Geodesic
On recurrence in $G$-spaces
Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 279-284.

Voir la notice de l'article provenant de la source Math-Net.Ru

We introduce and analyze the following general concept of recurrence. Let $G$ be a group and let $X$ be a G-space with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $\Delta_{\mathfrak{F}}(A)=\{g\in G\colon gB\subseteq A$ for some $B\in \mathfrak{F}$, $B\subseteq A\}$, and say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $R\bigcap \Delta_{\mathfrak{F}} (A)\neq\emptyset$ for each $A\in \mathfrak{F}$.
Keywords: $G$-space, recurrent subset, ultrafilters, Stone-Čech compactification. 
@article{ADM_2017_23_2_a9,
     author = {Igor Protasov and Ksenia Protasova},
     title = {On recurrence in $G$-spaces},
     journal = {Algebra and discrete mathematics},
     pages = {279--284},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a9/}
}
TY  - JOUR
AU  - Igor Protasov
AU  - Ksenia Protasova
TI  - On recurrence in $G$-spaces
JO  - Algebra and discrete mathematics
PY  - 2017
SP  - 279
EP  - 284
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a9/
LA  - en
ID  - ADM_2017_23_2_a9
ER  - 
%0 Journal Article
%A Igor Protasov
%A Ksenia Protasova
%T On recurrence in $G$-spaces
%J Algebra and discrete mathematics
%D 2017
%P 279-284
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a9/
%G en
%F ADM_2017_23_2_a9
Igor Protasov; Ksenia Protasova. On recurrence in $G$-spaces. Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 279-284. http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a9/

[1] V. Bergelson, N. Hindman, “Quotient sets and density recurrent sets”, Trans. Amer. Math. Soc., 364 (2012), 4495–4531 | DOI | MR | Zbl

[2] H. Furstenberg, “Poincare recurrence and number theory”, Bulletin Amer. Math. Soc., 5:3 (1981), 211–234 | DOI | MR | Zbl

[3] N. Hindman, D. Strauss, Algebra in the Stone-$\check{C}$ech compactification, de Gruyter, Berlin, 1998 | MR

[4] I. Protasov, “Filters and topologies on groups”, Mat. Stud., 3 (1994), 15–28 | MR | Zbl

[5] E. Reznichenko, O. Sipacheva, Discrete subsets in topologicaln groups and countable extremally disconnected groups, 2016, arXiv: 1608.03546v2

[6] P. Zakrzewski, “On the complexity of the ideal of absolute null sets”, Ukr. Math. J., 64 (2012), 306–308 | DOI | MR | Zbl