tailieunhanh - Chaos-related properties on the product of semiflows

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. | Turk J Math (2017) 41: 1323 – 1336 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Chaos-related properties on the product of semiflows Alica MILLER∗, Chad MONEY Department of Mathematics, University of Louisville, Louisville, KY, USA Received: • Accepted/Published Online: • Final Version: Abstract: 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. Key words: Product of semiflows, directional monoid, psp monoid, sip monoid, Devaney chaos 1. Introduction In this paper we generalize the continuous versions of the following statements from the papers [3] by Deˇgirmenci and Ko¸cak (published in 2010) and [9] by Li and Zhou (published in 2013) to the case of the most general possible acting monoids: Lemmas 1–4 and Theorems 1–3 from [3], Lemmas (1), (1)–(3), (1)–(3), (1)–(3), (1)–(4) and Theorems , , from [9]. However, we have not explicitly stated all these statements as corollaries of our statements, especially when they are immediate corollaries. The papers in which the semiflows do not necessarily have continuous actions are relatively rare in the literature. In our own research we exclusively deal with continuous actions and so we assume the continuity of all the actions. In the papers [3] and [9] some actions are not assumed to be continuous. However, the papers [3] and [9] deal, respectively, with discrete (. N0 ) and continuous .