Force recurrence of semigroup actions

Abstract

We investigate the sets of countable discrete semigroups that force recurrence, that is, the recurrent properties of a point along a subset of a countable semigroup action. We show that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken IP-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite IP-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence.

1 Introduction

By a topological dynamical system (or dynamical system for short) we mean a pair (X, G), where X is a compact metric space with a metric d and G is a topological group or semigroup acting continuously on X. Throughout the paper, the sets of integers, non-negative integers and positive integers are denoted by ${\mathbb {Z}}$, ${\mathbb {Z}}_+$ and ${\mathbb {N}}$, respectively. When $G={\mathbb {Z}}$ (resp. ${\mathbb {Z}}_+$) the action is generated by a homeomorphism (resp. a continuous map) $T: X \rightarrow X$, and we usually denote the dynamical system by (X, T), which is called a cascade.

Recurrence is a basic property of topological dynamical systems. Let (X, T) be a cascade. Recall that a point $x \in X$ is a recurrent point if there is some sequence of positive integers $n_i \rightarrow \infty$ such that $T^{n_i}x \rightarrow x$. Let Rec(X, T) denote the set of all recurrent points of (X, T). The following result is the famous Birkhoff theorem (see [10, 19, 34] for example).

Theorem 1.1

(Birkhoff) Rec(X, T) is non-empty for every cascade (X, T).

An important problem in topological dynamical systems is to investigate the recurrence of a point along some subset of ${\mathbb {N}}$. A subset A of ${\mathbb {Z}}_+$ is called a topological recurrence set if for every cascade (X, T), there is some sequence $\{n_i\}_{i=1}^{\infty }$ in A such that $n_i \rightarrow \infty$ and $T^{n_i}x \rightarrow x$ for some $x \in X$. Birkhoff’s Theorem means that ${\mathbb {Z}}_+$ is a topological recurrence set. Amazingly, the notion of topological recurrence sets in topological dynamical systems are close related to the coloring problem in combinatorial mathematics, see e.g. [26, 34]. In [34], Weiss obtained an important characterization of a topological recurrence set, which clarified the relationship between recurrence sets and difference sets of syndetic sets.

Let (X, T) be a cascade. It is natural to ask that what conditions are satisfied to $S \subset {\mathbb {N}}$ such that there must be some recurrent point in the closure of $\{T^nx: n \in S\}$ for every point $x \in X$. This topic has been studied by some authors, see, e.g., [11, 17, 21, 29]. Such a set is said to force recurrence in [11]. A well-know result implies that if a subset of ${\mathbb {N}}$ has positive upper density, then it forces recurrence (see [11,  Theorem 2]), where the upper density of $S \subset {\mathbb {N}}$ is defined as

$$\begin{aligned} {\overline{d}}(S)=\limsup _{N \rightarrow \infty }\frac{|S \cap [1,N]|}{N}. \end{aligned}$$

A celebrated theorem of Furstenberg shows that the recurrence of topological dynamical systems is closely related to IP-sets [19,  Theorem 2.17]. This terminology was derived from Furstenberg and Weiss in [20]. Following the idea of Furstenberg, Blokh and Fieldsteel [11] showed that a subset of ${\mathbb {N}}$ forces recurrence if and only if it contains a broken IP-set.

The theory of group or semigroup actions has attracted a lot of attention by many authors, for example, see works related to size and combinatorial properties [6], recurrence [1, 7, 8, 13, 15, 31], Lyapunov stability [9], topological entropy [25], sensitivity and chaos [28, 30, 33], transitivity and mixing [12, 27, 32, 35, 36], etc. Especially, Bergelson and McCutcheon [7] extended the notion of topological recurrence sets from the additive semigroup ${\mathbb {N}}$ to arbitrary countable semigroups, and explored their relationships with combinatorics.

In this paper, we focus on investigating the sets of countable discrete semigroups that force recurrence, which following the idea in [11]. We introduce the notion of force recurrence set for general semigroups. We prove that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken IP-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite IP-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence.

2 Force recurrence

Throughout this paper, we let G be an infinite countable discrete semigroup. A semigroup G is a monoid if it has an identity e, and then we write $G^+=G\setminus \{e\}$. By a topological dynamical system we mean that a triple $(X, G, \pi )$ (simple for (X, G)), where X is a compact metric space with the metric d and $\pi : G \times X \rightarrow X, (g,x) \mapsto gx$ is a continuous mapping satisfying

• (1) $\pi (e,x)=x$ for each $x \in X$ if G has an identity e;

• (2) $\pi (s, \pi (t,x))=\pi (st,x)$ for each $s,t \in G$ and $x \in X$.

If a non-empty compact subset $Y \subseteq X$ is G-invariant (i.e., $gy \in Y$ for any $g \in G$ and $y \in Y$), then (Y, G) is called a subsystem of (X, G).

For two dynamical systems (X, G) and (Y, G), their product system $(X \times Y, G)$ is defined by the diagonal action: $g(x,y)=(gx,gy)$ for all $x \in X$, $y \in Y$ and $g \in G$.

Let (X, G) be a dynamical system. A point $x \in X$ is called a recurrent point if $N_+(x, U)$ is non-empty for any neighborhood U of x, where

$$\begin{aligned} N_+(x, U)=\{g \in G^+: gx \in U\} \end{aligned}$$

is called the set of return times of the point x to U. Let Rec(X, G) denote the set of recurrent points of (X, G).

Definition 2.1

We say that a set $S \subseteq G$ forces recurrence if whenever (X, G) is a dynamical system and $K \subseteq X$ is compact, and for some $x \in X$ and all $s \in S$, $sx \in K$, we have $K \cap Rec(X, G)\ne \emptyset$.

In this section, we mainly provide a characterization of subsets of the semigroup that force recurrence. Let G be a semigroup. For $g \in G$ and $S \subset G$, denote

$$\begin{aligned} g^{-1}S=\{h \in G: gh \in S\} \text{ and } Sg^{-1}=\{h \in G: hg \in S\}. \end{aligned}$$

Theorem 2.2

Let ${\mathcal {P}}$ be a non-empty family of infinite subsets of the semigroup G such that

• (1) for all $S \in {\mathcal {P}}$, there is some $g \in G^+$ such that $g^{-1}S \cap S \in {\mathcal {P}}$;

• (2) ${\mathcal {P}}$ has the Ramsey property, that is, $S_1 \cup S_2 \in {\mathcal {P}}$ implies $S_1 \in {\mathcal {P}}$ or $S_2 \in {\mathcal {P}}$.

Then S forces recurrence for all $S \in {\mathcal {P}}$.

Proof

We shall follow the idea of the proof of [11,  Theorem 3]. Let $S \in {\mathcal {P}}$, (X, G) be a dynamical system, $K \subseteq X$ is compact, and $x \in X$ satisfies $sx \in K$ for all $s \in S$. We will show that $K \cap Rec(X, G) \ne \emptyset$.

Set $K_1=K$ and $S_1=S$. Then we can find $p_1 \in G^+$ such that $p_1^{-1}S_1 \cap S_1 \in {\mathcal {P}}$. This implies $sx \in K_1 \cap p_1^{-1}K_1$ for every $s \in p_1^{-1}S_1 \cap S_1$. It follows that $K_1 \cap p_1^{-1}K_1$ is a non-empty compact subset of $K_1$. Let

$$\begin{aligned} K_1 \cap p_1^{-1}K_1=\bigcup _{i=1}^{n_1}K_{1,i}, \end{aligned}$$

where each $K_{1,i}$ is a non-empty compact subset of X with $\mathrm {diam}(K_{1,i})<1/2$. For $i=1, 2, \cdots , n_1$, let $S_{1,i}=\{s \in p_1^{-1}S_1 \cap S_1: sx \in K_{1,i}\}$, then we have

$$\begin{aligned} p_1^{-1}S_1 \cap S_1=\bigcup _{i=1}^{n_1}S_{1,i}. \end{aligned}$$

Since ${\mathcal {P}}$ has the Ramsey property, one has $S_{1, i_1} \in {\mathcal {P}}$ for some $i_1$. Set

$$\begin{aligned} S_2=S_{1, i_1} \text{ and } K_2=K_{1, i_1}. \end{aligned}$$

Then we have $S_2 \in {\mathcal {P}}$, $K_2 \subseteq K_1$, $\mathrm {diam}(K_2)<1/2$ and $p_1(K_2) \subseteq K_1$.

We continue inductively. Assume that $S_n$, $K_n$ and $p_{n-1}$ have been found such that $S_n \in {\mathcal {P}}$, $K_n \subseteq K_{n-1}$, $\mathrm {diam}(K_n)<1/n$, $sx \in K_n$ for all $s \in S_n$ and $p_{n-1}(K_n) \subseteq K_{n-1}$. Then we apply the above argument to $S_n$ and $K_n$, there is $p_n \in G^+$ such that $p_n^{-1}S_n \cap S_n \in {\mathcal {P}}$. By the construction of $S_n$ and $K_n$, we know that $sx \in K_n \cap p_n^{-1}K_n$ for any $s \in p_n^{-1}S_n \cap S_n$. Let

$$\begin{aligned} K_n \cap p_n^{-1}K_n=\bigcup _{i=1}^{m_n} K_{n,i}, \end{aligned}$$

where each $K_{n,i}$ is a non-empty compact subset of X with $\mathrm {diam}(K_{n,i})<1/(n+1)$. Let $S_{n,i}=\{s \in p_n^{-1}S_n \cap S_n: sx \in K_{n,i}\}$. Then

$$\begin{aligned} p_n^{-1}S_n \cap S_n=\bigcup _{i=1}^{m_n}S_{n,i}, \end{aligned}$$

which follows that $S_{n, i_n} \in {\mathcal {P}}$ for some $i_n$. Set

$$\begin{aligned} S_{n+1}=S_{n, i_n} \text{ and } K_{n+1}=K_{n, i_n}. \end{aligned}$$

This completes the inductive process.

By induction, we obtain a sequence of non-empty compact sets $\{K_n\}_{n=1}^{\infty }$ and a sequence $\{p_n\}_{n=1}^{\infty }$ of $G^+$ such that

• $K_1 \supseteq K_2 \supseteq \cdots \supseteq K_n \supseteq \cdots$;

• $\mathrm {diam}(K_n)<1/n$ for each $n \ge 2$;

• $p_n(K_{n+1}) \subseteq K_n$ for each $n \ge 1$.

Let y be the single point in $\bigcap _{n=1}^\infty K_n$. Then we have for all m, $p_m y \in K_m$. This shows that $y \in K \cap Rec(X, G)$. $\square$

Next we give a characterization of the sets of semigroups that force recurrence. Before that, let us recall some notations. For a set A, denote by ${\mathcal {P}}_f(A)$ the set of all finite non-empty subsets of A.

Definition 2.3

Let G be a semigroup. Given a sequence $\{p_n\}_{n=1}^{\infty }$ in G, the IP-set generated by the sequence is defined by

$$\begin{aligned} FP(\{p_n\}_{n=1}^{\infty })=\left\{ \prod _{n \in F}p_n: F \in {\mathcal {P}}_f({\mathbb {N}})\right\} , \text{ where } \prod _{n \in F}p_n=p_{n_1} \cdot p_{n_2} \cdot \ldots \cdot p_{n_k} \end{aligned}$$

for $F=\{n_1, n_2, \ldots , n_k\} \in {\mathcal {P}}_f({\mathbb {N}})$ with $n_1<n_2<\cdots <n_k$.

For each $L \in {\mathbb {N}}$, the initial L-segment of $FP(\{p_n\}_{n=1}^{\infty })$ is defined as

$$\begin{aligned} FP(\{p_n\}_{n=1}^L)=\left\{ \prod _{n \in F}p_n: F \in {\mathcal {P}}_f(\{1, \ldots , L\})\right\} . \end{aligned}$$

A subset S of G contains a broken IP-set if there is a sequence $\{p_n\}_{n=1}^{\infty }$ in G such that for each $L \in {\mathbb {N}}$, there is $s_L \in G$ with $FP(\{p_n\}_{n=1}^L) \cdot s_L \subseteq S$.

Remark 2.4

The most interesting IP-sets are the infinite ones. However, If u is an idempotent, then $\{u\}$ is an IP-set. And even if G is a group, there may be many finite IP-sets. For example any finite subgroup of G is an IP-set.

Let (X, G) be a dynamical system. For $x \in X$ and $U \subseteq X$, let $N(x,U)=\{g \in G: gx \in U\}$. Following the idea of [19,  Theorem 2.17], we can obtain

Lemma 2.5

Let (X, G) be a dynamical system. If $x \in Rec(X, G)$, then N(x, U) contains an IP-set for every neighborhood U of x.

Proof

Suppose x is a recurrence point for (X, G) and U is a neighborhood of U. Let $p_1 \in G^+$ satisfy

$$\begin{aligned} p_1x \in U. \end{aligned}$$

(2.1)

Now we find a neighborhood $U_1$ of x such that $U_1 \subset U$ and

$$\begin{aligned} z \in U_1 \Rightarrow p_1z \in U. \end{aligned}$$

(2.2)

For such $U_1$ we can find $p_2 \in G^+$ such that

$$\begin{aligned} p_2x \in U_1. \end{aligned}$$

(2.3)

Combining (2.1), (2.2) and (2.3), we have

$$\begin{aligned} gx \in U \text{ for } g=p_1, p_2 \text{ and } p_1 \cdot p_2. \end{aligned}$$

(2.4)

We continue inductively. Assume that different elements $p_1, p_2, \cdots , p_n$ in $G^+$ have been found such that (2.4) is valid for all $g=p_{n_1} \cdot p_{n_2} \cdot \ldots \cdot p_{n_k}$ with $1 \le n_1<n_2<\cdots <n_k \le n$. Then we find a neighborhood $U_{n+1}$ of x such that $U_{n+1} \subset U$ and

$$\begin{aligned} z \in U_{n+1} \Rightarrow gz \in U \end{aligned}$$

(2.5)

for all of the above mentioned g. Thus, if $p_{n+1} \in G^+$ is defined such that

$$\begin{aligned} p_{n+1}x \in U_{n+1}, \end{aligned}$$

(2.6)

then (2.4) will be valid whenever g replaced by $g \cdot p_{n+1}$ or by $p_{n+1}$. This completes the inductive process, and it is easy to check that $FP((p_n)_{n \in {\mathbb {N}}}) \subseteq N(x, U)$. $\square$

The following theorem is the product version of the Hindman’s theorem [22].

Theorem 2.6

(The finite product theorem, [4, 5, 37]) Let G be a semigroup. If $S \subset G$ is an IP-set, $r \in {\mathbb {N}}$ and $S=\bigcup _{i=1}^r C_i$, then there is $i \in \{1, 2, \ldots , r\}$ such that $C_i$ contains an IP-set.

For a subset A of a topological space X, we denote $cl_X(A)$ the closure of A in X.

Lemma 2.7

Let G be a monoid, let $S \subseteq G$, and let $\Sigma _2=\{0,1\}^G$ be the product space endowed with the product topology. Define the shift action of G on $\Sigma _2$ by $h\omega (g)=\omega (gh)$ for all $g, h \in G$ and all $\omega \in \Sigma _2$. Then $(\Sigma _2, G)$ is a dynamical system. Define $1_S \in \Sigma _2$ by $1_S(s)=1$ if and only if $s \in S$. Let $X=cl_{\Sigma _2}\{g1_S: g \in G\}$. Then X is an invariant closed subset of $\Sigma _2$ and (X, G) is a subsystem of $(\Sigma _2, G)$. Let $K=\{x \in X: x(e)=1\}$. Then K is a nonempty open and closed subset of X and $s1_S \in K$ for all $s \in S$.

Proof

Since G is countable, $\Sigma _2$ is a compact metric space. Sets of the form $\{\omega \in \Sigma _2: \omega (g)=i\}$ for $g \in G$ and $i \in \{0,1\}$ form a subbasis for the topology on $\Sigma _2$ so K is open and closed. It is routine to verify the rest of the assertions in the lemma. $\square$

Theorem 2.8

Suppose that G is a monoid. Then a subset S of $G^+$ forces recurrence if and only if it contains a broken IP-set.

Proof

Let ${\mathcal {P}}_{\mathrm {bip}}$ denote the collection of all subsets of G that contains a broken IP-set. To prove S forces recurrence all $S \in {\mathcal {P}}_{\mathrm {bip}}$, we only show that the family ${\mathcal {P}}_{\mathrm {bip}}$ satisfies the conditions of Theorem 2.2.

Let $S \in {\mathcal {P}}_{\mathrm {bip}}$ and suppose that S contains a broken $FP(\{p_n\}_{n=1}^{\infty })$. Fix $M \in {\mathbb {N}}$. Then for each $L \in {\mathbb {N}}$ with $L>M$, we can choose $s_L \in G$ such that

$$\begin{aligned} FP(\{p_n\}_{n=1}^L)\cdot s_L \subseteq S, \end{aligned}$$

which implies

$$\begin{aligned} p_M^{-1}S \cap S\supseteq & {} \left( p_M^{-1} \cdot FP(\{p_n\}_{n=1}^L) \cdot s_L\right) \cap \left( FP(\{p_n\}_{n=1}^L) \cdot s_L\right) \\\supseteq & {} \left[ \left( p_M^{-1}\cdot FP(\{p_n\}_{n=1}^L)\right) \cap \left( FP(\{p_n\}_{n=1}^L) \right) \right] \cdot s_L\\\supseteq & {} FP(\{p_n\}_{n=M+1}^L) \cdot s_L. \end{aligned}$$

This shows that $p_M^{-1}S \cap S$ contains a broken $FP(\{p_n\}_{n=M+1}^{\infty })$.

Next, we will show that ${\mathcal {P}}_{\mathrm {bip}}$ has the Ramsey property. Suppose that $S \in {\mathcal {P}}_{\mathrm {bip}}$ contains a broken $FP(\{p_n\}_{n=1}^{\infty })$ and $S=S_1 \cup S_2$. Without loss of generality, we may suppose that $S_1 \cap S_2=\emptyset$. Let x be a point in $\{1, 2\}^S$ that defined by

$$\begin{aligned} x(s)=i \text{ if } \text{ and } \text{ only } \text{ if } s \in S_i. \end{aligned}$$

(2.7)

For each $N \in {\mathbb {N}}$, there exists some $s_N \in G$ such that

$$\begin{aligned} FP(\{p_n\}_{n=1}^N) \cdot s_N \subset S. \end{aligned}$$

(2.8)

Let $x_N \in \{1, 2\}^{FP(\{p_n\}_{n=1}^{\infty })}$ be defined by

$$\begin{aligned} x_N(s)=\left\{ \begin{array}{cl} x(s \cdot s_N), &{} s \in FP(\{p_n\}_{n=1}^N),\\ \\ 1, &{} \text{ otherwise }.\end{array}\right. \end{aligned}$$

Since $\{1, 2\}^{FP(\{p_n\}_{n=1}^{\infty })}$ is a compact metric space, we can choose a subsequence $\{N_j\}_{j=1}^{\infty }$ such that $x_{N_j}$ converges to some $y \in \{1, 2\}^{FP(\{p_n\}_{n=1}^{\infty })}$. Write

$$\begin{aligned} C_i=\{s \in FP(\{p_n\}_{n=1}^{\infty }): y(s)=i\} \text{ for } i=1,2. \end{aligned}$$

By Theorem 2.6, there exists some $i \in \{1, 2\}$ such that $C_i$ is an IP-set, i.e., it contains $FP(\{q_n\}_{n=1}^{\infty })$ for some sequence $\{q_n\}_{n=1}^{\infty }$ in G. For each $L \in {\mathbb {N}}$, we can find some sufficiently large j(L) such that $FP(\{q_n\}_{n=1}^L) \subseteq FP(\{p_n\}_{n=1}^{N_{j(L)}})$ and $x_{N_{j(L)}}(s)=y(s)=i$ for all $s \in FP(\{q_n\}_{n=1}^L)$. This implies

$$\begin{aligned} x(s \cdot s_{N_{j(L)}})=i \text{ for } \text{ all } s \in FP(\{q_n\}_{n=1}^L). \end{aligned}$$

Thus we have $FP(\{q_n\}_{n=1}^L) \cdot s_{N_{j(L)}} \subseteq S_i$. Therefore, $S_i$ contains a broken $FP(\{q_n\}_{n=1}^{\infty })$.

Conversely, suppose that S is a set that forces recurrence. Let $\Sigma _2, X, K$ and $1_S$ be as in Lemma 2.7. Then there is some point $y \in K \cap Rec(X, G)$. By Lemma 2.5, we know that N(y, K) is an IP-set. So there is a sequence $\{p_n\}_{n=1}^{\infty }$ of G such that $FP(\{p_n\}_{n=1}^{\infty }) \subseteq N(y, K)$. This implies

$$\begin{aligned} y(s)=1 \text{ for } \text{ all } s \in FP(\{p_n\}_{n=1}^{\infty }). \end{aligned}$$

(2.9)

For each $L \in {\mathbb {N}}$, there exists $s_L \in G$ such that $s_L1_S \in V$, where

$$\begin{aligned} V=\{x \in X: x(s)=y(s) \text{ for } \text{ all } s \in FP(\{p_n\}_{n=1}^L)\}. \end{aligned}$$

Thus, for every $s \in FP(\{p_n\}_{n=1}^L)$, one has

$$\begin{aligned} 1_S(s\cdot s_L)=s_L1_S(s)=y(s)=1. \end{aligned}$$

This shows that

$$\begin{aligned} FP(\{p_n\}_{n=1}^L) \cdot s_L \subseteq S. \end{aligned}$$

(2.10)

Therefore, S contains a broken $FP(\{p_n\}_{n=1}^{\infty })$. $\square$

3 Force recurrence via Furstenberg family

In this section, we will consider more general forms of recurrence for semigroup actions via Furstenberg family. Let ${\mathcal {P}}$ be a non-empty collection of subsets of the semigroup G. We say that ${\mathcal {P}}$ is a Furstenberg family (or family for short) if it is hereditary upward, i.e., $S_1 \in {\mathcal {P}}$ and $S_1 \subseteq S_2$ implies $S_2 \in {\mathcal {P}}$.

For a family ${\mathcal {P}}$, the block family of ${\mathcal {P}}$, denote by $b{\mathcal {P}}$, is the family consisting of sets $S \subset G$ for which there exists some $P \in {\mathcal {P}}$ such that for every finite subset F of P one has $F \cdot s_F \subseteq S$ for some $s_F \in G$. It is easy to check that

$$\begin{aligned} b{\mathcal {P}}=\{S \subseteq G: (\exists P \in {\mathcal {P}}) (\forall F \in {\mathcal {P}}_f(G))(\exists s_F \in G) \text{ such } \text{ that } (P \cap F) \cdot s_F \subseteq S\}. \end{aligned}$$

3.1 Force family recurrence

Let ${\mathcal {P}}$ be a family of the semigroup G and (X, G) be a topological dynamical system. A point $x \in X$ is called a ${\mathcal {P}}$-recurrent point if $N(x, U) \in {\mathcal {P}}$ for any neighborhood U of x. Denote the set of all ${\mathcal {P}}$-recurrent points of (X, G) by $Rec_{{\mathcal {P}}}(X, G)$. We note that the recurrence in Sect. 2 can be regard as ${\mathcal {P}}_+$-recurrence, where ${\mathcal {P}}_+$ denote the family of all non-empty subsets of G that have non-identity elements of G.

Definition 3.1

Let ${\mathcal {P}}$ be a non-empty family of the semigroup G. We say that a set $S \subseteq G$ forces ${\mathcal {P}}$-recurrence if whenever (X, G) is a dynamical system and $K \subseteq X$ is compact, and for some $x \in X$ and all $s \in S$, $sx \in K$, we have $K \cap Rec_{{\mathcal {P}}}(X, G)\ne \emptyset$.

Following the idea of Theorem 2.8, we have the following general result.

Theorem 3.2

Let ${\mathcal {P}}$ be a non-empty family of the monoid G. If S is a subset of G that forces ${\mathcal {P}}$-recurrence, then $S \in b{\mathcal {P}}$.

Proof

Let $\Sigma _2, X, K$ and $1_S$ be as in Lemma 2.7. Clearly, $s1_S \in K$ for all $s \in S$. Thus there exists a ${\mathcal {P}}$-recurrent point $y \in K$. Notice that K is also a non-empty open subset of X. Let $P=N(y, K)$. Then $P \in {\mathcal {P}}$. For each non-empty finite subset F of P, there exists $s_F \in G$ such that $s_F1_S \in V$, where

$$\begin{aligned} V=\{x \in X: x(s)=y(s) \text{ for } \text{ all } s \in F\}. \end{aligned}$$

Thus, for every $s \in F$, one has

$$\begin{aligned} 1_S(s\cdot s_L)=s_L1_S(s)=y(s)=1. \end{aligned}$$

This shows that $F \cdot s_L \subseteq S$. Therefore, $S \in b {\mathcal {P}}$. $\square$

Let ${\mathcal {P}}$ be a family of the semigroup G. Denote by ${\mathcal {P}}_{\mathrm {force}}$ the collection of all subsets of G that force ${\mathcal {P}}$-recurrence. It is easy to see that ${\mathcal {P}}_{\mathrm {force}}$ is a family, and it is not empty if and only if $Rec_{{\mathcal {P}}}(X, G)$ is non-empty for every topological dynamical system (X, G). In addition, a subset S of G forces ${\mathcal {P}}$-recurrence if and only if whenever (X, G) is a dynamical system and $x \in X$, $cl_X\{gx: g \in S\} \cap Rec_{{\mathcal {P}}}(X, G) \ne \emptyset$.

Theorem 3.3

Let ${\mathcal {P}}$ be a family of the monoid G. If ${\mathcal {P}}_{\mathrm {force}}$ is not empty, then we have

• (1) ${\mathcal {P}}_{\mathrm {force}}$ has the Ramsey property, that is, $S_1 \cup S_2 \in {\mathcal {P}}_{\mathrm {force}}$ implies $S_1 \in {\mathcal {P}}_{\mathrm {force}}$ or $S_2 \in {\mathcal {P}}_{\mathrm {force}}$;

• (2) ${\mathcal {P}}_{\mathrm {force}}=b {\mathcal {P}}_{\mathrm {force}}$.

Proof

(1) Let $S \in {\mathcal {P}}_{\mathrm {force}}$ and $S=S_1 \cup S_2$. If neither $S_1$ nor $S_2$ forces ${\mathcal {P}}$-recurrence, then there exist topological dynamical systems (X, G), (Y, G) and $x \in X, y \in Y$ such that neither $K_1=cl_X\{gx: g \in S_1\}$ nor $K_2=cl_Y\{gy: g \in S_2\}$ contains ${\mathcal {P}}$-recurrence points. Consider the product system $(X \times Y, G)$ and $K=cl_{X \times Y}\{(gx, gy): g \in S\}$. Since S forces ${\mathcal {P}}$-recurrence, there is some ${\mathcal {P}}$-recurrence point $(z_1,z_2) \in K$. Without loss of generality, we may assume that $(z_1, z_2) \in cl_{X \times Y}\{(gx, gy): g \in S_1\}$. Then $z_1 \in K_1$ is a ${\mathcal {P}}$-recurrence point of (X, G), which is a contradiction. Thus, ${\mathcal {P}}_{\mathrm {force}}$ has the Ramsey property.

(2) It is obvious that ${\mathcal {P}}_{\mathrm {force}} \subseteq b {\mathcal {P}}_{\mathrm {force}}$. Let $S \in b {\mathcal {P}}_{\mathrm {force}}$. Then there exists some ${\widetilde{S}} \in {\mathcal {P}}_{\mathrm {force}}$ such that for every non-empty finite subset F of G, there exists $s_F \in G$ such that $({\widetilde{S}} \cap F) \cdot s_F \subseteq S$.

Let (X, G) be a topological dynamical system, K a compact subset of X and $x \in X$ such that $sx \in K$ for all $s \in S$. Since G is countable, we can find an increasing sequence $\{F_n\}_{n=1}^{\infty }$ of non-empty finite subsets of G such that

$$\begin{aligned} F_1 \subset F_2 \subset \cdots \subset F_n \subset \cdots \text{ and } \bigcup _{n=1}^{\infty } F_n=G. \end{aligned}$$

Let $z_n=s_{F_n}x$ for all $n \in {\mathbb {N}}$. Since X is a compact metric space, we can find $z \in X$ and a subsequence $\{n_i\}_{i=1}^{\infty }$ such that $z_{n_i}$ convergence to z. Given $g \in {\widetilde{S}}$, then $gs_{F_{n_i}} \in S$, and thus $gz_{n_i} \in K$, for all sufficiently large i. By the continuity of g, we have $gz_{n_i} \rightarrow gz \in K$. This shows that $gz \in K$ for all $g \in {\widetilde{S}}$. Since ${\widetilde{S}}$ forces ${\mathcal {P}}$-recurrence, there is some ${\mathcal {P}}$-recurrence point $y \in K$. Thus, $S \in {\mathcal {P}}_{\mathrm {force}}$. $\square$

Denote by ${\mathcal {P}}_{\mathrm {ip}}$ the family of all sets that contains some IP-set. It is obvious that $b{\mathcal {P}}_{\mathrm {ip}}={\mathcal {P}}_{\mathrm {bip}}$. Thus, by Lemma 2.5, Theorems 2.8, 3.2 and 3.3, we have

Corollary 3.4

Suppose that G is a monoid and S is a subset of $G^+$. Then the following conditions are equivalent:

• (1) S forces recurrence;

• (2) S forces ${\mathcal {P}}_{\mathrm {ip}}$-recurrence;

• (3) $S \in b{\mathcal {P}}_{\mathrm {ip}}$.

Furthermore, we have

$$\begin{aligned} {\mathcal {P}}_{+,\mathrm {force}}={\mathcal {P}}_{\mathrm {ip}, \mathrm {force}}= b{\mathcal {P}}_{+,\mathrm {force}}=b{\mathcal {P}}_{\mathrm {ip},\mathrm {force}}=b{\mathcal {P}}_{\mathrm {ip}}. \end{aligned}$$

3.2 Force infinite recurrence

Let (X, G) be a dynamical system. A point $x \in X$ is called a infinite recurrent point if N(x, U) is infinite for any neighborhood U of x. Denote by ${\mathcal {P}}_{\mathrm {inf}}$ the family of all infinite subsets of G. Then x is an infinite recurrence point if and only if it is a ${\mathcal {P}}_{\mathrm {inf}}$-recurrent point.

The following lemma can be found in [35,  Lemma 3.18].

Lemma 3.5

Let (X, G) be a dynamical system. If x is an infinite recurrence point, then N(x, U) contains an infinite IP-set for every neighborhood U of x.

Similar to the proof of Theorem 2.2, we have the following result.

Theorem 3.6

Let ${\mathcal {P}}$ be a non-empty family of infinite subsets of the semigroup G such that

• (1) for all $S \in {\mathcal {P}}$, there exist infinitely many $g \in G$ such that $g^{-1}S \cap S \in {\mathcal {P}}$;

• (2) ${\mathcal {P}}$ has the Ramsey property, that is, $S_1 \cup S_2 \in {\mathcal {P}}$ implies $S_1 \in {\mathcal {P}}$ or $S_2 \in {\mathcal {P}}$.

Then S forces ${\mathcal {P}}_{\mathrm {inf}}$-recurrence for all $S \in {\mathcal {P}}$.

Proof

Let $S \in {\mathcal {P}}$, (X, G) be a dynamical system, $K \subseteq X$ is compact, and $x \in X$ satisfies $sx \in K$ for all $s \in S$. We will show that $K \cap Rec_{{\mathcal {P}}_{\mathrm {inf}}}(X, G) \ne \emptyset$.

Set $K_1=K$ and $S_1=S$. Then we can find $p_1 \in G$ such that $p_1^{-1}S_1 \cap S_1 \in {\mathcal {P}}$. This implies $sx \in K_1 \cap p_1^{-1}K_1$ for every $s \in p_1^{-1}S_1 \cap S_1$. It follows that $K_1 \cap p_1^{-1}K_1$ is a non-empty compact subset of $K_1$. Let

$$\begin{aligned} K_1 \cap p_1^{-1}K_1=\bigcup _{i=1}^{n_1}K_{1,i}, \end{aligned}$$

where each $K_{1,i}$ is a non-empty compact subset of X with $\mathrm {diam}(K_{1,i})<1/2$. For $i=1, 2, \cdots , n_1$, let $S_{1,i}=\{s \in p_1^{-1}S_1 \cap S_1: sx \in K_{1,i}\}$, then we have

$$\begin{aligned} p_1^{-1}S_1 \cap S_1=\bigcup _{i=1}^{n_1}S_{1,i}. \end{aligned}$$

Since ${\mathcal {P}}$ has the Ramsey property, one has $S_{1, i_1} \in {\mathcal {P}}$ for some $i_1$. Set

$$\begin{aligned} S_2=S_{1, i_1} \text{ and } K_2=K_{1, i_1}. \end{aligned}$$

Then we have $S_2 \in {\mathcal {P}}$, $K_2 \subseteq K_1$, $\mathrm {diam}(K_2)<1/2$ and $p_1(K_2) \subseteq K_1$.

We continue inductively. Assume that $S_n$, $K_n$ and $p_{n-1}$ have been found such that $S_n \in {\mathcal {P}}$, $K_n \subseteq K_{n-1}$, $\mathrm {diam}(K_n)<1/n$, $sx \in K_n$ for all $s \in S_n$ and $p_{n-1}(K_n) \subseteq K_{n-1}$. Then we apply the above argument to $S_n$ and $K_n$, by Condition (1), there is $p_n \ne p_i$ for $i=1, 2, \cdots , n-1$, such that $p_n^{-1}S_n \cap S_n \in {\mathcal {P}}$. By the construction of $S_n$ and $K_n$, we know that $sx \in K_n \cap p_n^{-1}K_n$ for any $s \in p_n^{-1}S_n \cap S_n$. Let

$$\begin{aligned} K_n \cap p_n^{-1}K_n=\bigcup _{i=1}^{m_n} K_{n,i}, \end{aligned}$$

where each $K_{n,i}$ is a non-empty compact subset of X with $\mathrm {diam}(K_{n,i})<1/(n+1)$. Let $S_{n,i}=\{s \in p_n^{-1}S_n \cap S_n: sx \in K_{n,i}\}$. Then

$$\begin{aligned} p_n^{-1}S_n \cap S_n=\bigcup _{i=1}^{m_n}S_{n,i}, \end{aligned}$$

which follows that $S_{n, i_n} \in {\mathcal {P}}$ for some $i_n$. Set

$$\begin{aligned} S_{n+1}=S_{n, i_n} \text{ and } K_{n+1}=K_{n, i_n}. \end{aligned}$$

This completes the inductive process.

By induction, we obtain a sequence of non-empty compact sets $\{K_n\}_{n=1}^{\infty }$ and a sequence $\{p_n\}_{n=1}^{\infty }$ of G such that

• $K_1 \supseteq K_2 \supseteq \cdots \supseteq K_n \supseteq \cdots$;

• $\mathrm {diam}(K_n)<1/n$ for each $n \ge 2$;

• $p_n(K_{n+1}) \subseteq K_n$ for each $n \ge 1$;

• $p_i \ne p_j$ for each $i \ne j$.

Let y be the single point in $\bigcap _{n=1}^\infty K_n$. Then we have for all m, $p_m y \in K_m$. This shows that $y \in K \cap Rec_{{\mathcal {P}}_{\mathrm {inf}}}(X, G)$. $\square$

Next, we provide a characterization of subsets of the semigroup that force infinite recurrence via infinite IP-sets. Let ${\mathcal {P}}_{\mathrm {inf,ip}}$ denote the family of all subsets of the semigroup G that contains some infinite IP-set. We have the following lemma:

Lemma 3.7

Let G be a semigroup which is either right or left cancellative. Then $b {\mathcal {P}}_{\mathrm {inf, ip}}$ has the Ramsey property.

Proof

This is established in Corollary 5.4 in the Appendix. $\square$

The idea is that the proof involves results about the Stone–Čech compactification of G which are not needed for the rest of the results of this paper, so we leave it to an Appendix.

Theorem 3.8

Suppose that G is a monoid and $S \subseteq G$. Statements (1) and (2) are equivalent and imply statement (3). If G is either right or left cancellative, then all three statement are equivalent.

• (1) S forces ${\mathcal {P}}_{\mathrm {inf}}$-recurrence;

• (2) S forces ${\mathcal {P}}_{\mathrm {inf,ip}}$-recurrence;

• (3) $S \in b{\mathcal {P}}_{\mathrm {inf,ip}}$,

Proof

It follows directly from Lemma 3.5 and Theorem 3.2 that (1) $\Leftrightarrow$ (2) $\Rightarrow$ (3). Now we only show that (3) $\Rightarrow$ (1) if G is either right or left cancellative.

By Theorem 3.6 and Lemma 3.7, it suffices to prove for all $S \in {\mathcal {P}}_{\mathrm {inf, ip}}$, there are infinitely many $g \in G$ such that $g^{-1}S \cap S \in b {\mathcal {P}}_{\mathrm {inf, ip}}$. Let $S \in b{\mathcal {P}}_{\mathrm {inf, ip}}$ and suppose that S contains a broken infinite IP-set $FP(\{p_n\}_{n=1}^{\infty })$. Fix $M \in {\mathbb {N}}$. Then for each $L \in {\mathbb {N}}$ with $L>M$, we can choose $s_L \in G$ such that

$$\begin{aligned} FP(\{p_n\}_{n=1}^L)\cdot s_L \subseteq S, \end{aligned}$$

which implies

$$\begin{aligned} g^{-1}S \cap S\supseteq & {} \left( g^{-1} \cdot FP(\{p_n\}_{n=1}^L) \cdot s_L\right) \cap \left( FP(\{p_n\}_{n=1}^L) \cdot s_L\right) \\\supseteq & {} \left[ \left( g^{-1}\cdot FP(\{p_n\}_{n=1}^L)\right) \cap \left( FP(\{p_n\}_{n=1}^L) \right) \right] \cdot s_L\\\supseteq & {} FP(\{p_n\}_{n=M+1}^L) \cdot s_L. \end{aligned}$$

for all $g \in FP(\{p_n\}_{n=1}^M)$, and thus $g^{-1}S \cap S$ contains a broken infinite IP-set $FP(\{p_n\}_{n=M+1}^{\infty })$. Therefore, $g^{-1}S \cap S \in b{\mathcal {P}}_{\mathrm {inf, ip}}$ for all $g \in FP(\{p_n\}_{n=1}^{\infty })$. $\square$

3.3 Force minimality

Recall that a dynamical system (X, G) is called minimal if it contains no proper subsystem, i.e., the orbit $\mathrm {orb}(x,G)=\{gx: g \in G\}$ of x is dense in X for all $x \in X$. A point x is called a minimal point if it belonging to some minimal subsystem of (X, G). Note that x is a minimal point of (X, G) if and only if $cl_X\{gx: g \in G\}$ is minimal.

Let G be a semigroup. A subset $S \subseteq G$ is called syndetic if there exists a finite subset F of G such that $F^{-1}S=\bigcup _{g \in F} g^{-1}S=G$. Denote by ${\mathcal {P}}_s$ the family of all syndetic sets in G.

It is a routine Zorn’s Lemma argument to show that any dynamical system contains a minimal dynamical system. The proof of the following lemma can be found in [15,  Proposition 5.21] with the caution that they use the left-right switches of both the definition of syndetic and the action of G on X.

Lemma 3.9

Let (X, G) be a dynamical system and $x \in X$. Then x is a minimal point if and only if it is an ${\mathcal {P}}_s$-recurrent point.

Definition 3.10

We say that a set $S \subset G$ forces minimality if whenever (X, G) is a dynamical system and $K \subseteq X$ is compact, and for some $x \in X$ and all $s \in S$, $sx \in K$, there exists a minimal subset non-disjoint from K.

Now we prove the following theorem.

Theorem 3.11

Let G be a monoid and S a subset of G. Then the following conditions are equivalent:

• (1) S forces ${\mathcal {P}}_s$-recurrence;

• (2) S forces minimality;

• (3) $S \in b{\mathcal {P}}_s$;

Proof

(1) $\Rightarrow$ (2) Let S be a set that forces ${\mathcal {P}}_s$-recurrence. Suppose that (X, G) is a dynamical system, K is a compact subset of X, and $x \in X$ is a point such that $sx \in K$ for all $s \in S$. Then there exists a ${\mathcal {P}}_s$-recurrence point $z \in K$. By Lemma 3.9, one has z is a minimal point so $z \in cl_X\{gx: g \in G\} \cap K$. Therefore, S forces minimality.

(2) $\Rightarrow$ (3) Let S be a set that forces minimality. Let $\Sigma _2, X, K$ and $1_S$ be as in Lemma 2.7. Clearly, $s1_S \in K$ for all $s \in S$. By the force minimality, there exists a minimal point $y \in K$. Notice that K is also a non-empty open subset of X. By Lemma 3.9, one has $N(y, K) \in {\mathcal {P}}_{\mathrm {s}}$. For each $F \in {\mathcal {P}}_f(G)$, there exists $s_F \in G$ such that $s_F1_S \in V$, where

$$\begin{aligned} V=\{x \in X: x(s)=y(s) \text{ for } \text{ all } s \in F\}. \end{aligned}$$

Thus, for every $s \in N(y, K) \cap F$, one has

$$\begin{aligned} 1_S(s\cdot s_L)=s_L1_S(s)=y(s)=sy(e)=1. \end{aligned}$$

This shows that

$$\begin{aligned} (N(x,K) \cap F) \cdot s_L \subseteq S. \end{aligned}$$

(3.1)

Therefore, $S \in b{\mathcal {P}}_{\mathrm {s}}$.

(3) $\Rightarrow$ (1) Suppose that $S \in b{\mathcal {P}}_{\mathrm {s}}$. Then there exists some ${\widetilde{S}} \in {\mathcal {P}}_{\mathrm {s}}$ such that for every $F \in {\mathcal {P}}_f(G)$, there exists $s_F \in G$ such that

$$\begin{aligned} ({\widetilde{S}} \cap F) \cdot s_F \subseteq S. \end{aligned}$$

Now let (X, G) be a dynamical system, K a compact subset of X and $x \in X$ such that $sx \in K$ for all $s \in S$. Since G is countable, we can find an increasing sequence $\{F_n\}_{n=1}^{\infty }$ of finite subsets of G such that

$$\begin{aligned} F_1 \subset F_2 \subset \cdots \subset F_n \subset \cdots \text{ and } \bigcup _{n=1}^{\infty }F_n=G. \end{aligned}$$

Without loss of generality, we may assume that $F_1 \cap {\widetilde{S}} \ne \emptyset$. Pick some $r \in F_1 \cap {\widetilde{S}}$, and let $z_n=s_{F_n}x \in r^{-1}K$ for all $n \in {\mathbb {N}}$. By the compactness of K and the continuity of r, we can find a subsequence $\{n_i\}_{i=1}^{\infty }$ such that $z_{n_i}$ convergence to $z \in r^{-1}K$. Given $g \in r^{-1}{\widetilde{S}}$. Then we have $rg \in {\widetilde{S}}$, which implies for all sufficiently large i,

$$\begin{aligned} rgz_{n_i}=(rg \cdot s_{F_{n_i}})x \in K. \end{aligned}$$

By the continuity of rg we have $rgz \in K$. This shows that $gz \in r^{-1}K$ for all $g \in r^{-1}{\widetilde{S}}$.

Let F be a finite subset of G such that $F^{-1}{\widetilde{S}}=G$. Choose a finite subset H of G such that $F=rH$. Then we can obtain

$$\begin{aligned} cl_X\{gz: g \in G\} \subseteq \bigcup _{h \in H} (rh)^{-1}K. \end{aligned}$$

(3.2)

Indeed, for each $g \in G$, there is $h \in H$ such that $rhg \in {\widetilde{S}}$, which implies $hg \in r^{-1}{\widetilde{S}}$, and thus $gz \in (rh)^{-1}K$. For the closed invariant subset $cl_X\{gz: g \in G\}$ we can find a non-empty minimal subset $Y \subseteq cl_X\{gz: g \in G\}$. Furthermore, we know that every point in Y is ${\mathcal {P}}_{\mathrm {s}}$-recurrent by Lemma 3.9. Last, we only show that $Y \cap K \ne \emptyset$. Pick $y \in Y$, choose a sequence $\{g_n\}_{n=1}^{\infty }$ of G such that $g_nz \rightarrow y$. By (3.2), we know that for each n, there exists some $h_n \in H$ such that $rh_ng_nz \in K$. Since H is finite, we may assume that $h_n$ is constantly equal to h. It follows that

$$\begin{aligned} rhg_nz \rightarrow rhy \in K. \end{aligned}$$

Thus, $rhy \in Y \cap K$. This completes the proof. $\square$

3.4 Force non-wandering

In this subsection, we study the non-wandering for semigroup actions. Let (X, G) be a dynamical system. For a non-empty family ${\mathcal {P}}$ of non-empty sets of the semigroup G, we say that a point $x \in X$ is ${\mathcal {P}}$-non-wandering if $N(U, U) \in {\mathcal {P}}$ for every neighborhood U of x, where

$$\begin{aligned} N(U, U)=\{g \in G: U \cap g^{-1}U \ne \emptyset \}. \end{aligned}$$

Denote the set of all ${\mathcal {P}}$-non-wandering points of (X, G) by $\Omega _{{\mathcal {P}}}(X, G)$.

Definition 3.12

We say that a subset $S \subseteq G$ forces ${\mathcal {P}}$-non-wandering if whenever (X, G) is a dynamical system and $K \subset X$ is compact, and for some $x \in X$ and all $s \in S$, $sx \in K$, then we have $K \cap \Omega _{{\mathcal {P}}}(X, G)\ne \emptyset$.

Theorem 3.13

Let ${\mathcal {P}}$ be a non-empty family of non-empty sets of the semigroup G such that

• (1) right shift invariant: $S \in {\mathcal {P}}$ implies $Sg^{-1} \in {\mathcal {P}}$ for all $g \in G$;

• (2) ${\mathcal {P}}$ has the Ramsey property.

Then S forces ${\mathcal {P}}$-non-wandering for all $S \in {\mathcal {P}}$.

Proof

Let $S \in {\mathcal {P}}$ and K be a compact set in a dynamical system (X, G) such that for some point $x \in X$ and all $s \in S$, $sx \in K$. Write $K=\bigcup _{i=1}^{n_1}K_{1,i}$, where each $K_{1, i}$ is a non-empty compact subset with $\mathrm {diam}(K_{1,i})<1$. For $i=1, 2, \ldots , n_1$, let $S_{1,i}=\{s \in S: sx \in K_{1,i}\}$. Then we have $S=\bigcup _{i=1}^{n_1} S_{1,i}$. Since ${\mathcal {P}}$ has the Ramsey property, one has $S_{1, i_1} \in {\mathcal {P}}$ for some $i_1 \in \{1, 2, \ldots , n_1\}$. Set

$$\begin{aligned} S_1=S_{1, i_1} \text{ and } K_1=K_{1, i_1}. \end{aligned}$$

By induction, we obtain a sequence of non-empty compact sets $\{K_n\}_{n=1}^{\infty }$ and a sequence $\{S_n\}_{n=1}^{\infty } \subset {\mathcal {P}}$ such that

• $K \supseteq K_1 \supseteq K_2 \supseteq \cdots \supseteq K_n \supseteq \cdots$;

• $S \supseteq S_1 \supseteq S_2 \supseteq \cdots \supseteq S_n \supseteq \cdots$;

• $\mathrm {diam}(K_n)<1/n$ for each $n \ge 1$;

• $sx \in K_n$ for all $s \in S_n$.

Let y be the single point in $\bigcap _{n=1}^{\infty }K_n$. Then for every neighborhood U of y, there exists some $n_U$ such that $K_{n_U} \subset U$. Pick some $s \in S_{n_U}$, then we have $sx \in K_{n_U}\subset U$ and $h(sx)=(hs)x \in K_{n_U} \subseteq U$ for all $h \in S_{n_U}s^{-1} \in {\mathcal {P}}$. This implies $S_{n_U}s^{-1} \subset N(U, U)$, and hence $N(U, U) \in {\mathcal {P}}$. $\square$

4 Density of group and force recurrence

The notions of upper Banach density of group have been studied from several points of view (see, for example, [2, 14]). Let G be a countable discrete infinite semigroup. For a subset A in G and a finite set $F \subset G$, define

$$\begin{aligned} {\overline{D}}_F(A)=\sup _{g \in G} \frac{|A \cap Fg|}{|F|} . \end{aligned}$$

The upper Banach density of A is defined by

$$\begin{aligned} BD^*(A)=\inf _{F \in {\mathcal {P}}_f(G)}{\overline{D}}_{F}(A) \end{aligned}$$

(4.1)

Recall that an infinite countable discrete group G is called amenable if there exists a sequence of finite subsets $F_n \subset G$ such that for every $g \in G$,

$$\begin{aligned} \lim _{n \rightarrow +\infty } \frac{|g F_n \triangle F_n|}{|F_n|}=0, \end{aligned}$$

(4.2)

where $|\cdot |$ denotes the cardinality of a set and $\triangle$ stands for the symmetric difference of sets. A sequence satisfying condition (4.2) is called a Følner sequence (see [18]). The basic example of an amenable group is the group $G={\mathbb {Z}}^d$ for some $d \in {\mathbb {N}}$, and $\{F_n=[0, n-1]^d: n \in {\mathbb {N}}\}$ is a Følner sequence of G.

Lemma 4.1

Let G be a countably infinite discrete amenable group, let $A \subseteq G$ such that $BD^*(A)>0$, and let F be a finite subset of G. There exists $g \in G \setminus F$ such that $BD^*(A \cap g^{-1}A)>0$.

Proof

This follows immediately from Proposition 2.2 (ii) of [3]. $\square$

The proof of Lemma 4.2 is adapted from the proof of [23,  Theorem 11.11].

Lemma 4.2

Let G be a countably infinite discrete amenable group and let $S \subseteq G$ such that $BD^*(S)>0$. Then $S \in b {\mathcal {P}}_{\mathrm {inf, ip}}$.

Proof

Let e be the identity of G and let $D_1=S$. By Lemma 4.1, pick $g_1 \in G \setminus \{e\}$ such that $BD^*(D_1 \cap g_1^{-1}D_1)>0$.

Let $n \in {\mathbb {N}}$ and assume we have chosen $(D_k)_{k=1}^n$ and $(g_k)_{k=1}^n$ such that for $k \in \{1, 2, \cdots , n\}$,

• (1) $g_k \in G$ and $D_k \subset G$;

• (2) $BD^*(D_k \cap g_k^{-1}D_k)>0$;

• (3) if $k<n$, then $D_{k+1}=D_k \cap g_k^{-1}D_k$; and

• (4) if $k<n$, then $g_{k+1} \notin FP(\{g_t\}_{t=1}^k)$.

Let $D_{n+1}=D_n \cap g_n^{-1}D_n$ and let $F=FP(\{g_t\}_{t=1}^n) \cup \{e\}$. Pick by Lemma 4.1 some $g_{n+1} \in G \setminus F$ such that $BD^{*}(D_{n+1}\cap g_{n+1}^{-1}D_{n+1})>0$. One easily shows by induction that for each n,

$$\begin{aligned} D_{n+1}=S \cap \left( \bigcap _{g \in FP(\{g_t\}_{t=1}^n)}g^{-1}S\right) . \end{aligned}$$

Let $P=FP(\{g_t\}_{t=1}^{\infty })$. By hypothesis (4), P is an infinite IP-set. Given finite non-empty $F \subset P$ pick $n \in {\mathbb {N}}$ such that $F \subseteq FP(\{g_t\}_{t=1}^n)$ and pick $s_F \in D_{n+1}$. Then $F \cdot s_F \subseteq S$ so $S \in b {\mathcal {F}}_{\mathrm {inf, ip}}$. $\square$

Theorem 4.3

Let G be an infinite countable discrete amenable group. If $S \subseteq G$ has positive upper Banach density, then S forces ${\mathcal {P}}_{\mathrm {inf}}$-recurrence.

Proof

Since G is a group, this is an immediate consequence of Lemma 4.2 and Theorem 3.8. $\square$

Question 4.4

Let G be an arbitrary countable discrete group or semigroup and let S be a subset of G with positive upper Banach density. Must S force ${\mathcal {P}}_{\mathrm {inf}}$-recurrence?

Appendix: When \(b {\mathcal {P}}_{\mathrm {inf, ip}}\) has the Ramesey property

In this section we utilize the algebraic structure of the Stone–Čech compactification $\beta G$ of a discrete semigroup $(G, \cdot )$. We shall assume that the reader is familiar with the basic facts about this structure. For an elementary introduction see [24,  Part I].

We shall show that if $\beta G \setminus G$ is a subsemigroup of $\beta G$, in particular if G is either right or left cancellative, then both ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property. An exact characterization of when $\beta G \setminus G$ is a subsemigroup of $\beta G$ is given in [24,  Theorem 4.28].

Lemma 5.1

Assume that G is a semigroup, p is an idempotent in $\beta G \setminus G$, and $A \in p$. Then A contains an infinite IP-set. In fact there exists an injective sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })\subseteq A$.

Proof

Let $A^{\star }=\{x \in A: x^{-1}A \in p\}$. By [24,  Lemma 4.14] if $x \in A^{\star }$, then $x^{-1}A^{\star } \in p$. Choose $x_1 \in A^{\star }$. Inductively let $n \in {\mathbb {N}}$ and assume we have chosen injective $(x_t)_{t=1}^n$ in G such that $E =FP(\{x_t\}_{t=1}^n) \subseteq A^{\star }$. Since $p \in \beta G \setminus G$, $G \setminus \{x_1, x_2, \ldots , x_n\} \in p$ so

$$\begin{aligned} \left( A^{\star } \cap \bigcap _{y \in E}y^{-1}A^{\star }\right) \setminus \{x_1, x_2, \ldots , x_n\} \in p. \end{aligned}$$

Pick $x_{n+1} \in \left( A^{\star } \cap \bigcap _{y \in E}y^{-1}A^{\star }\right) \setminus \{x_1, x_2, \ldots , x_n\}$. $\square$

Lemma 5.2

Assume that G is a semigroup, $\beta G \setminus G$ is a subsemigroup of $\beta G$, and $A \subseteq G$. If A contains an infinite IP-set $FP(\{x_n\}_{n=1}^{\infty })$, then there is an idempotent $p \in \beta G \setminus G$ such that for every $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty }) \in p$.

Proof

We claim that for each $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty })$ is infinite. To see this let $m>1$ and let $E=FP(\{x_n\}_{n=1}^{m-1})$. Then

$$\begin{aligned} FP(\{x_n\}_{n=1}^{\infty })=E \cup FP(\{x_n\}_{n=m}^{\infty }) \cup \bigcup _{y \in E} y \cdot FP(\{x_n\}_{n=m}^{\infty }) \end{aligned}$$

so one of the listed sets is infinite and thus $FP(\{x_n\}_{n=m}^{\infty })$ is infinite.

Let ${\mathcal {A}}=\{FP(\{x_n\}_{n=m}^{\infty }): m \in {\mathbb {N}}\}$. Then ${\mathcal {A}}$ is a nested family of infinite sets so by [24,  Corollary 3.14], there is some $q \in \beta G \setminus G$ such that ${\mathcal {A}} \subseteq q$. That is, $(\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })} \ne \emptyset$. By [24,  Lemma 5.11], $\bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$ is a semigroup so $(\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$ is a compact right topological semigroup so by [16,  Lemma 1], there is an idempotent $p \in (\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$. $\square$

Theorem 5.3

Assume that G is a semigroup, $\beta G \setminus G$ is a subsemigroup of $\beta G$, and $A \subseteq G$.

• (1) $A \in {\mathcal {P}}_{\mathrm {inf, ip}}$ if and only if there is an idempotent $p \in \beta G \setminus G$ such that $A \in p$;

• (2) $A \in b {\mathcal {P}}_{\mathrm {inf, ip}}$ if and only if there exists an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $A \in p \cdot q$.

Proof

(1) The necessity follows from Lemma 5.2 and the sufficiency follows from Lemma 5.1.

(2) To establish the necessity, pick a sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })$ is infinite and for each $m \in {\mathbb {N}}$ there exists $s_m \in G$ such that $FP(\{x_n\}_{n=1}^m)\cdot s_m \subseteq A$. Pick by Lemma 5.2 an idempotent $p \in \beta G \setminus G$ such that for every $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty }) \in p$. Pick $q \in \beta G$ such that $\left\{ \{s_m: m>n\}: n \in {\mathbb {N}}\right\} \subseteq q$. We claim that $A \in p \cdot q$. To see this, it suffices to show that $FP(\{x_n\}_{n=1}^{\infty }) \subseteq \{y \in G: y^{-1}A \in q\}$ by [24,  Lemma 4.12] so let $z \in FP(\{x_n\}_{n=1}^{\infty })$ and pick $F \in {\mathcal {P}}_f({\mathbb {N}})$ such that $z=\prod _{t \in F} x_t$. Let $n =\max F$. Then $\{s_m: m>n\} \in q$ and $\{s_m: m>n\} \subseteq z^{-1}A$.

For the sufficiency, pick an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $A \in p \cdot q$. Let $B=\{y \in G: y^{-1}A \in q\}$. Then $B \in p$ by [24,    Lemma 4.12] so pick by Lemma 5.1 an injective sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })\subseteq B$. Now let $m \in {\mathbb {N}}$. It suffices to show that there exists $s_m \in G$ such that $FP(\{x_n\}_{n=1}^m) \cdot s_m \subseteq A$. Let $E=FP(\{x_n\}_{n=1}^m)$. Then E is a finite subset of B so we may pick $s_m \in \bigcap _{y \in E} y^{-1}A$. $\square$

Corollary 5.4

Assume that G is a semigroup and $\beta G \setminus G$ is a subsemigroup of $\beta G$. Then ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property. In particular if G is infinite and is either right cancellative or left cancellative, then ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property.

Proof

Let $S_1$ and $S_2$ be subsets of G. If $S_1 \cup S_2 \in {\mathcal {P}}_{\mathrm {inf,ip}}$, pick an idempotent $p \in \beta G \setminus G$ such that $S_1 \cup S_2 \in p$. Since p is an ultrafilter, either $S_1 \in p$ or $S_2 \in p$. If $S_1 \cup S_2 \in b{\mathcal {P}}_{\mathrm {inf,ip}}$, pick an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $S_1 \cup S_2 \in p \cdot q$. Since $p \cdot q$ is an ultrafilter, either $S_1 \in p \cdot q$ or $S_2 \in p \cdot q$. The “in particular" conclusions follow from [24,  Corollary 4.29]. $\square$