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We study the properties of \begin{document}$ \Phi $\end{document} -irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of \begin{document}$ \Phi $\end{document} -irregular set in terms of entropy on chain recurrent classes and prove that \begin{document}$ \Phi $\end{document} -irregular sets of full entropy are typical. We also consider \begin{document}$ \Phi $\end{document} -level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

Recently Bernardes and Darji provided a very nice characterization of a residual set of maps of Cantor set in terms of covers of special type. Using their characterization, we provide a more direct description of this class. This way we are able to provide a further characterization of dynamical properties (e.g. shadowing properties, nullness) of maps in the class and further study of what features (e.g. prescribed minimal sets or the values of topological entropy) we can get by small perturbations of a given homeomorphism.

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

Motivated by a recent result of Ciesielski and Jasinski we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954.

Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph metric the hyperspace F1(X) is a continuum, and then for a topological graph X we show how, using the inverse limit approach of Barge and Martin, the inverse limit of a fuzzy dynamical system on X can be realized as an attractor of a fuzzy dynamical system on a manifold.

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that, for any integer$n\geq 1$, there is a continuum of topological dimension$n$supporting a transitive completely scrambled homeomorphism and an$n$-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15-year-old open problem.

The goal of this paper is to establish connections between two entropy concepts, namely the fuzzy entropy of a fuzzy set, which appears in fuzzy set theory, and the topological entropy of Zadeh's extensions of continuous maps defined on metric compact spaces, which is a topological dynamics concept. We develop several concepts that allow us to simultaneously address the two entropy concepts. Except for stating several basic properties, we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments. Such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp, discrete dynamical systems.

New model of linguistic variables based on fuzzy partition and its subpartitions.System allowing to choose something in between interpretability and precision.Rules extracted with various fuzzy partitions and fuzzy association measures.Prediction by fuzzy IF-THEN rules in implicative fuzzy inference engine. We propose a new generalized model of linguistic variables based on fuzzy partition and its subpartitions. We use this new model for mining relationships between linguistic variables (linguistic associations) from a data set. These relationships can be interpreted as fuzzy IF-THEN rules in the implicative fuzzy inference engine, which is an extended version of the implicative inference called Perception-based Logical Deduction. We show that our extension leads to statistically significant improvements with respect to the previous model used with the help of original and successful Perception-based Logical Deduction. We perform the comparison with different measures of rule quality and five datasets. We can obtain improvements in prediction precision while retaining the interpretability of the models. We also compare our method with the classical machine learning methods and obtain a similar quality of precision, which is very encouraging because interpretability usually leads to worse precision.

We show that for any map f on an arc-like continuum X, the induced map f ˆ on the hyperspace of subcontinua C ( X ) fixes a point in any f ˆ -invariant subcontinuum of C ( X ) . This extends a result of Robatian [21] , who proved it for the arc. However, as we show, the result does not extend to tree-like continua. We conclude with a list of related problems. Our proof builds on Hamilton's proof of the fixed point property of arc-like continua [12] .

In this paper we provide several results concerning the approximation of Zadeh's extension of a given function. We provide several simple observations due to which we provide a general procedure allowing to approximate Zadeh's extension of any continuous map. Among other things, we demonstrated that the technique of F-transform can also be used to approximate this extension. Further, we studied how the choice of the metric on the space of fuzzy sets (resp. fuzzy numbers) can affect the quality of the approximation, and we clarified that, in comparison to recent habits, the choice of the levelwise metric need not be the most appropriate one, and that the approximation process could be simplified. Our results are general in the sense that we mostly require nothing but the continuity of the original map.

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadeh's extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadeh's extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

Let X be a compact metric space , let φ be a continuous self-map on X, and let F ( X ) denote the space of fuzzy sets on X equipped with the levelwise topology. In this paper we study relations between a given (crisp) dynamical system ( X , φ ) and its g-fuzzification ( F ( X ) , Φ g ) . Among other things we study various (weak, strong, mild etc.) mixing properties and also several kinds of chaotic behaviors (Li–Yorke chaos, ω -chaos, distributional chaos, topological chaos etc.). We specified subspaces of fuzzy sets which makes sense to study further. Additionally, we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension, while many global characteristics are preserved in the opposite direction. Thus, among other things, we obtained easy-to-check criteria for checking absence of complex behavior in induced fuzzy dynamical systems.

This paper is a contribution to theoretical background of data mining, more precisely to fuzzy association analysis. We consider three the most commonly used confirmation measures and we study relations among found and known associations given by them. Good understanding of such relationships is necessary for creating more efficient algorithms or for subsequent work with found associations as well as for cooperation with the consumer of the data mining process. Even if our motivation to this work arose from mining of linguistic associations, found properties that coincide with semantics of mined associations are valid in general. Additionally, some examples showing how to use obtained properties are also contained in this paper.

It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e. a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e. periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by Roman-Flores and Chalco-Cano [Some chaotic properties of Zadeh's extension, Chaos, Solitons and Fractals 35(3) (2008) 452-459].

Let X denote a locally compact metric space and @f : X->X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X,@f), i.e., to obtain a map @F for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277-283] to generalize Zadeh's original extension principle. In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems. Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F ( x , y ) = ( f ( x ) , g x ( y ) ) , satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987–997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.

Fuzzy association analysis extracts relationships from data. The result of fuzzy association analysis depends on a chosen t-norm that is used for calculating confidence and support measures of mined association rules. We show that the set of mined association rules might change depending on the t-norm. We measure the distances of sets of mined rules with different t-norms and also with set of rules mined by crisp association analysis. We experiment with various datasets and partitioning methods to examine relationships of mined rules by different t-norms. Our experiments shed new light on application of fuzzy association mining and confirm that fuzzy association analysis usually brings signifficantly different results when compared to results given by crisp (non-fuzzy) association analysis.

Within this contribution we establish a theoretical background necessary for studying inverse limits of fuzzy dynamical systems induced by crisp (non-fuzzy) ones. First, we elaborate topological properties of the space of fuzzy sets, such as connectedness and compactness. Second, we focus our attention on dynamical conditions sufficient for the existence of indecomposable continua in the inverse limit space.

We show that for every positive integer k there exists an interval map $f:I\to I$ such that (1) f is Li-Yorke chaotic, (2) the inverse limit space $I_{f}=\lim_{\leftarrow}\{f,I\}$ does not contain an indecomposable subcontinuum, (3) f is $C^{k}$ -smooth, and (4) f is not $C^{k+1}$ -smooth. We also show that there exists a $C^{\infty}$ -smooth f that satisfies (1) and (2). This answers a recent question of Oprocha and the first author from (Proc. Am. Math. Soc. 143(8):3659-3670, 2015), where the result was proved for $k=0$ . Our study builds on the work of Misiurewicz and Smital of a family of zero entropy weakly unimodal maps. With the help of a result of Bennett, as well as Blokh’s spectral decomposition theorem, we are also able to show that each $I_{f}$ contains, for every integer i, a subcontinuum $C_{i}$ with the following two properties: (i) $C_{i}$ is $2^{i}$ -periodic under the shift homeomorphism, and (ii) $C_{i}$ is a compactification of a topological ray. Finally, we prove that the chaotic attractors we construct are topologically distinct from the one presented by P Oprocha and the first author.