Transitive points via Furstenberg family

Abstract

Let ( X , T ) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z + with hereditary upward property). A point x ∈ X is called an F -transitive one if { n ∈ Z + : T n x ∈ U } ∈ F for every non-empty open subset U of X; the system ( X , T ) is called F -point transitive if there exists some F -transitive point. In this paper, we aim to classify transitive systems by F -point transitivity. Among other things, it is shown that ( X , T ) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive). It is shown that every weakly mixing system is F i p -point transitive, while we construct an F i p -point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ ⁎ ( F w t ) -transitive if and only if it is weakly disjoint from every P-system.