Unpredictable points and stronger versions of Ruelle-Takens and Auslander-Yorke chaos (Q1755427)

The two main goals of the paper are: (1) The generalization of the notion of unpredictable point to a general semiflow \((T,X,\pi)\) consisting of a noncompact abelian topological monoid \(T\), a metric space \((X,d)\) and a continuous monoid action \(\pi:T\times X\rightarrow X\) such that \(\pi(0,x)=x\) and \(\pi(s,\pi(t,x))=\pi(s+t,x)\) for any \(s,t\in T\) and \(x\in X\). (2) The introduction of the so-called strong Ruelle-Takens chaoticity and strong Auslander-Yorke chaoticity; these notions are motivated by the notion of a Poincaré chaotic semiflow (a semiflow having an unpredictable transitive point). The author presents a series of results dealing with the relationship of these two types of strong chaoticity with the Poincaré chaoticity, the Devaney chaoticity, and involving the notion of unpredictable point. It can be noticed that the statements could not hold for all acting monoids, but the author has found large classes of monoids for which they apply (almost difference closed monoids, directional monoids, \textit{psp} monoids,...). Finally, it is worth mentioning that the paper is very well written and well structured.