Diagonal points having dense orbit

Abstract

Let f : X → X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ N, f × f 2 × • • • × f m : X m → X m is transitive, (ii) for each m ∈ N, there exists x ∈ X such that the diagonal m-tuple (x, x,. .. , x) has a dense orbit in X m under the action of f × f 2 × • • • × f m. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).

The sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities

“…It is universally acknowledged that if both of two discrete dynamical systems are syndetically transitive and weakly mixing, then the product dynamical system of them is also syndetically transitive and weakly mixing (see [6] for details). In the above result, we generalize the corresponding classical result to the dynamical systems of semigroup actions by making use of the topology on the semigroup S defined at the beginning of the paper.…”

Point transitivity, -transitivity and multi-minimality

“…, x) has a dense orbit in X m under the action of f × f 2 × · · · × f m . Following [26], we will say that (X, f ) is multi-transitive if it satisfies (1) and that (X, f ) is -transitive if it satisfies (2). It is shown in [26] that weak mixing, multi-transitivity and -transitivity are equivalent for minimal systems.…”

“…Following [26], we will say that (X, f ) is multi-transitive if it satisfies (1) and that (X, f ) is -transitive if it satisfies (2). It is shown in [26] that weak mixing, multi-transitivity and -transitivity are equivalent for minimal systems. Moothathu asked whether there are implications between multi-transitivity and weak mixing for general (non-minimal) systems.…”

“…It is shown in [26] that -transitivity implies weak mixing, but there exists a strongly mixing system which is not -transitive. Another natural problem is the following question.…”

“…In 2010, Moothathu [26] provided a simplified proof of Theorem 1.2 without resorting to the heavy machinery of the structure theory of minimal systems. He also asked whether (X, f ) should possess the following two properties: (1) for each m ∈ N, (X m , f × f 2 × · · · × f m ) is transitive; (2) for each m ∈ N, there exists a residual subset Y of X such that for every point…”