On the transitivity of multifunctions and density of orbits in generalized topological spaces

Abstract

The connection between transitivity and existence of a dense orbit for multifunctions φ : X X in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topological spaces are investigated.

On Baire generalized topological spaces and some problems connected with discrete dynamical systems

“…In the paper [31] one can find the example showing that the converse statement is not true. However one can prove the following fact:…”

“…We shall say that A is a nowhere dense set if int γ (cl γ (A)) = / 0 (e.g. [6], [31]). It is easy to see that a subset of a nowhere dense set is nowhere dense.…”

“…More information about Baire spaces can be found in [31] and [25]. A deeper analysis of this topic is beyond the scope of this study, so we will omit it.…”

“…One can ask if we can consider, in the above theorem, a set of all points x ∈ X such that there exists a dense orbit of x under Φ instead of the set {x ∈ X : cl γ ( Θ a Φ (x)) = X}. The following example shows (see [31]) that answer to this question is negative.…”

“…This chapter is mainly based on the results contained in the papers [31], [30] and [25]. If we give statements from other sources, this will be marked by giving references to the relevant article.…”