All the common notions about dynamics in cascades-topological transitivity, periodic points, sensitive dependence, and so forth-can be formulated in the context of a general abelian semiflow. Many intricate results, such as the redundancy of Devaney chaos, remain true (with very minor qualifications) in this wider context. However, when we examine general monoid actions on a product space, it turns out that the topological and algebraic structure of N 0 plays a large role in the preservation of chaotic properties. In order to obtain meaningful results in that arena, new ideas such as "directional" and "synnrec" are introduced, then applied. v TABLE OF CONTENTS
In this paper we generalize some results about the chaos-related properties on the product of two semiflows, which appeared in the literature in the last few years, to the case of the most general possible acting monoids. In order to do that we introduce some new notions, namely the notions of a directional, psp and sip monoid, and the notion of a strongly transitive semiflow. In particular, we obtain a sufficient condition for the Devaney chaoticity of a product, which works for the (very large) class of the psp acting monoids.
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