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The natural consequence of the existence of different kinds of chaos is the study of their mutual dependence and the relationship between these concepts and the entropy of systems. This observation also applies to the local approach to this issue. In this article, we will focus on this problem in the context of “points focusing chaos.” We aim to show their mutual independence by considering the sets of appropriate periodic dynamical systems in the space of discrete dynamical systems.

In this paper we consider the issues of local entropy for a finite family of generators (that generates the semigroup). Our main aim is to show that any continuous function can be approximated by s-chaotic family of generators.

The aim of the paper is to examine topological properties of disruptions of nonautonomous discrete dynamical system by other “close” systems. Our considerations will be connected with entropy of a system at a point and with stable points of a system.

In this paper, we investigate topological properties of the sets of almost continuous functions f: [0,1] → [0,1] which locally approximate a fixed function. This approximation is associated with the certain value of entropy of a function at a fixed point of the primal function.

At the end of the twentieth century, Á. Császár pointed out the possibility of the transition from topology to generalized topology and vice versa. One of the tools used for this purpose are semi-open sets. In this paper, we investigate properties of functions in GTS in the context of semi-open sets, and problems connected with transfer of properties of functions defined and assuming their values in primal spaces to functions considered as mappings in the spaces generated by primal spaces.

We investigate some properties of functions belonging to the Zahorski classes. In particular, we answer the question: Is it possible to find close to a function from ith Zahorski class (i = 1, 2, 3, 4, 5) a function belonging to exactly the same Zahorski class and having a special (local) properties connected with the entropy?

In this paper, we introduce a new notion: extremely chaotic point and, moreover, a dual notion: family of functions having extremely chaotic point property. The basic properties of such a kind of families of functions will be considered.

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: ConnC (which is a subfamily of the class Conn introduced in [Korczak-Kubiak. E.. Paw- lak. R.J.: Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions. Czechoslovak Math. J. 63 (2013). 679-700]). The main result of the paper Is a theorem saying that for any function ƒ ∈ ConnC and any point x0 ∈ Fix(ƒ) there exists a ring R ⊂ ConnC containing function ƒ and in the intersection of any “graph neighbourhood of ƒ” and “neighbourhood of ƒ in topology of uniform convergence”, one can find functions ξ,Ψ ∈ R having a strong entropy point y0 located close to the point x0 and being a discontinuity point of the function ξ and a continuity point of the function Ψ.

Cellulose aggregate fibrils (CAFs) with dimensions of 100,000×3000×300 nm from unbleached kraft pulp (KP), oxygen delignified KP (KPox.del), and fully bleached kraft pulp (BKP) were liberated by a series of high shear and fractionation operations. The CAFs served as microfibril model material to evaluate their dimensional and hygroexpansive behaviors when submitted to variable relative humidities (RHs). The atomic force microscopy images of CAFs from different sources of kraft fibers were obtained during a RH cycle from 50% RH to 78% RH and then to 21% RH while being kept at 23°C. The resulting images were analyzed to determine dimensional changes in length, its concurrent cross-sectional area, width, and height. The mean value of changes in length and width was in the range of 2.3–3.2% and 1.9–3.3%, respectively. The changes in area and height were in the range of 14.5–18.2% and 12.4–17.3%, respectively. The length of CAFs showed a negative correlation with RH, whereas cross-sectional area, width, and height changes correlated positively with RHs. In the out-of-plane direction, such as area and height, the hygroexpansivity was one order of magnitude larger than the in-plane hygroexpansivity, i.e., in terms of length and width.

Some results connected with dynamics of quasi-continuous functions or functions possessing a dense orbit and belonging to the class d-Conn are presented.