On recurrence in G-spaces

TL;DR

This paper introduces a generalized notion of recurrence in G-spaces, defining $rak{F}$-recurrence based on group actions and families of subsets, providing a new framework for analyzing recurrence phenomena.

Contribution

It proposes a novel, general concept of recurrence in G-spaces using families of subsets and group actions, extending previous notions.

Findings

Defined $rak{F}$-recurrence in G-spaces.

Characterized properties of $rak{F}$-recurrent sets.

Provided a framework for future recurrence analysis.

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 $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $\Delta_{\mathfrak{F}}(A)=\{g\in G: gB\subseteq A$ 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 $A\in \mathfrak{F}$.

Equations

$${\Delta }_{\mathfrak{F}}(A)=\{g\in G:gB\subseteq A\text{ for some }B\in \mathfrak{F},B\subseteq A\}.$$

$${\Delta }_{\mathfrak{F}}(A)=\{g\in G:gA\cap A\in \mathfrak{F}\},{\Delta }_{\mathfrak{F}}(A)=({\Delta }_{\mathfrak{F}}(A){)}^{-1}.$$

Full text

Let $G$ be a group with the identity $e$ and let $X$ be a $G$-space, a set with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. 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 $\mathfrak{F}$ of subset of $X$ and $A\in\mathfrak{F}$, we denote

[TABLE]

Clearly, $e\in\Delta_{\mathfrak{F}}(A)$ and if $\mathfrak{F}$ is upward directed $(A\in\mathfrak{F}$, $A\subseteq C$ imply $C\in\mathfrak{F})$ and if $\mathfrak{F}$ is $G$-invariant $(A\in\mathfrak{F},$ $g\in G$ imply $gA\in\mathfrak{F}$) then

[TABLE]

If $X$ is a left regular $G$-space and $\emptyset\notin\mathfrak{F}$ then $\Delta_{\mathfrak{F}}(A)\subseteq AA^{-1}.$

For a $G$-space $X$ and a family $\mathfrak{F}$ of subsets of $X$, we say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $\Delta_{\mathfrak{F}}(A)\cap R\neq\emptyset$ for every $A\in\mathfrak{F}$. We denote by $\mathfrak{R}_{\mathfrak{F}}$ the filter on $G$ with the base $\cap\{\Delta_{\mathfrak{F}}(A):A\in\mathfrak{F}^{\prime}\},$ where $\mathfrak{F}^{\prime}$ is a finite subfamily of $\mathfrak{F}$, and note that, for an ultrafilter $p$ on $G$, $\mathfrak{R}_{\mathfrak{F}}\in p$ if and only if each member of $p$ is $\mathfrak{F}$-recurrent.

The notion of an $\mathfrak{F}$-recurrent subset is well-known in the case in which $G$ is an amenable group, $X$ is a left regular $G$-space and $\mathfrak{F}=\{A\subseteq X:\mu(A)>0$ for some left invariant Banach measure $\mu$ on $X\}$. See [1] and [2] for historical background.

Now we endow $G$ with the discrete topology and identity the Stone-$\check{C}$ech compactification $\beta G$ of $G$ with the set of all ultrafilters on $G$. Then the family $\{\overline{A}:A\subseteq G\}$, where $\overline{A}=\{p\in\beta G:A\in p\}$, forms a base for the topology of $\beta G$. Given a filter $\varphi$ on $G$, we denote $\overline{\varphi}=\cap\{\overline{A}:A\in\varphi\}$.

We use the standard extension [3] of the multiplication on $G$ to the semigroup multiplication on $\beta G$. We take two ultrafilters $p,q\in\beta G$, choose $P\in p$ and, for each $x\in P$, pick $Q_{x}\in q$. Then $\cup_{x\in P}xQ_{x}\in pq$ and the family of these subsets forms a base of the ultrafilter $pq$.

We recall [4] that a filter $\varphi$ on a group $G$ is left topological if $\varphi$ is a base at the identity $e$ for some (uniquely at defined) left translation invariant (each left shift $x\longmapsto gx$ is continuous) topology on $G$. If $\varphi$ is left topological then $\overline{\varphi}$ is a subsemigroup of $\beta G$ [4]. If $G=X$ and a filter $\varphi$ is left topological then $\varphi=\mathfrak{R}_{\varphi}$.

**Proposition 1. *For every $G$-space $X$ and any family $\mathfrak{F}$ of subsets of $X$, the filter $\mathfrak{R}_{\mathfrak{F}}$ is left topological. ***

Proof. By [4], a filter $\varphi$ on a group $G$ is left topological if and only if, for every $\Phi\in\varphi$, there is $H\in\varphi$, $H\subseteq\Phi$ such that, for every $x\in H$, $xH_{x}\subseteq\Phi$ for some $H_{x}\in\varphi$.

We take an arbitrary $A\in\mathfrak{F}$, put $\Phi=\triangle_{\mathfrak{F}}(A)$ and, for each $g\in\triangle_{\mathfrak{F}}(A)$, choose $B_{g}\in\mathfrak{F}$ such that $gB_{g}\in A$. Then $g\triangle_{\mathfrak{F}}(B_{g})\subseteq\triangle_{\mathfrak{F}}(A)$ so put $H=\Phi$.

To conclude the proof, let \ \ A_{1},\ldots,A_{n}\in\mathfrak{F}\ \. We denote $\ \ \Phi_{1}=\triangle_{\mathfrak{F}}(A_{1}),\ldots,\Phi_{n}=\triangle_{\mathfrak{F}}(A_{n}),$ $\Phi=\Phi_{1}\cap\ldots\cap\Phi_{n}.$ We use the above paragraph, to choose $H_{1},\ldots,H_{n}$ corresponding to $\Phi_{1},\ldots,\Phi_{n}$ and put $H=H_{1}\cap\ldots\cap H_{n}$. $\ \ \ \ \Box$

Let $X$ be a $G$-space and let $\mathfrak{F}$ be a family of subsets of $X$. We say that a family $\mathfrak{F}^{\prime}$ of subsets of $X$ is $\mathfrak{F}$-disjoint if $A\cap B\notin\mathfrak{F}$ for any distinct $A,B\in\mathfrak{F}^{\prime}$.

A family $\mathfrak{F}^{\prime}$ of subsets of $X$ is called $\mathfrak{F}$-packing large if, for each $A\in\mathfrak{F}^{\prime}$, any $\mathfrak{F}$-disjoint family of subsets of $X$ of the form $gA,$ $g\in G$ is finite.

We say that a subset $S$ of a group $G$ is a $\ \triangle_{\omega}$-set if $\ e\in A$ and every infinite subset $Y$ of $G$ contains two distinct elements $x,y$ such that $x^{-1}y\in S$ and $y^{-1}x\in S$.

Proposition 2. Let $X$ be a $G$-space and let $\mathfrak{F}$ be a $G$-invariant upward directed family of subsets of $X$. Then $\mathfrak{F}$ is $\mathfrak{F}$-packing large if and only if, for each $A\in\mathfrak{F}$, the subset \ \triangle_{\mathfrak{F}}(A)\ of $\ G$ is a $\ \triangle_{\omega}$-set.

Proof. We assume that $\mathfrak{F}$ is $\mathfrak{F}$-packing large and take an arbitrary infinite subset $Y$ of $G$. Then we choose distinct $g,h\in Y$ such that $gA\cap hA\in\mathfrak{F}$, so $g^{-1}h\in\triangle_{\mathfrak{F}}(A)$, $hg\in\triangle_{\mathfrak{F}}(A)$ and $\triangle_{\mathfrak{F}}(A)$ is a $\triangle_{\omega}$-set.

Now we suppose that $\triangle_{\mathfrak{F}}(A)$ is a $\triangle_{\omega}$-set and take an arbitrary infinite subset $Y$ of $G$. Then there are distinct $g,h\in Y$ such that $g^{-1}h\in\triangle_{\mathfrak{F}}(A)$ so $g^{-1}hA\cap A\in\mathfrak{F}$ and $gA\cap hA\in\mathfrak{F}$. It follows that the family $\{gA:g\in Y\}$ is not $\mathfrak{F}$-disjoint. $\ \ \ \Box$

Proposition 3. For every infinite group $G$, the following statements hold

$(i)$* a subset $A\subseteq G$ is a $\triangle_{\omega}$-set if and only if $e\in A$ and every infinite subset $Y$ of $G$ contains an infinite subset $Z$ such that $x^{-1}y\in A$, $y^{-1}x\in A$ for any distinct $x,y\in Z$;*

$(ii)$* the family $\varphi$ of all $\triangle_{\omega}$-sets of $G$ is a filter;*

$(iii)$* if $A\in\varphi$ then $G=FA$ for some finite subset $F$ of $G$.*

Proof. $(i)$ We assume that $A$ is a $\triangle_{\omega}$-set and define a coloring $\chi$ of $[Y]^{2}$, $\chi:[Y]^{2}\longrightarrow\{0,1\}$ by the rule: $\chi(\{x,y\})=1$ if and only if $x^{-1}y\in A$, $y^{-1}x\in A$. By the Ramsey theorem, there is an infinite subset $Z$ of $Y$ such that $\chi$ is monochrome on $[Z]^{2}$. Since $A$ is a $\triangle_{\omega}$-set $\chi(\{x,y\})=1$ for all $\{x,y\}\in[Z]^{2}$.

$(ii)$ follows from $(i)$.

$(iii)$ We assume the contrary and choose an injective sequence $(x_{n})_{n\in\omega}$ in $G$ such that \ x_{n+1}\notin x_{i}A\ for each $\ \ i\in\{0,\ldots,n\}$, and denote $Y=\{x_{n}:n\in\omega\}$. Then x_{m}^{-1}x_{n}\in A\ \ for every $\ \ m,n$, $\ \ m<n$, so $A$ is not a $\triangle_{\omega}$-set. $\ \ \Box$

**Proposition 4. *Let $G$ be a infinite group and let $\varphi$ denotes the filter of all $\triangle_{\omega}$-sets of $G$. Then $\overline{\varphi}$ is the smallest closed subset of $\beta G$ containing all ultrafilters on $G$ of the form $q^{-1}q$, $q\in\beta G$, $g^{-1}=\{A^{-1}:A\in q\}$. ***

Proof. We denote by $Q$ the smallest closed subset of $\beta G$ containing all $q^{-1}q$, $\ \ q\in\beta G$. It follows directly from the definition of the multiplication in $\beta G$ that $p\in Q$ if and only if either $p$ is principal and $p=e$ or, for each $P\in p$, there is an injective sequence $(x_{n})_{n\in\omega}$ in $G$ such that $x_{m}^{-1}x_{n}\in P$ for all $m<n$.

Applying Proposition $3(i)$, we conclude that \ q^{-1}q\in\overline{\varphi}\ for each \ q\in\beta G\ so $\ Q\subseteq\overline{\varphi}$. On the other hand, if $p\notin\overline{\varphi}$ then there is $P\in p$ such that $G\setminus P$ is a $\triangle_{\omega}$-set. By above paragraph, $p\notin Q$ so $\overline{\varphi}\subseteq Q$. $\ \ \ \Box$

Now let $G$ be an amenable group, $X$ be a left regular $G$-space and $\mathfrak{F}=\{A\in X:\mu(A)>0$ for some left invariant Banach measure $\mu$ on $G\}$. For combinatorial characterization of $\mathfrak{F}$ see [6]. Clearly, $\mathfrak{F}$ is upward directed $G$-invariant and $\mathfrak{F}$-packing large. By Proposition 2, $\overline{\varphi}\subseteq\overline{\mathfrak{R}_{\mathfrak{F}}}$. By Proposition 4, $\overline{\mathfrak{R}_{\mathfrak{F}}}$ contains all ultrafilters of the form $q^{-1}q$, $q\in\beta G$, so we get Theorem 3.14 from [1].

We suppose that a $G$-space $X$ is endowed with a $G$-invariant probability measure $\mu$ defined on some ring of subsets of $X$. Then the family $\mathfrak{F}\{A\subseteq X:\mu(B)>0$ for some $B\subseteq A\}$ is $\mathfrak{F}$-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 $g\in G$, the mapping $X\longrightarrow X$, $(g,x)\longmapsto gx$ is continuous. We take a point $x\in X$, denote by $\mathfrak{F}$ the filter of all neighborhoods of $x$, and recall that $x$ is recurrent if, for every $U\in\mathfrak{F}$, there exists $g\in G\backslash\{e\}$ such that $gx\in U$. Clearly, $x$ is a recurrent point if and only if $G\setminus\{e\}$ if a set of $\mathfrak{F}$-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 $\triangle_{\omega}$-set of $G$ and let $\tau$ be a left translation invariant topology on $G$ with continuous inversion $x\longmapsto x^{-1}$ at the identity $e$. Then the closure $cl_{\tau}A$ is a neighborhood of $e$ in $\tau$.

Proof. On the contrary, we suppose that $cl_{\tau}A$ is not a neighborhood of $e$, put $U=G\setminus cl_{\tau}A$. Then $U$ is open and $e\in cl_{\tau}U$.

We take an arbitrary $x_{0}\in U$ and choose an open neighborhood $U_{0}$ of the identity such that $x_{0}U_{0}^{-1}\subseteq U$. Then we take $x_{1}\in U_{0}\cap U$ and choose an open neighborhood $U_{1}$ of $e$ such that $U_{1}\subseteq U_{0}$ and $x_{1}U_{1}^{-1}\subseteq U$. We take $x_{2}\in U_{1}\cap U$ and choose an open neighborhood $U_{0}$ of $e$ such that $U_{2}\subseteq U_{1}$ and $x_{2}U_{2}^{-1}\subseteq U$ and so on. After $\omega$ steps, we get a sequence $(x_{n})_{n\in\omega}$ in $G$ such that $x_{n}x_{m}^{-1}\in U$ for all $n<m$. We denote $Y=\{x_{n}^{-1}:n\in\omega\}$. Then $(x_{n}^{-1})^{-1}x_{m}^{-1}\in A$ for all $n<m$, so $A$ is not a $\triangle_{\omega}$-set. $\ \ \Box$

A subset $A$ of an infinite group $G$ is called a $\triangle_{<\omega}$-set if $e\in A$ and there exists a natural number $n$ such that every subset $Y$ of $G$, $\mid Y\mid=n$ contains two distinct $x,y\in Y$ such that $x^{-1}y\in A$, $y^{-1}x\in A$. 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 $g\in A$ such that $Fg\subseteq A$. The family $\psi$ of all $\triangle_{<\omega}$-sets of $G$ is a filter [5], clearly, $\psi\subseteq\varphi$. Every infinite group $G$ has a $\triangle_{\omega}$-set but not $\triangle_{<\omega}$-set $A$: it suffices to choose inductively a sequence $(X_{n})_{n\in\omega}$ of subsets of $G$, $\mid X_{n}\mid=n$ such that $\bigcup_{n\in\omega}X_{n}^{-1}X_{n}$ has no infinite subsets of the form $Y^{-1}Y$ and put $A=\{e\}\cup(G\backslash\bigcup_{n\in\omega}X_{n}^{-1}X_{n})$, so $\psi\subset\varphi$.

By analogy with Propositions 3 and 4, we can prove

Proposition 6. Let $G$ be an infinite group and let $\psi$ be the filter of all $\triangle_{<\omega}$-subsets of $G$. Then $p\in\overline{\psi}$ if and only if either $p$ is principal and $p=e$ or, for every $A\in p$, there exists a sequence $(X_{n})_{n\in\omega}$ of subsets of $G$, $|X_{n}|=n+1$, $X_{n}=\{x_{n0},\ldots,x_{nn}\}$ such that $x_{ni}^{-1}x_{n}j\in A$ for all $i<j\leq n$.

Let $A$ be a subset of a group $G$ such that $e\in A$, $A=A^{-1}$. We consider the Cayley graph $\Gamma_{A}$ with the set of vertices $G$ and the set of edges $\{\{x,y\}:x^{-1}y\in A,\ \ x\neq y\}$. 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 $\triangle_{\omega}$-set if and only if any independent set in $\Gamma_{A}$ is finite, and $A$ is $\triangle_{\omega}$-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.