On recurrence in G-spaces
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×X⟶X, (g,x)⟼gx. For a family F of subset of X and A∈F, we denote ΔF(A)={g∈G:gB⊆A for some B∈F, B⊆A}, and say that a subset R of G is F-recurrent if R⋂ΔF(A)≠∅ for each A∈F....
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Інститут прикладної математики і механіки НАН України
2017
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | On recurrence in G-spaces / I.V. Protasov, K.D. Protasova // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 279-284. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1560222019-06-18T01:31:28Z On recurrence in G-spaces Protasov, I.V. Protasova, K.D. 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×X⟶X, (g,x)⟼gx. For a family F of subset of X and A∈F, we denote ΔF(A)={g∈G:gB⊆A for some B∈F, B⊆A}, and say that a subset R of G is F-recurrent if R⋂ΔF(A)≠∅ for each A∈F. 2017 Article On recurrence in G-spaces / I.V. Protasov, K.D. Protasova // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 279-284. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC:37A05, 22A15, 03E05. http://dspace.nbuv.gov.ua/handle/123456789/156022 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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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×X⟶X, (g,x)⟼gx. For a family F of subset of X and A∈F, we denote ΔF(A)={g∈G:gB⊆A for some B∈F, B⊆A}, and say that a subset R of G is F-recurrent if R⋂ΔF(A)≠∅ for each A∈F. |
| format |
Article |
| author |
Protasov, I.V. Protasova, K.D. |
| spellingShingle |
Protasov, I.V. Protasova, K.D. On recurrence in G-spaces Algebra and Discrete Mathematics |
| author_facet |
Protasov, I.V. Protasova, K.D. |
| author_sort |
Protasov, I.V. |
| title |
On recurrence in G-spaces |
| title_short |
On recurrence in G-spaces |
| title_full |
On recurrence in G-spaces |
| title_fullStr |
On recurrence in G-spaces |
| title_full_unstemmed |
On recurrence in G-spaces |
| title_sort |
on recurrence in g-spaces |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156022 |
| citation_txt |
On recurrence in G-spaces / I.V. Protasov, K.D. Protasova // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 279-284. — Бібліогр.: 6 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT protasoviv onrecurrenceingspaces AT protasovakd onrecurrenceingspaces |
| first_indexed |
2025-07-14T08:16:51Z |
| last_indexed |
2025-07-14T08:16:51Z |
| _version_ |
1837610082370584576 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 23 (2017). Number 2, pp. 279–284
c© Journal “Algebra and Discrete Mathematics”
On recurrence in G-spaces
Igor Protasov and Ksenia Protasova
To the memory of Vitaly Sushchansky
Abstract. 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 × X −→ X, (g, x) 7−→ gx. For a family F of
subset of X and A ∈ F, we denote ∆F(A) = {g ∈ G : gB ⊆ A for
some B ∈ F, B ⊆ A}, and say that a subset R of G is F-recurrent
if R
⋂
∆F(A) 6= ∅ for each A ∈ F.
Let G be a group with the identity e and let X be a G-space, a set
with the action G × X −→ X, (g, x) 7−→ gx. If X = G and gx is the
product of g and x then X is called a left regular G-space.
Given a G-space X, a family F of subset of X and A ∈ F, we denote
∆F(A) = {g ∈ G : gB ⊆ A for some B ∈ F, B ⊆ A}.
Clearly, e ∈ ∆F(A) and if F is upward directed (A ∈ F, A ⊆ C imply
C ∈ F) and if F is G-invariant (A ∈ F, g ∈ G imply gA ∈ F) then
∆F(A) = {g ∈ G : gA ∩A ∈ F}, ∆F(A) = (∆F(A))−1.
If X is a left regular G-space and ∅ /∈ F then ∆F(A) ⊆ AA−1.
For a G-space X and a family F of subsets of X, we say that a subset
R of G is F-recurrent if ∆F(A) ∩R 6= ∅ for every A ∈ F. We denote by
RF the filter on G with the base ∩{∆F(A) : A ∈ F′}, where F′ is a finite
subfamily of F, and note that, for an ultrafilter p on G, RF ∈ p if and
only if each member of p is F-recurrent.
2010 MSC: 37A05, 22A15, 03E05.
Key words and phrases: G-space, recurrent subset, ultrafilters, Stone-Čech
compactification.
280 On recurrence in G-spaces
The notion of an F-recurrent subset is well-known in the case in which
G is an amenable group, X is a left regular G-space and F = {A ⊆ X :
µ(A) > 0 for some left invariant Banach measure µ on X}. See [1] and
[2] for historical background.
Now we endow G with the discrete topology and identity the Stone-
Čech compactification βG of G with the set of all ultrafilters on G. Then
the family {A : A ⊆ G}, where A = {p ∈ βG : A ∈ p}, forms a base for
the topology of βG. Given a filter ϕ on G, we denote ϕ = ∩{A : A ∈ ϕ}.
We use the standard extension [3] of the multiplication on G to the
semigroup multiplication on βG. We take two ultrafilters p, q ∈ βG,
choose P ∈ p and, for each x ∈ P , pick Qx ∈ q. Then ∪x∈PxQx ∈ pq and
the family of these subsets forms a base of the ultrafilter pq.
We recall [4] that a filter ϕ on a group G is left topological if ϕ is
a base at the identity e for some (uniquely at defined) left translation
invariant (each left shift x 7−→ gx is continuous) topology on G. If ϕ is
left topological then ϕ is a subsemigroup of βG [4]. If G = X and a filter
ϕ is left topological then ϕ = Rϕ.
Proposition 1. For every G-space X and any family F of subsets of X,
the filter RF is left topological.
Proof. By [4], a filter ϕ on a group G is left topological if and only if, for
every Φ ∈ ϕ, there is H ∈ ϕ, H ⊆ Φ such that, for every x ∈ H, xHx ⊆ Φ
for some Hx ∈ ϕ.
We take an arbitrary A ∈ F, put Φ = △F(A) and, for each g ∈ △F(A),
choose Bg ∈ F such that gBg ∈ A. Then g△F(Bg) ⊆ △F(A) so put
H = Φ.
To conclude the proof, let A1, . . . , An ∈F. We denote
Φ1 = △F(A1), . . . , Φn = △F(An), Φ = Φ1 ∩ . . . ∩ Φn.
We use the above paragraph, to choose H1, . . . ,Hn corresponding to
Φ1, . . . ,Φn and put H = H1 ∩ . . . ∩Hn.
Let X be a G-space and let F be a family of subsets of X. We say
that a family F′ of subsets of X is F-disjoint if A∩B /∈ F for any distinct
A,B ∈ F′.
A family F′ of subsets of X is called F-packing large if, for each A ∈ F′,
any F-disjoint family of subsets of X of the form gA, g ∈ G is finite.
I . Protasov, K. Protasova 281
We say that a subset S of a group G is a △ω-set if e ∈ A and
every infinite subset Y of G contains two distinct elements x, y such that
x−1y ∈ S and y−1x ∈ S.
Proposition 2. Let X be a G-space and let F be a G-invariant upward
directed family of subsets of X. Then F is F-packing large if and only if,
for each A ∈ F, the subset △F(A) of G is a △ω-set.
Proof. We assume that F is F-packing large and take an arbitrary infinite
subset Y of G. Then we choose distinct g, h ∈ Y such that gA ∩ hA ∈ F,
so g−1h ∈ △F(A), hg ∈ △F(A) and △F(A) is a △ω-set.
Now we suppose that △F(A) is a △ω-set and take an arbitrary infinite
subset Y of G. Then there are distinct g, h ∈ Y such that g−1h ∈ △F(A)
so g−1hA∩A ∈ F and gA∩hA ∈ F. It follows that the family {gA : g ∈ Y }
is not F-disjoint.
Proposition 3. For every infinite group G, the following statements hold
(i) a subset A ⊆ G is a △ω-set if and only if e ∈ A and every infinite
subset Y of G contains an infinite subset Z such that x−1y ∈ A,
y−1x ∈ A for any distinct x, y ∈ Z;
(ii) the family ϕ of all △ω-sets of G is a filter;
(iii) if A ∈ ϕ then G = FA for some finite subset F of G.
Proof. (i) We assume that A is a △ω-set and define a coloring χ of [Y ]2,
χ : [Y ]2 −→ {0, 1} by the rule: χ({x, y}) = 1 if and only if x−1y ∈ A,
y−1x ∈ A. By the Ramsey theorem, there is an infinite subset Z of Y
such that χ is monochrome on [Z]2. Since A is a △ω-set χ({x, y}) = 1 for
all {x, y} ∈ [Z]2.
(ii) follows from (i).
(iii) We assume the contrary and choose an injective sequence (xn)n∈ω
in G such that xn+1 /∈ xiA for each i ∈ {0, . . . , n}, and denote Y = {xn :
n ∈ ω}. Then x−1
m xn ∈ A for every m,n, m < n, so A is not a △ω-set.
Proposition 4. Let G be a infinite group and let ϕ denotes the filter of
all △ω-sets of G. Then ϕ is the smallest closed subset of βG containing
all ultrafilters on G of the form q−1q, q ∈ βG, g−1 = {A−1 : A ∈ q}.
Proof. We denote by Q the smallest closed subset of βG containing all
q−1q, q ∈ βG. It follows directly from the definition of the multiplication
in βG that p ∈ Q if and only if either p is principal and p = e or, for each
P ∈ p, there is an injective sequence (xn)n∈ω in G such that x−1
m xn ∈ P
for all m < n.
282 On recurrence in G-spaces
Applying Proposition 3(i), we conclude that q−1q ∈ ϕ for each q ∈ βG
so Q ⊆ ϕ. On the other hand, if p /∈ ϕ then there is P ∈ p such that
G \ P is a △ω-set. By above paragraph, p /∈ Q so ϕ ⊆ Q.
Now let G be an amenable group, X be a left regular G-space and
F = {A ∈ X : µ(A) > 0 for some left invariant Banach measure µ on G}.
For combinatorial characterization of F see [6]. Clearly, F is upward
directed G-invariant and F-packing large. By Proposition 2, ϕ ⊆ RF. By
Proposition 4, RF contains all ultrafilters of the form q−1q, q ∈ βG, so
we get Theorem 3.14 from [1].
We suppose that a G-space X is endowed with a G-invariant proba-
bility measure µ defined on some ring of subsets of X. Then the family
F{A ⊆ X : µ(B) > 0 for some B ⊆ A} is F-packing large.
In particular, we can take a compact group X, endow X with the
Haar measure, choose an arbitrary subgroup G of X and endow G with
the discrete topology.
Another example: let a discrete group G acts on a topological space X
so that, for each g ∈ G, the mapping X −→ X, (g, x) 7−→ gx is continuous.
We take a point x ∈ X, denote by F the filter of all neighborhoods of x,
and recall that x is recurrent if, for every U ∈ F, there exists g ∈ G\{e}
such that gx ∈ U . Clearly, x is a recurrent point if and only if G \ {e} if
a set of F-recurrence, so by Proposition 1, x defines some non-discrete
left translation invariant topology on G.
Proposition 5. Let G be a infinite group, A be a △ω-set of G and let τ
be a left translation invariant topology on G with continuous inversion
x 7−→ x−1 at the identity e. Then the closure clτA is a neighborhood of e
in τ .
Proof. On the contrary, we suppose that clτA is not a neighborhood of e,
put U = G \ clτA. Then U is open and e ∈ clτU .
We take an arbitrary x0 ∈ U and choose an open neighborhood U0 of
the identity such that x0U
−1
0 ⊆ U . Then we take x1 ∈ U0 ∩U and choose
an open neighborhood U1 of e such that U1 ⊆ U0 and x1U
−1
1 ⊆ U . We take
x2 ∈ U1 ∩U and choose an open neighborhood U0 of e such that U2 ⊆ U1
and x2U
−1
2 ⊆ U and so on. After ω steps, we get a sequence (xn)n∈ω in G
such that xnx
−1
m ∈ U for all n < m. We denote Y = {x−1
n : n ∈ ω}. Then
(x−1
n )−1x−1
m ∈ A for all n < m, so A is not a △ω-set.
A subset A of an infinite group G is called a △<ω-set if e ∈ A and
there exists a natural number n such that every subset Y of G, | Y |= n
I . Protasov, K. Protasova 283
contains two distinct x, y ∈ Y such that x−1y ∈ A, y−1x ∈ A. These
subsets were introduced in [5] under name thick subsets, but thick subsets
are well-known in combinatorics with another meaning [3]: A is thick if,
for every finite subset F of, there is g ∈ A such that Fg ⊆ A. The family
ψ of all △<ω-sets of G is a filter [5], clearly, ψ ⊆ ϕ. Every infinite group
G has a △ω-set but not △<ω-set A: it suffices to choose inductively a
sequence (Xn)n∈ω of subsets of G, | Xn |= n such that
⋃
n∈ω X
−1
n Xn has
no infinite subsets of the form Y −1Y and put
A = {e} ∪ (G\
⋃
n∈ω
X−1
n Xn),
so ψ ⊂ ϕ.
By analogy with Propositions 3 and 4, we can prove
Proposition 6. Let G be an infinite group and let ψ be the filter of all
△<ω-subsets of G. Then p ∈ ψ if and only if either p is principal and
p = e or, for every A ∈ p, there exists a sequence (Xn)n∈ω of subsets
of G, |Xn| = n + 1, Xn = {xn0, . . . , xnn} such that x−1
ni xnj ∈ A for all
i < j 6 n.
Let A be a subset of a group G such that e ∈ A, A = A−1. We
consider the Cayley graph ΓA with the set of vertices G and the set of
edges {{x, y} : x−1y ∈ A, x 6= y}. We recall that a subset S of vertices of
a graph is independent if any two distinct vertices from S are not incident.
Clearly, A is a △ω-set if and only if any independent set in ΓA is finite,
and A is △ω-set if and only if there exists a natural number n such that
any independent set S is of size |S| < n.
Problem 1. Characterize all infinite graphs with only finite independent
set of vertices.
Problem 2. Given a natural number n, characterize all infinite graphs
such that any independent set S of vertices is of size |S| < n.
In the context of this note, above problems are especially interesting
in the case of Cayley graphs of groups.
References
[1] V. Bergelson, N. Hindman,Quotient sets and density recurrent sets, Trans. Amer.
Math. Soc. 364 (2012), 4495-4531.
[2] H. Furstenberg, Poincare recurrence and number theory, Bulletin Amer. Math. Soc.
5.3 (1981),211-234.
284 On recurrence in G-spaces
[3] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification, de Gruyter,
Berlin, 1998.
[4] I. Protasov, Filters and topologies on groups, Mat. Stud. 3 (1994),15-28.
[5] E. Reznichenko, O. Sipacheva, Discrete subsets in topologicaln groups and countable
extremally disconnected groups, preprint (arxiv: 1608, 03546v2) 2016.
[6] P. Zakrzewski, On the complexity of the ideal of absolute null sets, Ukr. Math. J.
64 (2012),306-308.
Contact information
I. Protasov,
K. Protasova
Kyiv University, Department of Cybernetics,
Kyiv National University, Volodimirska 64,
Kyiv 01033, Ukraine
E-Mail(s): i.v.protasov@gmail.com,
k.d.ushakova@gmail.com
Received by the editors: 04.02.2017.
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