Regionally proximal relation of order d along arithmetic progressions and nilsystems

Abstract

The regionally proximal relation of order d along arithmetic progressions, namely AP [d] for d ∈ N, is introduced and investigated. It turns out that if (X, T ) is a topological dynamical system with AP [d] = ∆, then each ergodic measure of (X, T ) is isomorphic to a d-step pro-nilsystem, and thus (X, T ) has zero entropy. Moreover, it is shown that if (X, T ) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic, then AP [d] = RP [d] for each d ∈ N. It follows that for a minimal ∞-pro-nilsystem, AP [d] = RP [d] for each d ∈ N. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

Topological characteristic factors and nilsystems

“…Finally, we confirm Conjecture 1.1 in [24]. In fact we show more, namely: Theorem F: Let (X , T ) be a minimal system which is an open extension of its maximal distal factor, then for any d ∈ N, AP [d] = RP [d] .…”

“…Since the averages in (1.2) is only related to τ d , it is natural to define a kind of regionally proximal relation of higher order, by using τ d directly. In [24] the Conjecture 1.1 of [24]: Let (X , T ) be a minimal distal system. Then RP [d] = AP [d] for d ∈ N.…”

A proof of Erdős's $B+B+t$ conjecture

“…The role played in this paper by Erdős progressions parallels the role played by Erdős cubes in in [14]. Various other notions of dynamical progressions, for example those in [8,14,10], have already been used for related questions, but the one we use does not seem to have been defined previously. We remark that in group rotations all the notions of dynamical progressions agree with the conventional notion of arithmetic progression.…”

Syndetic proximal set in topological transformation group

“…Proximally is one of the important subject in topological dynamics, where the proximal between points in different orbits and their dynamic properties are studied [3] . The idea of Proximally depends on the study of continuity and compaction between groups [4].In this paper been studied Proximally in the periodic points and the almost periodic point depende on syndetical set [4]. Relationship syndetical proximally point with some characteristic dynamical system (proximally point -extensively proximally point-almost periodic point-invariant set-minimal set-periodic point) presented ,also we studied syndetical proximally and extensively proximally independed on syndetical set and extensively set , and it will be give a necessary and sufficient condition syndetical proximally and extensively proximally to be proximally and we give some theorem a bout syndetical proximally point .We use symbol  to indication the end…”