualitative properties of topological dynamics by various tools coming from different fields such as functional and real analysis, measure-theory and topology itself. We aimed to present various aspects of such analysis, and when possible, present concrete applications of developed general (theoretical in nature) methodology. This should additionally highlight close connections between pure and applied mathematics.

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Specification properties and dense distributional chaos
Piotr Oprocha
The notion of distributional chaos was introduced by Schweizer and Smítal in [Trans. Amer. Math. Soc., 344 (1994) 737] for continuous maps of a compact interval. Further, this notion was generalized to three versions $d_1C$--$d_3C$ for maps acting on general compact metric spaces (see e.g. [Chaos Solitons Fractals, 23 (2005) 1581]). The main result of [ J. Math. Anal. Appl. , 241 (2000) 181] says that a weakened version of the specification property implies existence of a two points scrambled set which exhibits a $d_1 C$ version of distributional chaos. In this article we show that much more complicated behavior is present in that case. Strictly speaking, there exists an uncountable and dense scrambled set consisting of recurrent but not almost periodic points which exhibit uniform $d_1 C$ versions of distributional chaos.
keywords: distributional chaos specification property generalized specification property.
Coherent lists and chaotic sets
Piotr Oprocha
In this article we apply (recently extended by Kato and Akin) an elegant method of Iwanik (which adopts independence relations of Kuratowski and Mycielski) in the construction of various chaotic sets. We provide ''easy to track'' proofs of some known facts and establish new results as well. The main advantage of the presented approach is that it is easy to verify each step of the proof, when previously it was almost impossible to go into all the details of the construction (usually performed as an inductive procedure). Furthermore, we are able extend known results on chaotic sets in an elegant way. Scrambled, distributionally scrambled and chaotic sets with relation to various notions of mixing are considered.
keywords: chaotic set residual relation. scrambled set coherent list Distributional chaos
On averaged tracing of periodic average pseudo orbits
Piotr Oprocha Xinxing Wu

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same property as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicontinuous systems, and that it implies distributional chaos when measure center is nondegenerate.

keywords: Average shadowing property measure center mean equicontinuity distributional chaos distal pair

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