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We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$. [ABSTRACT FROM AUTHOR]

In the first parts of this paper we consider relationships between some aspects of continuity in generalized topological (metric) spaces. The last part is devoted to the dynamics of Darboux and quasi-continuous functions. The obtained results allow to answer the important questions connected with one dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted Conn*) contained between the families (widely described in literature) of Darboux Baire 1 functions (DB 1) and connectivity functions ( Conn). The solutions to our problems are based, among other, on the suitable construction of the ring, which turned out to be in some senses an “optimal construction”. These considerations concern mainly real functions defined on [0, 1] but in the last chapter we also extend them to the case of real valued iteratively H-connected functions defined on topological spaces. [ABSTRACT FROM AUTHOR]