Transitive points in CR-dynamical systems

Revisiting variations in topological transitivity

“…The proof is complete if we show (4) =⇒ (5). But this is just a consequence of the Furstenberg Intersection Lemma for CR-dynamical systems, and we see that the proof is similar to that for continuous maps.…”

“…We recall basics from [1,2] where dynamical systems are presented by closed relations, aided with some more rigourous discussions in the recent articles [4,5]. Though, our definitions are directed to develop our theory on transitivity for CR-dynamical systems.…”

“…However, both [1] and [2] overlook any study of the important and one of the oldest dynamical property of 'transitivity' for dynamics given by closed relations. Though, in some way such a study has been made recently in [4,5,6].…”

“…The another part is clear from that fact that for every n ∈ N and x−1 ∈ T − G (x), π n+1 (x) ∈ G −n (x). ( 6) =⇒ (5) The proof is clear from the fact that y ∈ G −n (x) implies that there is a backward orbit x−1 ∈ T − G (x) such that y = π (n+1) (x). ( 5) ⇐⇒ (7).…”