Questions in linear recurrence: From the $T\oplus T$-problem to lineability

by

Sophie Grivaux, Antoni López-Martínez & Alfred Peris2020 Mathematics Subject
Classification: 47A16, 37B20, 37B02.
Key words and phrases: Linear dynamical systems, hypercyclicity, recurrence,
quasi-rigidity, lineability.

###### Abstract

We study, for a continuous linear operator $T$ on an F-space $X$, when the
direct sum operator $T\oplus T$ is recurrent on $X\oplus X$. In particular: we
establish the analogous notion, for recurrence, to that of (topological) weak-
mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we
construct a recurrent but not quasi-rigid operator on each separable infinite-
dimensional Banach space, solving the $T\oplus T$-recurrence problem in the
negative way. The quasi-rigidity notion is closely related to the dense
lineability of the set of recurrent vectors, and using similar conditions we
study the lineability and dense lineability properties for the set of
$\mathcal{F}$-recurrent vectors, under very weak assumptions on the
Furstenberg family $\mathcal{F}$.

##  1 Introduction

This paper focuses on some aspects of Linear Dynamics and the general setting
is the following: a (real or complex) linear dynamical system $(X,T)$ is a
pair formed by a continuous linear operator $T:X\longrightarrow X$ on a (real
or complex) separable infinite-dimensional F-space (i.e. $X$ is a completely
metrizable topological vector space). We will denote by $\mathcal{L}(X)$ the
set of continuous linear operators acting on such a space $X$, and
$\mathbb{K}$ will stand for either the real or complex field, $\mathbb{R}$ or
$\mathbb{C}$.

Given a linear dynamical system $(X,T)$, the $T$-orbit of a vector $x\in X$ is
the set

$\textup{Orb}(x,T):=\\{x,Tx,T^{2}x,T^{3}x,...\\}=\\{T^{n}x:n\in\mathbb{N}_{0}\\},$

where $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. We say that a vector $x\in X$
is hypercyclic for $T$ if $\textup{Orb}(x,T)$ is a dense set in $X$; and that
the operator $T$ is hypercyclic if it admits a hypercyclic vector. The so-
called Birkhoff transitivity theorem (see for instance [7, Theorem 1.2]) shows
the equivalence between hypercyclicity and the notion of topological
transitivity: a system $(X,T)$ is called topologically transitive if for each
pair of non-empty open subsets $U,V\subset X$ one can find some (and hence
infinitely many) $n\in\mathbb{N}_{0}$ such that $T^{n}(U)\cap
V\neq\varnothing$. It follows that the set $\textup{HC}(T)$ of hypercyclic
vectors for $T$ is a dense $G_{\delta}$ subset of $X$ as soon as $T$ is
hypercyclic.

Hypercyclicity has been, historically, the main and most studied notion in
Linear Dynamics. For instance, the following long-standing problem, which we
call the $T\oplus T$-hypercyclicity problem, was posed in 1992 by D. Herrero
[33]:

######  Question 1.1 (The $T\oplus T$-hypercyclicity problem).

Let $T$ be a hypercyclic operator on the F-space $X$. Is the operator $T\oplus
T$ acting on the direct sum $X\oplus X$ hypercyclic?

Recall that an operator $T$ is said to be (topologically) weakly-mixing if and
only if $T\oplus T$ is transitive, so the question above asks if there exists
any transitive but not weakly-mixing operator. The active research on
hypercyclicity yielded many equivalent reformulations of Question 1.1, and the
approach which will interest us most is the one taken in 1999 by Bès and the
third author of this paper. Here is the main result they obtained in [10]:

  1. –

A continuous linear operator $T\in\mathcal{L}(X)$ is weakly-mixing if and only
if it satisfies the so-called Hypercyclicity Criterion.

In other words, $T\oplus T\in\mathcal{L}(X\oplus X)$ is a hypercyclic operator
if and only if the following holds: there exist two dense subsets
$X_{0},Y_{0}\subset X$, an increasing sequence of positive integers
$(n_{k})_{k\in\mathbb{N}}$, and a family of (not necessarily continuous)
mappings $S_{n_{k}}:Y_{0}\longrightarrow X$ such that

  1. (i)

$T^{n_{k}}x\to 0$ for each $x\in X_{0}$;

  2. (ii)

$S_{n_{k}}y\to 0$ for each $y\in Y_{0}$;

  3. (iii)

$T^{n_{k}}S_{n_{k}}y\to y$ for each $y\in Y_{0}$.

Question 1.1 was finally answered negatively in 2006 by De La Rosa and Read,
when they exhibited a hypercyclic operator on a particular Banach space whose
direct sum with itself is not hypercyclic (see [21]). The techniques used
there were later refined by Bayart and Matheron in order to construct
hypercyclic but not weakly-mixing operators, on each Banach space admitting a
normalized unconditional basis whose associated forward shift is continuous,
as for instance on the classical $c_{0}(\mathbb{N})$ and
$\ell^{p}(\mathbb{N})$ spaces, $1\leq p<\infty$ (see [5, 6]).

The aim of this paper is to study the previous problem, and to develop the
corresponding theory, for the notion of recurrence: we say that a vector $x\in
X$ is recurrent for $T$ if $x$ belongs to the closure of its forward orbit
$\overline{\textup{Orb}(Tx,T)}$; and that the operator $T$ is recurrent if its
set $\textup{Rec}(T)$ of recurrent vectors is dense in $X$. Note that, in
order to say that an operator $T$ has a recurrent behaviour, it is not enough
to assume that $\textup{Rec}(T)\neq\varnothing$ since the zero-vector is
recurrent for every linear map. It is shown in [19, Proposition 2.1] that the
notion of recurrence coincides with that of topological recurrence, i.e. the
property that for each non-empty open subset $U\subset X$ one can find some
(and hence infinitely many) $n\in\mathbb{N}$ such that $T^{n}(U)\cap
U\neq\varnothing$. In addition, and as in the hypercyclicity case, when $T$ is
recurrent its set $\textup{Rec}(T)$ of recurrent vectors is a dense
$G_{\delta}$ subset of $X$.

Recurrence is one of the oldest and most studied concepts in the Topological
Dynamics area of knowledge (see [24, 3, 36]), but in Linear Dynamics its
systematic study began with the 2014 paper of Costakis, Manoussos and Parissis
[19]. As one can naturally expect, many questions in linear recurrence remain
open and here we will be particularly interested in the so-called $T\oplus
T$-recurrence problem, which was stated in [19, Question 9.6]:

######  Question 1.2 (The $T\oplus T$-recurrence problem).

Let $T$ be a recurrent operator on the F-space $X$. Is the operator $T\oplus
T$ acting on the direct sum $X\oplus X$ recurrent?

This question has been recently restated as an open problem in the 2021 paper
[20] for C0-semigroups of operators. In the first part of this work we
identify the systems $(X,T)$ for which the recurrence of $T$ implies that of
$T\oplus T$. This characterization is given in terms of quasi-rigidity (see
Definition 2.2), which will be for recurrence, the analogous property to that
of weak-mixing for hypercyclicity (see Theorem 2.5). We then answer Question
1.2 negatively, by constructing some recurrent but not quasi-rigid operators
(see Section 3).

The second part of this paper is dedicated to the study of some properties
related to the notion of lineability: recall that a subset $Y$ of an F-space
$X$ is called (dense) lineable if $Y\cup\\{0\\}$ contains a (dense) infinite-
dimensional vector space. A well-known result (due to Herrero and Bourdon)
states that the set $\textup{HC}(T)$ is dense lineable for every hypercyclic
operator $T$ (see [31, Theorem 2.55]); moreover, one easily observes that if
$T$ is a quasi-rigid operator, then $\textup{Rec}(T)$ is also dense lineable
(see Proposition 2.7). Here we generalize these results to a rather general
class of Furstenberg families $\mathcal{F}$: first we show that it makes sense
to study quasi-rigidity from the $\mathcal{F}$-recurrence point of view (see
Proposition 4.5); and then we check that, for an $\mathcal{F}$-recurrent
operator $T$, the set $\mathcal{F}\textup{Rec}(T)$ of $\mathcal{F}$-recurrent
vectors is always lineable and usually even dense lineable (see Theorems 5.5
and 5.8). As a consequence we obtain the Herrero-Bourdon theorem for
$\mathcal{F}$-hypercyclicity (see Subsection 5.3).

The paper is organized as follows. Section 2 is devoted to define and study
quasi-rigidity, which is the analogous property, for recurrence, to that of
weak-mixing for hypercyclicity. We show in Section 3 that there exist
recurrent but not quasi-rigid operators, answering Question 1.2 in a negative
way and solving the $T\oplus T$-recurrence problem. In Section 4 we recall the
definition of $\mathcal{F}$-recurrence, showing that quasi-rigidity is a
particular case of such a concept, and studying both the weakest and strongest
possible $\mathcal{F}$-recurrence notions. The lineability and dense
lineability properties, for the set $\mathcal{F}\textup{Rec}(T)$ of
$\mathcal{F}$-recurrent vectors, are studied in Section 5. We finally gather,
in Section 6, some left open problems.

We refer the reader to the textbooks [7, 31] for any unexplained but standard
notion about hypercyclicity, or more generally, about Linear Dynamics. Along
the paper we use the environment “Question” to state the different problems
that we solve, and we use the word “Problem(s)” to state the left open
problems.

##  2 Quasi-rigid dynamical systems

In this section we introduce and study the concept of quasi-rigidity, which is
naturally defined in the broader framework of Topological Dynamics (i.e.
without assuming linearity): a pair $(X,T)$ is called a dynamical system if
$T$ is a continuous self-map on a Hausdorff topological space $X$. An
important class of dynamical systems, which we will repeatedly use, is the
family of Polish systems: a pair $(X,T)$ is said to be a Polish dynamical
system whenever $T$ is a continuous self-map of a separable completely
metrizable space $X$.

######  Definition 2.1.

Given any dynamical system $(X,T)$ and any $N\in\mathbb{N}$ we will denote by
$T_{(N)}:X^{N}\longrightarrow X^{N}$ the $N$-fold direct product of $T$ with
itself, i.e. $(X^{N},T_{(N)})$ is the dynamical system

$T_{(N)}:=\underbrace{T\times\cdots\times
T}_{N}:\underbrace{X\times\cdots\times
X}_{N}\longrightarrow\underbrace{X\times\cdots\times X}_{N}.$

where $X^{N}:=\underbrace{X\times\cdots\times X}_{N}$ is the $N$-fold product
of $X$ and $T_{(N)}(x_{1},...,x_{N}):=(Tx_{1},...,Tx_{N})$.

If $(X,T)$ is linear, then $(X^{N},T_{(N)})$ will refer to the $N$-fold direct
sum linear dynamical system

$T_{(N)}:=\underbrace{T\oplus\cdots\oplus
T}_{N}:\underbrace{X\oplus\cdots\oplus
X}_{N}\longrightarrow\underbrace{X\oplus\cdots\oplus X}_{N},$

and in this case $X^{N}:=\underbrace{X\oplus\cdots\oplus X}_{N}$ is the
$N$-fold direct sum of $X$.

###  2.1 Quasi-rigidity: definition and equivalences

By a well-known theorem of Furstenberg, once a dynamical system $(X,T)$ is
weakly-mixing (i.e. if $T\times T$ is topologically transitive) then so is
every $N$-fold direct product system $(X^{N},T_{(N)})$. See [31, Theorem
1.51]. Hence, in order to completely answer Question 1.2 we should first study
the recurrent dynamical systems $(X,T)$ for which every $N$-fold direct
product is again recurrent. A first attempt would be to rely on the notion of
rigidity: a dynamical system $(X,T)$ is said to be rigid if there exists an
increasing sequence of natural numbers $(n_{k})_{k\in\mathbb{N}}$ such that
$T^{n_{k}}x\to x$ for every $x\in X$.

Rigidity has been studied in different contexts such as measure theoretic
recurrence [25], dynamical systems on topological spaces [27] and also for
linear systems [22, 19]. It is a really strong form of recurrence which
implies that $X^{N}=\textup{Rec}(T_{(N)})$ for all $N\in\mathbb{N}$, so this
is not the exact notion we are looking for. Nonetheless, it motivates the
definition:

######  Definition 2.2 (Quasi-rigidity).

Let $(X,T)$ be a dynamical system and let $(n_{k})_{k\in\mathbb{N}}$ be a
strictly increasing sequence of positive integers. We say that $T$ is:

  1. –

quasi-rigid with respect to the sequence $(n_{k})_{k\in\mathbb{N}}$, if there
exists a dense subset $Y$ of $X$ such that $T^{n_{k}}x\to x$, as $k\to\infty$,
for every $x\in Y$.

  2. –

topologically quasi-rigid with respect to $(n_{k})_{k\in\mathbb{N}}$, if for
every non-empty open subset $U\subset X$ there exists some
$k_{U}\in\mathbb{N}$ such that $T^{n_{k}}(U)\cap U\neq\varnothing$ for every
$k\geq k_{U}$.

Our aim is to show that, given a dynamical system $(X,T)$, the previous
notions characterize the recurrent behaviour of every $N$-fold direct product
system $(X^{N},T_{(N)})$ under very weak assumptions (dynamically speaking) on
the underlying space $X$. The following result shows that, when the underlying
space $X$ is second-countable, topological quasi-rigidity is for recurrence
the analogous property to that of weak-mixing for transitivity:

######  Lemma 2.3.

Let $(X,T)$ be a dynamical system on a second-countable space $X$. The
following are equivalent:

  1. (i)

$T$ is topologically quasi-rigid;

  2. (ii)

$T_{(N)}$ is topologically recurrent for every $N\in\mathbb{N}$.

###### Proof.

(i) $\Rightarrow$ (ii): suppose that $T$ is topologically quasi-rigid with
respect to $(n_{k})_{k\in\mathbb{N}}$, let $N\in\mathbb{N}$ and consider any
finite sequence of non-empty open sets $U_{1},...,U_{N}\subset X$. Using now
the topological quasi-rigidity assumption we can find $k_{0}\in\mathbb{N}$
such that

$T^{n_{k}}(U_{j})\cap U_{j}\neq\varnothing\quad\text{ for every }1\leq j\leq
N\text{ and all }k\geq k_{0}.$

Then, for the open set $U=U_{1}\times\cdots\times U_{N}\subset X^{N}$ we have
that $T_{(N)}^{n_{k}}(U)\cap U\neq\varnothing$ for all $k\geq k_{0}$, so
$(X^{N},T_{(N)})$ is topologically recurrent.

(ii) $\Rightarrow$ (i): assume that $T_{(N)}$ is topologically recurrent for
every $N\in\mathbb{N}$ and let $(U_{s})_{s\in\mathbb{N}}$ be a countable base
of (non-empty) open sets for $X$. We can construct recursively an increasing
sequence of positive integers $(n_{k})_{k\in\mathbb{N}}$ such that
$T^{n_{k}}(U_{s})\cap U_{s}\neq\varnothing$ for every $s,k\in\mathbb{N}$ with
$1\leq s\leq k$: given any $k\in\mathbb{N}$, the set
$U:=U_{1}\times\cdots\times U_{k}$ is a non-empty open subset of $X^{k}$ and
by the topological recurrence of the map $T_{(k)}$ we can pick
$n_{k}\in\mathbb{N}$ sufficiently large with

$T_{(k)}^{n_{k}}(U)\cap U\neq\varnothing\quad\text{ and hence }\quad
T^{n_{k}}(U_{s})\cap U_{s}\neq\varnothing\quad\text{ for all }1\leq s\leq k.$

It is easy to check that $T$ is topologically quasi-rigid with respect to
$(n_{k})_{k\in\mathbb{N}}$. ∎

As one would expect, if the underlying space is completely metrizable, we can
identify the “pointwise” quasi-rigidity notion with that of topological quasi-
rigidity:

######  Proposition 2.4.

Let $(X,T)$ be a Polish dynamical system. The following are equivalent:

  1. (i)

$T$ is quasi-rigid;

  2. (ii)

$T$ is topologically quasi-rigid.

###### Proof.

The implication (i) $\Rightarrow$ (ii) is straightforward: assume that there
is a dense subset $Y\subset X$ and an increasing sequence
$(n_{k})_{k\in\mathbb{N}}$ such that $T^{n_{k}}x\to x$ for every $x\in Y$. For
each non-empty open subset $U\subset X$ we can select $x\in U\cap Y$, and then

$T^{n_{k}}x\in T^{n_{k}}(U)\cap U\quad\text{ for every sufficiently large
}k\in\mathbb{N}.$

Hence $T$ is topologically quasi-rigid with respect to
$(n_{k})_{k\in\mathbb{N}}$. Let us prove (ii) $\Rightarrow$ (i): assume that
$d(\cdot,\cdot)$ is a metric defining the complete topology of $X$, let
$(U_{s})_{s\in\mathbb{N}}$ be a countable base of open sets for $X$ and set
$V_{s,s}:=U_{s}$ for each $s\in\mathbb{N}$. By (ii) we can find
$n_{1}\in\mathbb{N}$ such that

$T^{n_{1}}(U_{1})\cap U_{1}=T^{n_{1}}(V_{1,1})\cap V_{1,1}\neq\varnothing.$

The continuity of $T$ ensures the existence of a non-empty open subset
$V_{1,2}\subset X$ of diameter (with respect to $d$) less than
$\frac{1}{2^{2}}$ for which $\overline{V_{1,2}}\subset V_{1,1}$ and
$T^{n_{1}}\left(\overline{V_{1,2}}\right)\subset V_{1,1}$. Suppose now that
for some $k\in\mathbb{N}$ we have already constructed:

  1. –

finite sequences $(V_{s,j})_{s\leq j\leq k}$ of open subsets of $X$, for each
$1\leq s\leq k-1$;

  2. –

and a finite increasing sequence of positive integers $(n_{j})_{1\leq j\leq
k-1}$;

with the properties that $V_{s,j}$ has $d$-diameter less than
$\frac{1}{2^{j}}$ when $s<j$, and also that

$\overline{V_{s,j}}\subset V_{s,j-1}\quad\text{ and }\quad
T^{n_{j-1}}\left(\overline{V_{s,j}}\right)\subset V_{s,j-1}\quad\text{ for all
}1\leq s<j\leq k.$

Then, considering the open subsets $V_{1,k},V_{2,k},...,V_{k,k}\subset X$ and
using again (ii) we can select a positive integer $n_{k}\in\mathbb{N}$ with
$n_{k}>n_{k-1}$ for which $T^{n_{k}}(V_{s,k})\cap V_{s,k}\neq\varnothing$ for
all $1\leq s\leq k$. Again the continuity ot $T$ ensures the existence, for
each $1\leq s\leq k$, of a non-empty open subset $V_{s,k+1}\subset X$ of
$d$-diameter less than $\frac{1}{2^{k+1}}$ and such that

$\overline{V_{s,k+1}}\subset V_{s,k}\quad\text{ and }\quad
T^{n_{k}}\left(\overline{V_{s,k+1}}\right)\subset V_{s,k}\quad\text{ for all
}1\leq s\leq k.$

A recursive argument gives us an increasing sequence of positive integers
$(n_{k})_{k\in\mathbb{N}}$ and, for each $s\in\mathbb{N}$, a sequence of non-
empty open sets $(V_{s,k})_{k=s}^{\infty}$ such that each set $V_{s,k}$ has
$d$-diameter less than $\frac{1}{2^{k}}$ when $s<k$, but also satisfying

$\overline{V_{s,k+1}}\subset V_{s,k}\quad\text{ and }\quad
T^{n_{k}}\left(\overline{V_{s,k+1}}\right)\subset V_{s,k}\quad\text{ for all
}s,k\in\mathbb{N}\text{ with }s\leq k.$

By the Cantor intersection theorem, for each $s\in\mathbb{N}$ there is a
unique vector $y_{s}\in X$ such that

$\\{y_{s}\\}=\bigcap_{k\geq s}\overline{V_{s,k}}\subset V_{s,s}=U_{s}.$

We deduce that $Y:=\\{y_{s}:s\in\mathbb{N}\\}$ is a dense subset of $X$. Since
for each $s\in\mathbb{N}$ we have that $T^{n_{k}}y_{s}\in
T^{n_{k}}\left(\overline{V_{s,k+1}}\right)\subset V_{s,k}$ for all $k>s$, we
get that

$\limsup_{k\to\infty}d(T^{n_{k}}y_{s},y_{s})\leq\limsup_{k\to\infty}\left(\textup{diam}_{d}(V_{s,k})\right)\leq\limsup_{k\to\infty}\frac{1}{2^{k}}=0,$

and hence $T$ is quasi-rigid with respect to the sequence
$(n_{k})_{k\in\mathbb{N}}$. ∎

The proof of the implication (i) $\Rightarrow$ (ii) in Proposition 2.4 shows
that every quasi-rigid map is topologically quasi-rigid even if $X$ is not a
metrizable space. The converse fails in general (consider for instance [31,
Example 12.9]). Combining the previous results we get:

######  Theorem 2.5.

Let $(X,T)$ be a Polish dynamical system. The following are equivalent:

  1. (i)

$T$ is quasi-rigid;

  2. (ii)

$T$ is topologically quasi-rigid;

  3. (iii)

$T_{(N)}$ is topologically recurrent for every $N\in\mathbb{N}$;

  4. (iv)

$T_{(N)}$ is recurrent for every $N\in\mathbb{N}$.

###### Proof.

The implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) (and also (i)
$\Rightarrow$ (iv) $\Rightarrow$ (iii)) are trivial even if the space $X$ is
not second-countable neither complete. When $X$ is second-countable we get the
equivalence (ii) $\Leftrightarrow$ (iii) by Lemma 2.3. If we just assume
completeness, then we have the equivalence (iii) $\Leftrightarrow$ (iv) by
[19, Proposition 2.1]. Finally, using the Polish hypothesis we get the
equivalence (i) $\Leftrightarrow$ (ii) by Proposition 2.4. ∎

######  Remark 2.6.

Even though we have worked in the general setting of Polish dynamical systems,
it is worth mentioning that Theorem 2.5 holds true for every:

  1. –

compact dynamical system, i.e. when $T$ is a continuous self-map of a compact
metrizable space $X$ (many of references just study this class of systems, see
for instance [3, 24, 36]);

  2. –

continuous linear operator $T$ acting on any separable F-space $X$. In other
words, the previous result holds true for linear dynamical systems as defined
in the Introduction.

In the linear setting we have got the following relations between the already
exposed concepts:

$\begin{CD}\text{{\em satisfying the HC}}@>{\text{Theorem\leavevmode\nobreak\
\ref{The:Juan}}}>{}>\text{{\em quasi-rigidity}}\\\ @V{}V{}V@V{}V{}V\\\
\text{{\em hypercyclicity}}@>{\text{already known}}>{}>\text{{\em
recurrence}}\end{CD}$

i.e. the quasi-rigidity notion is for recurrence, the analogous property to
that of satisfying the Hypercyclicity Criterion (and hence to that of weak-
mixing) for hypercyclicity (see [10]).

###  2.2 Quasi-rigid operators

It is now time to discuss how the quasi-rigidity notion influences the
structure of the set of recurrent vectors for a linear dynamical system
$(X,T)$. We start by showing that quasi-rigidity implies the dense lineability
of the set of recurrent vectors:

######  Proposition 2.7.

If $T:X\longrightarrow X$ is a quasi-rigid operator, then $\textup{Rec}(T)$ is
dense lineable.

###### Proof.

Assume that $Y\subset X$ is a dense subset such that there exists a sequence
$(n_{k})_{k\in\mathbb{N}}$ of integers with $T^{n_{k}}x\to x$ for all $x\in
Y$. Then any $z\in Z:=\textup{span}(Y)$ satisfies that $T^{n_{k}}z\to z$, so
$Z\subset X$ is an infinite-dimensional dense vector space contained in
$\textup{Rec}(T)$. ∎

######  Remark 2.8.

Let $T:X\longrightarrow X$ be a quasi-rigid operator with respect to a
sequence $(n_{k})_{k\in\mathbb{N}}$ and consider $Y:=\left\\{x\in
X:T^{n_{k}}x\to x\text{ as }k\to\infty\right\\}$. This is a dense subset of
$X$ which can be either of first or second Baire category, depending on the
operator $T$:

  1. (a)

If $T$ is a rigid operator, then $Y=X$. This is the case of the identity
operator, but there also exist examples of rigid systems which are even
(Devaney) chaotic (see [22]).

  2. (b)

If $X$ is a Banach space and $Y$ is a second category set, then
$\sup_{k\in\mathbb{N}}\|T^{n_{k}}\|<\infty$ by the Banach-Steinhaus theorem.
Consider the operator $T:=\lambda B$ with $|\lambda|>1$, where $B$ is the
(unilateral) backward shift on $c_{0}(\mathbb{N})$ or any
$\ell^{p}(\mathbb{N})$ ($1\leq p<\infty$). This operator is weakly-mixing and
hence quasi-rigid, but the set $Y$ as defined before has to be of first
category since $x=(\frac{1}{n^{2}})_{n\in\mathbb{N}}\in X$ has the property
that $\|T^{n}x\|\to\infty$ as $n\to\infty$.

Now we give some (usually fulfilled) sufficient conditions for a dynamical
system to be quasi-rigid. Recall that a vector $x\in X$ is called cyclic for
an operator $T:X\longrightarrow X$ if

$\textup{span}(\textup{Orb}(x,T))=\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}=\left\\{p(T)x:p\text{
polynomial}\right\\},$

is a dense set in $X$; and an operator $T$ is called cyclic as soon as it
admits a cyclic vector. Moreover, a point $x\in X$ is called periodic for a
map $T$ whenever $T^{p}x=x$ for some positive integer $p\in\mathbb{N}$. See
[7, 31] for more on cyclicity and periodicity.

######  Proposition 2.9.

Let $(X,T)$ be a dynamical system for which any of the following holds:

  1. (a)

the set $\operatorname{Per}(T)$ of periodic points for $T$ is dense in $X$;

  2. (b)

$T$ admits a recurrent vector $x\in\textup{Rec}(T)$ for which the following
set is dense

$\\{Sx:ST=TS,S:X\longrightarrow X\text{ continuous mapping}\\};$

then $T$ is quasi-rigid. In particular, a continuous linear operator $T$
acting on a Hausdorff topological vector space $X$ is quasi-rigid whenever any
of the following holds:

  1. –

$T$ has dense periodic vectors;

  2. –

$T$ admits a recurrent and cyclic vector;

  3. –

$T$ admits a hypercyclic vector.

###### Proof.

To check (a) consider $Y=\operatorname{Per}(T)$ and
$(n_{k})_{k\in\mathbb{N}}=(k!)_{k\in\mathbb{N}}$. For (b) consider

$Y=\\{Sx:ST=TS,S:X\longrightarrow X\text{ continuous mapping}\\}.$

Indeed, since $x$ is recurrent there exists $(n_{k})_{k\in\mathbb{N}}$ such
that $T^{n_{k}}x\to x$, but then, given any point $y=Sx\in Y$ we get that
$T^{n_{k}}(Sx)=S(T^{n_{k}}x)\to Sx$. Finally, recall that every recurrent and
cyclic (and hence every hypercyclic) vector fulfills condition (b). ∎

Proposition 2.9 together with [19, Corollary 9.2] show that recurrence plus
cyclicity imply quasi-rigidity when $X$ is a complex Banach space. This result
extends to (real and complex) linear dynamical systems as defined in the
Introduction via the following lemma, in which the concept of complexification
is used (see Remark 5.2). We write $\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$.

######  Lemma 2.10.

Let $T$ be a topologically recurrent continuous linear operator acting on a
(real or complex) Hausdorff topological vector space $X$. The following
statements hold:

  1. (a)

For every (real or complex) number $\lambda\in\mathbb{K}\setminus\mathbb{T}$
the operator $(T-\lambda)$ has dense range.

  2. (b)

If $X$ is a real space, then for every
$\lambda\in\mathbb{C}\setminus\mathbb{T}$ the complexified operator
$(\widetilde{T}-\lambda)$, acting on the complexification $\widetilde{X}$ of
$X$, has dense range.

As a consequence, for any (real or complex) polynomial $p$ without unimodular
roots, the operator $p(T)$ has dense range on the (real or complex) space $X$.

The proof of this lemma is based on [31, Lemma 12.13], a well-known result of
Wengenroth for topologically transitive operators acting on Hausdorff
topological vector spaces. We just show statement (a), since (b) follows
exactly as in [31, Exercise 12.2.6] (Hint: when $X$ is a real space define the
complexification $\widetilde{T}:\widetilde{X}\longrightarrow\widetilde{X}$ as
it is done in Remark 5.2 for operators on F-spaces, and use the fact that for
each non-empty open subset $U\subset X$ there are infinitely many positive
integers $n\in\mathbb{N}$ with
$\widetilde{T}^{n}(U+iU)\cap(U+iU)\neq\varnothing$).

###### Proof of Lemma 2.10.

Given $\lambda\in\mathbb{K}\setminus\mathbb{T}$ let
$E:=\overline{(T-\lambda)(X)}$, which is a closed subspace of $X$, and suppose
that $E\neq X$. The quotient space $X/E$ is then a (real or complex) Hausdorff
topological vector space (since $E$ is closed). If $q:X\longrightarrow X/E$ is
the quotient map we have that $q((T-\lambda)x)=0$ and hence $q(Tx)=\lambda
q(x)$ for every $x\in X$. The operator $S$ on $X/E$ given by $S[x]=\lambda[x]$
is easily seen to be topologically recurrent since so is $T$.

However, if $|\lambda|<1$ we can choose $[x]\in X/E\setminus\\{0\\}$ and a
balanced $0$-neighbourhood $W$ such that $[x]\notin\overline{W}$. Since
$\lambda^{n}[x]\to 0$ and $\lambda W\subset W$ there exists a neighbourhood
$U$ of $[x]$ and a natural number $N\in\mathbb{N}$ for which $U\cap
W=\varnothing$ and $\lambda^{n}U\subset W$ for every $n\geq N$. Hence
$S^{n}(U)\cap U=\varnothing$ for all $n\geq N$ so $S$ is not topologically
recurrent, which is a contradiction. If $|\lambda|>1$ we can consider the map
$S^{-1}:X/E\longrightarrow X/E$ for which $S^{-1}[x]=\lambda^{-1}[x]$. By the
previous argument $S^{-1}$ is not topologically recurrent, but then $S$ is not
topologically recurrent (see [29, Section 6]), which is again a contradiction.

Let now $p$ be any (real or complex) polynomial without unimodular roots: if
$X$ is a complex space then $p(T)$ has dense range since, splitting the
polynomial, $p(T)$ can be written as a composition of dense range operators by
(a); and if $X$ is a real space then $p(\widetilde{T})$ also has dense range
in $\widetilde{X}$ via the same argument as above but using (b). Finally, if
$p$ has real coefficients, then $p(\widetilde{T})=p(T)+ip(T)$ and $p(T)$ has
dense range in the real space $X$. ∎

The proof of [19, Corollary 9.2] is highly based on the following result of D.
Herrero: given an operator $T$ on a complex Banach space $X$, if
$\sigma_{p}(T^{*})$ has empty interior then the set of cyclic vectors for $T$
is dense, and hence it is a dense $G_{\delta}$-set (see [32]). By using Lemma
2.10 we obtain this density result in the more general setting of operators on
F-spaces:

######  Corollary 2.11 (Extension of [19, Corollary 9.2]).

Let $(X,T)$ be a (real or complex) linear dynamical system. If $T$ is
recurrent and cyclic then the set of cyclic vectors for $T$ is a dense
$G_{\delta}$-set. In particular, if $T$ is recurrent and cyclic, then it is
quasi-rigid.

###### Proof.

The set of cyclic vectors is always a $G_{\delta}$-set. To check its density
let $x\in X$ be a cyclic vector for $T$ and pick a non-empty open set
$U\subset X$. Then there is a (real or complex) polynomial $p$ such that
$p(T)x\in U$. Splitting $p$ we have that

$\textstyle p(T)=\prod_{j=1}^{N}(T-\lambda_{j})\quad\text{ for some
}(\lambda_{j})_{j=1}^{N}\subset\mathbb{C}.$

Choose a small perturbation
$(\mu_{j})_{j=1}^{N}\subset\mathbb{C}\setminus\mathbb{T}$ of
$(\lambda_{j})_{j=1}^{N}\subset\mathbb{C}$ with the property that the
polynomial

$\textstyle\widehat{p}(T):=\prod_{j=1}^{N}(T-\mu_{j})\quad\text{ is still such
that }\widehat{p}(T)x\in U.$

Note that if $p$ has real coefficients, then
$(\mu_{i})_{i=1}^{N}\subset\mathbb{C}\setminus\mathbb{T}$ can be chosen such
that $\widehat{p}$ has still real coefficients. Finally, $\widehat{p}(T)x$ is
a cyclic vector for $T$ since

$\widehat{p}(T)\left(\\{q(T)x:q\text{
polynomial}\\}\right)=\\{q(T)(\widehat{p}(T)x):q\text{ polynomial}\\}$

is a dense set by Lemma 2.10. Then $\textup{Rec}(T)$ and the set of cyclic
vectors for $T$ are both residual, so $T$ admits a recurrent and cyclic
vector. Proposition 2.9 ends the proof. ∎

##  3 Existence of recurrent but not quasi-rigid operators

After having studied in Section 2 the dynamical systems $(X,T)$ for which all
their $N$-fold direct sums $(X^{N},T_{(N)})$ are recurrent, we are now ready
to answer negatively Question 1.2 showing that there exist recurrent but not
quasi-rigid operators. If for the moment we continue working in the (not
necessarily linear) dynamical systems setting, the following are natural
questions in view of the theory already developed in the previous section:

######  Question 3.1.

Let $(X,T)$ be a recurrent dynamical system:

  1. (a)

Is $(X\times X,T\times T)$ again a recurrent dynamical system?

  2. (b)

Assume that $(X\times X,T\times X)$ is recurrent. Is then $(X,T)$ quasi-rigid?

If we change the “recurrence” assumption into that of “topological
transitivity” then the respective answers are:

  1. (a)

No, and in particular:

    1. –

for non-linear systems consider any irrational rotation, see [31, Example
1.43];

    2. –

for linear maps we already mentioned in the Introduction the references [21,
5, 6].

  2. (b)

Yes, in both linear and non-linear cases $(X^{N},T_{(N)})$ is transitive for
every $N\in\mathbb{N}$ if and only if $T\times T$ is transitive (see [31,
Theorem 1.51]).

However, the answer to both questions is no in the recurrence setting, and
indeed: in [3, Lemma 9 and Example 4] it is shown that for each
$N\in\mathbb{N}$ there exists a non-linear dynamical system
$f:X\longrightarrow X$ on a compact metric space $X$ for which

$(X^{N},f_{N})\text{ is recurrent }\quad\text{ while
}\quad(X^{N+1},f_{N+1})\text{ is not recurrent}.$

This construction is highly non-linear since each compact space $X$ is the
disjoint union of $N+1$ sub-shifts of the well-known shift on two symbols. It
turns out that the answer to Question 3.1 in the framework of Linear Dynamics
is also negative. Let us state the main result of this section:

######  Theorem 3.2.

Let $X$ be any (real or complex) separable infinite-dimensional Banach space.
For each $N\in\mathbb{N}$ there exists an operator $T\in\mathcal{L}(X)$ such
that

$T_{(N)}:X^{N}\longrightarrow X^{N}\text{ is recurrent, and even
}\textup{Rec}(T_{(N)})=X^{N},$

but for which the operator $T_{(N+1)}:X^{N+1}\longrightarrow X^{N+1}$ is not
recurrent any more.

Considering $N=1$ we obtain examples of recurrent operators
$T\in\mathcal{L}(X)$ for which $T\oplus T$ is not recurrent. Thus we get a
negative answer to the $T\oplus T$-recurrence problem, i.e. we answer [19,
Question 9.6] and hence Question 1.2 in the negative.

Our proof of Theorem 3.2 relies heavily on a construction of Augé in [2] of
operators $T$ on (infinite-dimensional, separable) Banach spaces $X$ with wild
dynamics: the two sets

$A_{T}=\left\\{x\in X:\lim_{n\to\infty}\|T^{n}x\|=\infty\right\\}\quad\text{
and }\quad B_{T}=\left\\{x\in X:\liminf_{n\to\infty}\|T^{n}x-x\|=0\right\\}$

have non-empty interior and form a partition of $X$. Since
$B_{T}=\textup{Rec}(T)$, these operators have plenty (but not a dense set) of
recurrent vectors. In the remainder of this section we prove the complex
version of Theorem 3.2 by modifying the construction of [2]. The real case
follows in a really similar way by using the same arguments as in [2, Section
3.2].

###  3.1 Necessary prerequisites

Assume that $X$ is a complex separable infinite-dimensional Banach space,
denote by $X^{*}$ its topological dual space and by $B_{X^{*}}$ the unit ball
of such a space $X^{*}$. Moreover, given $(x,x^{*})\in X\times X^{*}$ we will
denote by $\langle x^{*},x\rangle=x^{*}(x)$ the dual evaluation. As in [2],
our operator will be built using a bounded biorthogonal system of $X$. The
following result is well-known (see [37, Vol I, Section 1.f] or [42]):

  1. –

Given a separable Banach space $X$ one can find
$(e_{n},e_{n}^{*})_{n\in\mathbb{N}}\subset X\times X^{*}$ such that:

     * •

$\textup{span}\\{e_{n}:n\in\mathbb{N}\\}$ is dense in $X$;

     * •

$\langle e_{n}^{*},e_{m}\rangle=\delta_{n,m}$ where $\delta_{n,m}=0$ if $n\neq
m$ and $1$ if $n=m$;

     * •

for each $n\in\mathbb{N}$ we have that $\|e_{n}\|=1$, and
$K:=\sup_{n\in\mathbb{N}}\|e_{n}^{*}\|<\infty$.

Once we have fixed a sequence $(e_{n},e_{n}^{*})_{n\in\mathbb{N}}\subset
X\times X^{*}$ with the previous properties we set

$c_{00}:=\textup{span}\\{e_{n}:n\in\mathbb{N}\\}.$

Given $x\in X$ we will write $x_{k}:=\langle e_{k}^{*},x\rangle$ for each
$k\in\mathbb{N}$, and we will repeatedly use that

$\|x_{k}e_{k}\|\leq K\|x\|\quad\text{ for each }k\in\mathbb{N},$ (1)

since

$\|x\|:=\sup_{x^{*}\in B_{X^{*}}}|\langle x^{*},x\rangle|\geq\frac{|\langle
e_{k}^{*},x\rangle|}{\|e_{k}^{*}\|}=\frac{|x_{k}|}{\|e_{k}^{*}\|}=\frac{\|x_{k}e_{k}\|}{\|e_{k}^{*}\|}\geq\frac{\|x_{k}e_{k}\|}{K},$

for each $k\in\mathbb{N}$, and hence that

$\|x-y\|<\varepsilon\ \text{ implies }\ |x_{k}-y_{k}|<K\varepsilon\quad\text{
for all }x,y\in X\text{ and }k\in\mathbb{N}.$ (2)

######  Remark 3.3.

If $(e_{n})_{n\in\mathbb{N}}$ is not a Schauder basis of $X$, the above
inequalities are still true but a vector $x\in X$ cannot in general be written
as a convergent series $x=\sum_{k\in\mathbb{N}}x_{k}e_{k}$. However, for the
vectors in $c_{00}=\textup{span}\\{e_{n}:n\in\mathbb{N}\\}$ this equality will
be true: for each $x\in c_{00}$ there is some $n\in\mathbb{N}$ such that

$x=\sum_{k=1}^{n}\langle e_{k}^{*},x\rangle e_{k}=\sum_{k=1}^{n}x_{k}e_{k}.$

From now on we fix a natural number $N\in\mathbb{N}$. We are going to
construct

  1. –

a projection $P$ from $X$ into itself;

  2. –

a sequence of functionals $(w_{k}^{*})_{k\geq N+2}\subset X^{*}$;

  3. –

two sequences $(m_{k})_{k\in\mathbb{N}}\in\mathbb{N}^{\mathbb{N}}$ and
$(\lambda_{k})_{k\in\mathbb{N}}\in\mathbb{T}^{\mathbb{N}}$;

  4. –

and two operators $R$ and $T$ on $X$;

in such a way that $T_{(N)}$ is recurrent while $T_{(N+1)}$ is not.

###  3.2 The projection $P$ and the sequences $(w_{k}^{*})_{k\geq N+2}$,
$(m_{k})_{k\in\mathbb{N}}$ and $(\lambda_{k})_{k\in\mathbb{N}}$

Denote by $P:X\longrightarrow\textup{span}\\{e_{1},e_{2},...,e_{N+1}\\}$ the
projection of $X$ onto the linear span of the first $N+1$ basis vectors
defined by

$Px:=\sum_{j=1}^{N+1}\langle e_{j}^{*},x\rangle e_{j}\quad\text{ for every
}x\in X.$

Note that $P$ is continuous. In fact $\|P\|\leq(N+1)K$. Let
$E:=\textup{span}\\{e_{1}^{*},e_{2}^{*},...,e_{N+1}^{*}\\}$ endowed with the
norm $\|\cdot\|^{*}$ of $X^{*}$, and consider on $E$ the equivalent norm
$\|\cdot\|_{\infty}$ defined in the following way for each
$\alpha_{1},\alpha_{2},...,\alpha_{N+1}\in\mathbb{C}$,

$\left\|\sum_{j=1}^{N+1}\alpha_{j}e_{j}^{*}\right\|_{\infty}:=\max\\{|\alpha_{1}|,|\alpha_{2}|,...,|\alpha_{N+1}|\\}.$

This makes sense because the vectors $e_{1}^{*},...,e_{N+1}^{*}$ are linearly
independent. There exist constants $M\geq m>0$ such that

$m\|w^{*}\|_{\infty}\leq\|w^{*}\|^{*}\leq M\|w^{*}\|_{\infty}\quad\text{ for
every }w^{*}\in E.$

Moreover, since the unit sphere $S_{E_{\infty}}:=\\{w^{*}\in
E:\|w^{*}\|_{\infty}=1\\}$ is a compact metrizable space there exists a
sequence $(w_{k}^{*})_{k\geq N+2}\subset S_{E_{\infty}}$, which is dense in
$S_{E_{\infty}}$. We include here a fact which will be used in the sequel:

######  Fact 3.2.1.

For each $w^{*}\in S_{E_{\infty}}$ there exists an index
$i\in\\{1,2,...,N+1\\}$ such that the following holds: for every
$\varepsilon>0$ and every $x\in X$ with $\|x-e_{i}\|<\varepsilon$ we have that

$\left|\langle w^{*},Px\rangle\right|>1-(N+1)K\varepsilon.$

###### Proof.

Since $w^{*}\in S_{E_{\infty}}$ there are coefficients
$(\alpha_{j})_{j=1}^{N+1}\in\mathbb{C}^{N+1}$ such that
$w^{*}=\sum_{j=1}^{N+1}\alpha_{j}e_{j}^{*}$, and there exists at least one
index $i\in\\{1,2,...,N+1\\}$ with $|\alpha_{i}|=1$. Given $\varepsilon>0$ and
$x\in X$ with $\|x-e_{i}\|<\varepsilon$, the inequality (2) implies that
$|x_{i}-1|<K\varepsilon$ while $|x_{j}|<K\varepsilon$ for every $1\leq j\leq
N+1$ with $j\neq i$. Since $0\leq|\alpha_{j}|\leq 1$ for every $1\leq j\leq
N+1$ we get that

$\left|\langle
w^{*},Px\rangle\right|=\left|\sum_{j=1}^{N+1}\alpha_{j}x_{j}\right|\geq\left|x_{i}\right|-\sum_{j=1,j\neq
i}^{N+1}|\alpha_{j}|\left|x_{j}\right|>(1-K\varepsilon)-NK\varepsilon=1-(N+1)K\varepsilon.\qed$

Let $(m_{k})_{k\in\mathbb{N}}\in\mathbb{N}^{\mathbb{N}}$ be a sequence of
positive integers with the following properties:

  1. (a)

$m_{k}\mid m_{k+1}$ for each $k\geq 1$;

  2. (b)

$m_{1}=m_{2}=\cdots=m_{N+1}=1$;

and starting from $k=N+2$, the sequence $(m_{k})_{k\geq N+2}$ grows fast
enough to satisfy:

  1. (c)

$\displaystyle\lim_{j\to\infty}\textstyle\left(m_{j}\cdot\sum_{k=j+1}^{\infty}\frac{1}{m_{k}}\right)=0$.

The sequence $(\lambda_{k})_{k\in\mathbb{N}}\in\mathbb{T}^{\mathbb{N}}$ is
then defined by setting $\lambda_{k}:=e^{2\pi i\frac{1}{m_{k}}}$ for each
$k\in\mathbb{N}$. Using the inequality $|e^{i\theta}-1|\leq|\theta|$, which is
true for every $\theta\in\mathbb{R}$, and also the fact that
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{N+1}=1$, we deduce the inequality

$\sum_{k=1}^{\infty}|\lambda_{k}-1|\leq
2\pi\sum_{k=N+2}^{\infty}\frac{1}{m_{k}}<\infty,$ (3)

since condition (c) on the sequence $(m_{k})_{k\in\mathbb{N}}$ implies that
the series $\sum_{k=1}^{\infty}\frac{1}{m_{k}}$ is convergent.

###  3.3 The operators $R$ and $T$

For each $x\in c_{00}=\textup{span}\\{e_{n}:n\in\mathbb{N}\\}$, which we write
as $x=\sum_{k=1}^{n}x_{k}e_{k}$, we set

$Rx:=\sum_{k=1}^{n}\lambda_{k}x_{k}e_{k}.$

This defines a bounded operator $R$ on $c_{00}$. Indeed, for every
$x=\sum_{k=1}^{n}x_{k}e_{k}\in c_{00}$,

$\|Rx\|\leq\|Rx-x\|+\|x\|\leq\sum_{k=1}^{n}|\lambda_{k}-1|\cdot\|x_{k}e_{k}\|+\|x\|\leq\left(K\sum_{k=1}^{\infty}|\lambda_{k}-1|+1\right)\|x\|$

by (1). Using the inequality (3) we see that the map $R$ extends to a bounded
operator on $X$ still denoted by $R$. We now define the operator $T$ on $X$ by
setting

$Tx:=Rx+\sum_{k=N+2}^{\infty}\frac{1}{m_{k-1}}\langle w_{k}^{*},Px\rangle
e_{k}\quad\text{ for every }x\in X.$ (4)

The second sum in the expression (4) defines a bounded operator by the
assumption (c) on the sequence $(m_{k})_{k\in\mathbb{N}}$, the fact that
$\|P\|\leq(N+1)K$ and also that $\|w_{k}^{*}\|^{*}\leq M$ for each $k\geq
N+2$. Indeed,

$\left\|\sum_{k=N+2}^{\infty}\frac{1}{m_{k-1}}\langle w_{k}^{*},Px\rangle
e_{k}\right\|\leq\sum_{k=N+2}^{\infty}\frac{\|w_{k}^{*}\|^{*}\cdot\|Px\|}{m_{k-1}}\leq\left(M\cdot(N+1)K\cdot\sum_{k=N+2}^{\infty}\frac{1}{m_{k-1}}\right)\|x\|,$

for every $x\in X$, where the last parenthesis has finite value by assumption
(c) on $(m_{k})_{k\in\mathbb{N}}$. It follows that $T$ is a continuous
operator on $X$. Let us now compute its $n$-th power:

######  Fact 3.3.1 (Modification of [2, Lemma 3.5]).

For every $x\in X$ and $n\geq 1$ we have that

$T^{n}x=R^{n}x+\sum_{k=N+2}^{\infty}\frac{\lambda_{k,n}}{m_{k-1}}\langle
w_{k}^{*},Px\rangle e_{k},$

where
$\lambda_{k,n}:=\sum_{l=0}^{n-1}\lambda_{k}^{l}=\frac{\lambda_{k}^{n}-1}{\lambda_{k}-1}$
for each $k\geq N+2$.

###### Proof.

Suppose that the formula holds for some $n\geq 1$. Then

$\displaystyle T^{n+1}x$
$\displaystyle=TR^{n}x+\sum_{k=N+2}^{\infty}\frac{\lambda_{k,n}}{m_{k-1}}\langle
w_{k}^{*},Px\rangle
Te_{k}=R^{n+1}x+\sum_{k=N+2}^{\infty}\frac{1}{m_{k-1}}\langle
w_{k}^{*},PR^{n}x\rangle e_{k}$
$\displaystyle+\sum_{k=N+2}^{\infty}\frac{\lambda_{k,n}}{m_{k-1}}\langle
w_{k}^{*},Px\rangle\left(Re_{k}+\sum_{j=N+2}^{\infty}\frac{1}{m_{j-1}}\langle
w_{j}^{*},Pe_{k}\rangle e_{j}\right)$
$\displaystyle=R^{n+1}x+\sum_{k=N+2}^{\infty}\frac{1+\lambda_{k,n}\cdot\lambda_{k}}{m_{k-1}}\langle
w_{k}^{*},Px\rangle
e_{k}=R^{n+1}x+\sum_{k=N+2}^{\infty}\frac{\lambda_{k,n+1}}{m_{k-1}}\langle
w_{k}^{*},Px\rangle e_{k}.$

In order to obtain these equalities we have used the fact that $PR^{n}x=Px$,
because of $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{N+1}=1$, and also that
$Pe_{k}=0$ for $k\geq N+2$. ∎

We will also need the following properties regarding the numbers
$\lambda_{k,n}$:

######  Fact 3.3.2 (Modification of [2, Fact 3.6]).

Let $n\geq 1$. Then:

  1. (i)

$|\lambda_{k,n}|\leq n$ for all $k\geq N+2$;

  2. (ii)

$\lambda_{k,m_{n}}=0$ whenever $n\geq k\geq N+2$;

  3. (iii)

$|\lambda_{k,n}|\geq\frac{2}{\pi}n>\frac{m_{k-1}}{\pi}$ whenever
$k=\min\\{j\geq N+2:2n\leq m_{j}\\}$.

###### Proof.

Each $\lambda_{k,n}$, for $k\geq N+2$, is the geometric sum

$\textstyle\lambda_{k,n}=\sum_{l=0}^{n-1}\lambda_{k}^{l}=\frac{\lambda_{k}^{n}-1}{\lambda_{k}-1}\quad\text{
with }\lambda_{k}=e^{2\pi i\frac{1}{m_{k}}};$

then (i) follows from the triangle inequality; (ii) since $n\geq k$ implies
that $\lambda_{k}^{m_{n}}=e^{2\pi i\frac{m_{n}}{m_{k}}}=1$ by condition (a) on
the sequence $(m_{k})_{k\in\mathbb{N}}$; and (iii) because

$|\lambda_{k,n}|=\left|\frac{e^{2\pi i\frac{n}{m_{k}}}-1}{e^{2\pi
i\frac{1}{m_{k}}}-1}\right|=\frac{\left|\sin(\pi\frac{n}{m_{k}})\right|}{\left|\sin(\pi\frac{1}{m_{k}})\right|}\geq\frac{2}{\pi}n>\frac{m_{k-1}}{\pi},$

using that $\sin(\theta)\geq\frac{2}{\pi}\theta$ for each
$\theta\in[0,\frac{\pi}{2}]$ and that $|\sin(\theta)|\leq|\theta|$ for every
$\theta\in\mathbb{R}$. ∎

Up to now, we have defined the operator $T$ and computed its powers $T^{n}$.
It remains to check that $T_{(N)}$ is recurrent while $T_{(N+1)}$ is not. We
start by showing that $R:X\longrightarrow X$ is a rigid operator (see
Subsection 2.1 or [19, Definition 1.2]) with respect to the (eventually)
increasing sequence of positive integers $(m_{k})_{k\in\mathbb{N}}$:

######  Fact 3.3.3 (Modification of [2, Lemma 3.4]).

We have $\displaystyle\lim_{k\to\infty}R^{m_{k}}x=x$ for all $x\in X$.

###### Proof.

Following [2, Lemma 3.4] we first show that the sequence
$(\|R^{m_{j}}\|)_{j\in\mathbb{N}}$ is bounded. In fact, if we fix
$x=\sum_{k=1}^{n}x_{k}e_{k}\in c_{00}$ such that $\|x\|=1$, for all $j\geq 1$
we have that

$\displaystyle\|R^{m_{j}}x-x\|$ $\displaystyle=$
$\displaystyle\left\|\sum_{k=1}^{n}(\lambda_{k}^{m_{j}}-1)x_{k}e_{k}\right\|$
$\displaystyle=$
$\displaystyle\left\|\sum_{k=j+1}^{n}(\lambda_{k}^{m_{j}}-1)x_{k}e_{k}\right\|\qquad\text{since
}\lambda^{m_{j}}=1\text{ for all }k\leq j,$ $\displaystyle\leq$ $\displaystyle
K\|x\|\sum_{k=j+1}^{n}|\lambda_{k}^{m_{j}}-1|=K\sum_{k=j+1}^{n}\left|e^{2\pi
i\frac{m_{j}}{m_{k}}}-1\right|\quad\text{ by \eqref{eq:1K.norma.x},}$
$\displaystyle\leq$ $\displaystyle 2\pi
K\sum_{k=j+1}^{\infty}\frac{m_{j}}{m_{k}}\qquad\text{since
}|e^{i\theta}-1|\leq|\theta|\text{ for every }\theta\in\mathbb{R},$

which is less than a constant independent of $j$ by condition (c) on
$(m_{k})_{k\in\mathbb{N}}$. The density of $c_{00}$ in $X$ implies that
$(\|R^{m_{j}}\|)_{j\in\mathbb{N}}$ is a bounded sequence. Now if given $x\in
X$ and $\varepsilon>0$ we find $y\in c_{00}$ with $\|y-x\|<\varepsilon$, then

$\|R^{m_{j}}x-x\|\leq\|R^{m_{j}}(x-y)\|+\|R^{m_{j}}y-y\|+\|y-x\|<\sup_{j\in\mathbb{N}}\|R^{m_{j}}\|\varepsilon+\|R^{m_{j}}y-y\|+\varepsilon.$

The claim follows since $R^{m_{j}}y=y$ for every $j\in\mathbb{N}$ large
enough, and $\varepsilon$ was arbitrary. ∎

Now we can prove the recurrence property of $T_{(N)}$:

######  Proposition 3.4.

The operator $T_{(N)}:X^{N}\longrightarrow X^{N}$ is recurrent since
$\textup{Rec}(T_{(N)})=X^{N}$.

###### Proof.

Fix $x^{(1)},x^{(2)},...,x^{(N)}\in X$ and set $x_{j}^{(i)}:=\langle
e_{j}^{*},x^{(i)}\rangle$ for each $1\leq i\leq N$ and $j\geq 1$. Choose
$(\alpha_{j})_{j=1}^{N+1}\in\mathbb{C}^{N+1}$ such that

$\begin{pmatrix}x_{1}^{(1)}&x_{2}^{(1)}&\cdots&x_{N+1}^{(1)}\\\
x_{1}^{(2)}&x_{2}^{(2)}&\cdots&x_{N+1}^{(2)}\\\ \vdots&\vdots&\cdots&\vdots\\\
x_{1}^{(N)}&x_{2}^{(N)}&\cdots&x_{N+1}^{(N)}\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\
\alpha_{2}\\\ \vdots\\\ \alpha_{N+1}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
\vdots\\\ 0\end{pmatrix}\quad\text{ and }\quad\max_{1\leq j\leq
N+1}|\alpha_{j}|=1.$

This is indeed possible since the system considered has $N$ equations and
$N+1$ unknowns. Set $w^{*}:=\sum_{j=1}^{N+1}\alpha_{j}e_{j}^{*}\in E$. Then
$w^{*}\in S_{E_{\infty}}$ and clearly $\langle w^{*},Px^{(i)}\rangle=0$ for
all $1\leq i\leq N$. Let $(k_{n})_{n\in\mathbb{N}}$ be an increasing sequence
of positive integers (all greater than $N+2$) such that
$\lim_{n\to\infty}w_{k_{n}}^{*}=w^{*}$ in $E$. By Fact 3.3.1, for each $1\leq
i\leq N$ we have that

$\displaystyle\left(T^{m_{k_{n}-1}}-R^{m_{k_{n}-1}}\right)x^{(i)}$
$\displaystyle=\sum_{k=N+2}^{\infty}\frac{\lambda_{k,m_{k_{n}-1}}}{m_{k-1}}\langle
w_{k}^{*},Px^{(i)}\rangle e_{k}=\underbrace{\sum_{N+2\leq
k<k_{n}}\frac{\lambda_{k,m_{k_{n}-1}}}{m_{k-1}}\langle
w_{k}^{*},Px^{(i)}\rangle e_{k}}_{(*)}$
$\displaystyle+\underbrace{\frac{\lambda_{k_{n},m_{k_{n}-1}}}{m_{k_{n}-1}}\langle
w_{k_{n}}^{*},Px^{(i)}\rangle
e_{k_{n}}}_{(**)}+\underbrace{\sum_{k_{n}<k}\frac{\lambda_{k,m_{k_{n}-1}}}{m_{k-1}}\langle
w_{k}^{*},Px^{(i)}\rangle e_{k}}_{(***)}.$

Note that for each $1\leq i\leq N$,

$(*)=\sum_{N+2\leq k<k_{n}}\frac{\lambda_{k,m_{k_{n}-1}}}{m_{k-1}}\langle
w_{k}^{*},Px^{(i)}\rangle e_{k}=0,$

since by (ii) of Fact 3.3.2 we have that $\lambda_{k,m_{k_{n}-1}}=0$ for
$k_{n}-1\geq k$. By (i) of Fact 3.3.2 we have that
$|\lambda_{k_{n},m_{k_{n}-1}}|\leq m_{k_{n}-1}$ and hence

$\|(**)\|=\left|\frac{\lambda_{k_{n},m_{k_{n}-1}}}{m_{k_{n}-1}}\langle
w_{k_{n}}^{*},Px^{(i)}\rangle\right|\leq|\langle
w_{k_{n}}^{*},Px^{(i)}\rangle|\underset{n\to\infty}{\longrightarrow}0,$

since $\lim_{n\to\infty}w_{k_{n}}^{*}=w^{*}$ and $\langle
w^{*},Px^{(i)}\rangle=0$ for every $1\leq i\leq N$. Moreover, by condition (c)
on the sequence $(m_{k})_{k\in\mathbb{N}}$ we get that, for each $1\leq i\leq
N$,

$\|(***)\|\leq\sum_{k_{n}<k}\frac{\left|\lambda_{k,m_{k_{n}-1}}\right|}{m_{k-1}}\|w_{k}^{*}\|^{*}\cdot\|P\|\cdot\|x^{(i)}\|\leq
M\cdot(N+1)K\cdot\|x^{(i)}\|\cdot\sum_{k_{n}<k}\frac{m_{k_{n}-1}}{m_{k-1}}\underset{n\to\infty}{\longrightarrow}0.$

Finally, for each $1\leq i\leq N$ we have that

$\displaystyle\|T^{m_{k_{n}-1}}x^{(i)}-x^{(i)}\|$ $\displaystyle\leq$
$\displaystyle\|\left(T^{m_{k_{n}-1}}-R^{m_{k_{n}-1}}\right)x^{(i)}\|+\|R^{m_{k_{n}-1}}x^{(i)}-x^{(i)}\|$
$\displaystyle\leq$
$\displaystyle\|(*)\|+\|(**)\|+\|(***)\|+\|R^{m_{k_{n}-1}}x^{(i)}-x^{(i)}\|\underset{n\to\infty}{\longrightarrow}0,$

where $\|R^{m_{k_{n}-1}}x^{(i)}-x^{(i)}\|\to 0$ by Fact 3.3.3. That is,
$(x^{(1)},x^{(2)},...,x^{(N)})\in\textup{Rec}(T_{(N)})$ and the arbitrariness
of the vectors $x^{(1)}$, $x^{(2)}$, …, $x^{(N)}$ implies that
$\textup{Rec}(T_{(N)})=X^{N}$. ∎

To end the proof of Theorem 3.2 we just have to show the following:

######  Proposition 3.5.

The operator $T_{(N+1)}:X^{N+1}\longrightarrow X^{N+1}$ is not recurrent.

###### Proof.

By contradiction suppose that $T_{(N+1)}$ is recurrent. Fix any positive value
$\varepsilon>0$ and consider the open balls
$B(e_{1},\varepsilon),B(e_{2},\varepsilon),...,B(e_{N+1},\varepsilon)\subset
X$ centred at each vector $e_{i}$, $1\leq i\leq N+1$, with radius
$\varepsilon$. Then there exists some $n\in\mathbb{N}$ such that

$T^{n}(B(e_{i},\varepsilon))\cap
B(e_{i},\varepsilon)\neq\varnothing\quad\text{ for every }i=1,2,...,N+1.$

Let $k_{n}:=\min\\{j\geq N+2:2n\leq m_{j}\\}\geq N+2$. By (iii) of Fact 3.3.2
we have that

$|\lambda_{k_{n},n}|>\tfrac{m_{k_{n}-1}}{\pi}.$ (5)

Moreover, Fact 3.2.1 applied to the functional $w_{k_{n}}^{*}$ yields the
existence of (at least) one index $i_{n}\in\\{1,2,...,N+1\\}$ for which

$\left|\langle w_{k_{n}}^{*},Px\rangle\right|>1-(N+1)K\varepsilon\quad\text{
whenever }\|x-e_{i_{n}}\|<\varepsilon.$ (6)

Picking any $x\in B(e_{i_{n}},\varepsilon)\cap
T^{-n}(B(e_{i_{n}},\varepsilon))\cap c_{00}$ we get the following inequalities

$\displaystyle\varepsilon$ $\displaystyle>$
$\displaystyle\|T^{n}x-e_{i_{n}}\|\geq\frac{1}{K}\left|\langle
e_{k_{n}}^{*},T^{n}x-e_{i_{n}}\rangle\right|=\frac{1}{K}\left|\langle
e_{k_{n}}^{*},T^{n}x\rangle\right|\qquad\text{ by \eqref{eq:1K.norma.x},}$
$\displaystyle=$
$\displaystyle\frac{1}{K}\left|\lambda_{k_{n}}^{n}x_{k_{n}}+\frac{\lambda_{k_{n},n}}{m_{k_{n}-1}}\langle
w_{k_{n}}^{*},Px\rangle\right|\qquad\qquad\qquad\qquad\quad\text{ by
Fact\leavevmode\nobreak\ \ref{Fact:T.power.n}.}$

Hence

$\displaystyle\varepsilon$ $\displaystyle>$
$\displaystyle\frac{1}{K}\left(\left|\frac{\lambda_{k_{n},n}}{m_{k_{n}-1}}\right|\cdot\left|\langle
w_{k_{n}}^{*},Px\rangle\right|-K\varepsilon\right)>\frac{1}{K\pi}\cdot|\langle
w_{k_{n}}^{*},Px\rangle|-\varepsilon\qquad\text{ by \eqref{eq:2K.eps} and
\eqref{eq:lambda_k_n},}$ $\displaystyle>$
$\displaystyle\frac{\left(1-(N+1)K\varepsilon\right)}{K\pi}-\varepsilon=\frac{1}{K\pi}-\left(\frac{N+1+\pi}{\pi}\right)\varepsilon\qquad\qquad\qquad\qquad\qquad\text{
by \eqref{eq:functional}}.$

The arbitrariness of $\varepsilon>0$ yields now a contradiction by letting
$\varepsilon\to 0$. ∎

The complex version of Theorem 3.2 is now proved. The construction can be
adapted to the real case by using the same arguments as in [2, Section 3.2],
so that every separable infinite-dimensional Banach space supports a recurrent
operator which is not quasi-rigid. Moreover, there are operators whose
$N$-fold direct sum is recurrent while its $N+1$-fold direct sum is not, and
taking $N=1$ we solve negatively the $T\oplus T$-recurrence problem. It is
worth mentioning that a slight variation of this construction can be used to
solve the respective problem for C0-semigroups of operators, which has been
recently stated in [20]. This was communicated to us by F. Bayart and Q. Menet
independently, and the details of this variation will appear in a forthcoming
work.

##  4 Furstenberg families for pointwise recurrence

In the last two decades hypercyclicity has been studied from the frequency of
visits point of view: one investigates “how often” the orbit of a vector
returns to every open subset of the space. The recent notions of frequent,
$\mathcal{U}$-frequent or reiterative hypercyclicity have emerged following
this line of thought (see [4, 43, 8, 9]). As shown in [8, 13], under some
natural assumptions this behaviour can be decomposed in two necessary
ingredients:

  1. 1)

usual hypercyclicity (we have to require that the operator under study has a
dense orbit);

  2. 2)

some kind of strong recurrence property (we have to require that each non-
empty open subset of the space contains an orbit returning to it with some
frequency).

Even though the study of linear recurrence began in 2014 with [19], stronger
recurrence notions were only systematically considered in the 2022 work [13],
and then in the following papers [29, 17] where the concepts of
$\mathcal{F}$-recurrence and $\mathcal{F}$-hypercyclicity with respect to a
Furstenberg family $\mathcal{F}$ are deeply studied. In this section we show
that quasi-rigidity can be studied from an $\mathcal{F}$-recurrence
perspective for a very particular kind of Furstenberg families called free
filters (see Proposition 4.5). Then we study both the weakest and strongest
possible $\mathcal{F}$-recurrence notions together with the Furstenberg
families associated to them (see Subsection 4.2), and we finish the section
recalling that $\mathcal{F}$-recurrence behaves well under homomorphisms,
quasi-conjugacies and commutants (see Subsection 4.3).

Let us start by recalling the precise definitions, as well as some known
examples.

###  4.1 Definitions, examples and $\mathcal{F}(A)$-recurrence

A collection of sets $\mathcal{F}\subset\mathscr{P}(\mathbb{N}_{0})$ is said
to be a Furstenberg family (a family for short) provided that: each set
$A\in\mathcal{F}$ is infinite; $\mathcal{F}$ is hereditarily upward (i.e.
$B\in\mathcal{F}$ whenever $A\in\mathcal{F}$ and $A\subset B$); and also
$A\cap[n,\infty[\in\mathcal{F}$ for all $A\in\mathcal{F}$ and
$n\in\mathbb{N}$. The dual family $\mathcal{F}^{*}$ for a given family
$\mathcal{F}$ is the collection of sets $A\subset\mathbb{N}_{0}$ such that
$A\cap B\neq\varnothing$ for every $B\in\mathcal{F}$. The rather vague notion
of frequency mentioned above will be defined in terms of families.

Given a dynamical system $(X,T)$, a point $x\in X$ and a non-empty (open)
subset $U\subset X$ we denote by

$N_{T}(x,U):=\\{n\in\mathbb{N}_{0}:T^{n}x\in U\\},$

the return set from $x$ to $U$. It will be denoted by $N(x,U)$ if no confusion
with the map arises. Following [8, 9, 12, 13, 15, 16, 17, 29] we can now
define:

######  Definition 4.1.

Let $(X,T)$ be a dynamical system and let $\mathcal{F}$ be a Furstenberg
family. A point $x\in X$ is said to be:

  1. –

$\mathcal{F}$-recurrent if $N(x,U)\in\mathcal{F}$ for every neighbourhood $U$
of $x$;

  2. –

$\mathcal{F}$-hypercyclic if $N(x,U)\in\mathcal{F}$ for every non-empty open
subset $U\subset X$.

Moreover, we will denote by:

  1. –

$\mathcal{F}\textup{Rec}(T)$ the set of $\mathcal{F}$-recurrent points, and we
will say that $T$ is $\mathcal{F}$-recurrent whenever the set
$\mathcal{F}\textup{Rec}(T)$ is dense in $X$;

  2. –

$\mathcal{F}\textup{HC}(T)$ the set of $\mathcal{F}$-hypercyclic points, and
we will say that $T$ is $\mathcal{F}$-hypercyclic whenever the set
$\mathcal{F}\textup{HC}(T)$ is non-empty.

######  Example 4.2.

The previous Linear Dynamics works [7, 8, 9, 12, 13, 15, 16, 17, 19, 29, 31]
have considered the Furstenberg families formed by:

  1. (a)

the infinite sets

$\mathcal{I}=\\{A\subset\mathbb{N}_{0}:A\text{ is infinite}\\}.$

The notion of $\mathcal{I}$-recurrence (resp. $\mathcal{I}$-hypercyclicity)
coincides with the concept of recurrence (resp. hypercyclicity) as defined in
the Introduction. We denote by $\textup{Rec}(T)$ (resp. $\textup{HC}(T)$) the
set of recurrent (resp. hypercyclic) vectors associated to this family.

  2. (b)

the sets containing arbitrarily long arithmetic progressions

$\mathcal{AP}:=\\{A\subset\mathbb{N}_{0}:A\text{ contains arbitrarily long
arithmetic progressions}\\}.$

In other words, $A\in\mathcal{AP}$ if $\forall l\in\mathbb{N},\ \exists
a,n\in\mathbb{N}$ such that $\\{a+kn:0\leq k\leq l\\}\subset A$, or
equivalently $A$ contains the arithmetic progression of length $l+1$, common
difference $n\in\mathbb{N}$ and initial term $a\in\mathbb{N}_{0}$. The concept
of $\mathcal{AP}$-recurrence is equivalent to the notion of (topological)
multiple recurrence, which requires that for each non-empty open subset
$U\subset X$ and each length $l\in\mathbb{N}$ there exists some
$n\in\mathbb{N}$ fulfilling that

$U\cap T^{-n}(U)\cap\cdots\cap T^{-ln}(U)\neq\varnothing.$

This concept was introduced for linear dynamical systems in [18] and further
developed in [36] and [16], where $\mathcal{AP}$-hypercyclicity is also
considered. We will denote by $\mathcal{AP}\textup{Rec}(T)$ (resp.
$\mathcal{AP}\textup{HC}(T)$) the set of $\mathcal{AP}$-recurrent (resp.
$\mathcal{AP}$-hypercyclic) vectors.

  3. (c)

the sets with positive upper Banach density
$\overline{\mathcal{BD}}:=\\{A\subset\mathbb{N}_{0}:\overline{\operatorname{Bd}}(A)>0\\}$,
where for each $A\subset\mathbb{N}_{0}$ its upper Banach density is defined as

$\overline{\operatorname{Bd}}(A):=\lim_{N\to\infty}\left(\max_{n\geq
0}\dfrac{\\#\ A\cap[n+1,n+N]}{N}\right).\quad\text{See
\cite[cite]{[\@@bibref{}{GTT2010}{}{}]} for alternative definitions.}$

The notion of $\overline{\mathcal{BD}}$-recurrence (resp.
$\overline{\mathcal{BD}}$-hypercyclicity) has been called reiterative
recurrence (resp. reiterative hypercyclicity), and $\operatorname{RRec}(T)$
(resp. $\operatorname{RHC}(T)$) is the set of reiteratively recurrent (resp.
reiteratively hypercyclic) vectors, see [8, 9, 13, 29].

  4. (d)

the sets with positive upper density
$\overline{\mathcal{D}}:=\\{A\subset\mathbb{N}_{0}:\overline{\operatorname{dens}}(A)>0\\}$,
where for each $A\subset\mathbb{N}_{0}$ its upper density is defined as

$\overline{\operatorname{dens}}(A):=\limsup_{N\to\infty}\dfrac{\\#\
A\cap[1,N]}{N}.$

The $\overline{\mathcal{D}}$-recurrence (resp.
$\overline{\mathcal{D}}$-hypercyclicity) notion has been called
$\mathcal{U}$-frequent recurrence (resp. $\mathcal{U}$-frequent
hypercyclicity) and $\operatorname{UFRec}(T)$ (resp. $\operatorname{UFHC}(T)$)
will denote the set of $\mathcal{U}$-frequent recurrent (resp.
$\mathcal{U}$-frequent hypercyclic) vectors, see [43, 8, 9, 13, 29].

  5. (e)

the sets with positive lower density
$\underline{\mathcal{D}}:=\\{A\subset\mathbb{N}_{0}:\underline{\operatorname{dens}}(A)>0\\}$,
where for each $A\subset\mathbb{N}_{0}$ its lower density is defined as

$\underline{\operatorname{dens}}(A):=\limsup_{N\to\infty}\dfrac{\\#\
A\cap[1,N]}{N}.$

The notion of $\underline{\mathcal{D}}$-recurrence (resp.
$\underline{\mathcal{D}}$-hypercyclicity) is that of frequent recurrence
(resp. frequent hypercyclicity), and $\operatorname{FRec}(T)$ (resp.
$\operatorname{FHC}(T)$) is the set of frequently recurrent (resp. frequently
hypercyclic) vectors, see [4, 8, 9, 13, 29].

  6. (f)

the syndetic sets

$\mathcal{S}:=\\{A\subset\mathbb{N}_{0}:A\text{ is syndetic}\\},$

i.e. $A\in\mathcal{S}$ whenever $\exists m\in\mathbb{N}$ such that $\forall
a\in\mathbb{N}_{0}$, $[a,a+m]\cap A\neq\varnothing$. The notion of
$\mathcal{S}$-recurrence has been called uniform recurrence and we denote by
$\operatorname{URec}(T)$ the set of uniformly recurrent vectors, see [24, 13,
29]. The family $\mathcal{S}$ coincides with the family of sets with positive
lower Banach density
$\underline{\mathcal{BD}}:=\\{A\subset\mathbb{N}_{0}:\underline{\textup{Bd}}(A)>0\\}$,
where for each $A\subset\mathbb{N}_{0}$ its lower Banach density is defined as

$\underline{\textup{Bd}}(A):=\lim_{N\to\infty}\left(\inf_{n\geq 0}\dfrac{\\#\
A\cap[n+1,n+N]}{N}\right).\quad\text{See
\cite[cite]{[\@@bibref{}{GTT2010}{}{}]} for alternative definitions.}$

  7. (g)

the IP-sets

$\mathcal{IP}:=\\{A\subset\mathbb{N}_{0}:A\text{ is an IP-set}\\},$

i.e. $A\in\mathcal{IP}$ whenever $\\{\sum_{k\in
F}n_{k}:F\subset\mathbb{N}\text{ finite}\\}\subset A$ for some sequence of
natural numbers $(n_{k})_{k\in\mathbb{N}}$. In this case we use the dual
family $\mathcal{IP}^{*}$, and for the $\mathcal{IP}^{*}$-recurrence notion we
denote by $\mathcal{IP}^{*}\textup{Rec}(T)$ the set
$\mathcal{IP}^{*}$-recurrent vectors, see [24, 26, 13, 29].

  8. (h)

the $\Delta$-sets

$\Delta:=\\{A\subset\mathbb{N}_{0}:A\text{ is an $\Delta$-set}\\},$

i.e. $A\in\Delta$ whenever $\exists B\in\mathcal{I}$ with
$(B-B)\cap\mathbb{N}\subset A$. In this case we again use the dual family
$\Delta^{*}$, and for the $\Delta^{*}$-recurrence notion we denote by
$\Delta^{*}\textup{Rec}(T)$ the set of $\Delta^{*}$-recurrent vectors, see
[24, 29].

We have the following relations between the families listed above:

$\Delta^{*}\subset\mathcal{IP}^{*}\subset\mathcal{S}=\underline{\mathcal{BD}}\subset\underline{\mathcal{D}}\subset\overline{\mathcal{D}}\subset\overline{\mathcal{BD}}\subset\mathcal{AP}\subset\mathcal{I}.$

Among these inclusions, some of them follow easily from the definitions while
others depend on deep theorems like the celebrated Szemerédi theorem, see [11,
16, 34, 44]. They imply the respective inclusions between the introduced sets
of recurrent vectors

$\displaystyle\Delta^{*}\textup{Rec}(T)$
$\displaystyle\subset\mathcal{IP}^{*}\textup{Rec}(T)\subset\operatorname{URec}(T)\subset...$
$\displaystyle...$
$\displaystyle\subset\operatorname{FRec}(T)\subset\operatorname{UFRec}(T)\subset\operatorname{RRec}(T)\subset\mathcal{AP}\textup{Rec}(T)\subset\textup{Rec}(T),$

and also between the sets of hypercyclic vectors

$\operatorname{FHC}(T)\subset\operatorname{UFHC}(T)\subset\operatorname{RHC}(T)\subset\mathcal{AP}\textup{HC}(T)\subset\textup{HC}(T).$

It is worth mentioning that the families $\mathcal{F}$ for which there exist
$\mathcal{F}$-hypercyclic operators are by far less common than those for
which $\mathcal{F}$-recurrence exists, which is not surprising, since having
an orbit distributed around the whole space is much more complicated than
having it just coming back around the initial point of the orbit (it is known
that there does not exist any $\Delta^{*}$, $\mathcal{IP}^{*}$ neither
$\mathcal{S}$-hypercyclic operator, see [8, 13]). Other families have been
considered from the $\mathcal{F}$-hypercyclicity perspective in the works [12,
23]. Our aim now is to show that a dynamical system is quasi-rigid if and only
if it is $\mathcal{F}$-recurrent with respect to a free filter $\mathcal{F}$.
Let us recall the following classical definitions (see [14]):

######  Definition 4.3.

Let $\mathcal{F}\subset\mathscr{P}(\mathbb{N}_{0})$ be a collection of sets of
natural numbers (note that $\mathcal{F}$ is not necessarily a Furstenberg
family here). We say that $\mathcal{F}$:

  1. –

has the finite intersection property, if
$\bigcap_{A\in\mathcal{A}}A\neq\varnothing$ for every
$\mathcal{A}\subset\mathcal{F}$ with $\\#\mathcal{A}<\infty$;

  2. –

is a filter, if $\mathcal{F}$ is hereditarily upward and for every
$A,B\in\mathcal{F}$ we have that $A\cap B\in\mathcal{F}$;

  3. –

is a free filter, if it is a filter and
$\bigcap_{A\in\mathcal{F}}A=\varnothing$.

Finally, given any infinite subset $A\subset\mathbb{N}_{0}$ we denote by
$\mathcal{F}(A)$ the free filter generated by $A$, which is the Furstenberg
family

$\mathcal{F}(A):=\left\\{B\subset\mathbb{N}_{0}:\\#(A\setminus
B)<\infty\right\\}.$

In the literature, $\mathcal{F}(A)$ has been called the Fréchet filter on $A$,
or also the eventuality filter generated by the increasing sequence of
integers $(n_{k})_{k\in\mathbb{N}}$ forming the set $A\subset\mathbb{N}_{0}$
(see [14]).

######  Remark 4.4.

It is clear that every free filter is a filter, and that every filter has the
finite intersection property. However, when $\mathcal{F}$ is a Furstenberg
family as defined in this paper the previous definitions have some extra (and
immediate) consequences:

  1. (a)

A Furstenberg family $\mathcal{F}$ has the finite intersection property if and
only if $\bigcap_{A\in\mathcal{A}}A\in\mathcal{I}$ for every
$\mathcal{A}\subset\mathcal{F}$ with $\\#\mathcal{A}<\infty$: indeed, if
$\mathcal{F}$ has the finite intersection property but for some
$\mathcal{A}\subset\mathcal{F}$ with $\\#\mathcal{A}<\infty$ we had that
$\bigcap_{A\in\mathcal{A}}A$ is finite, then for any fixed
$n>\max\left(\bigcap_{A\in\mathcal{A}}A\right)$ we would arrive to the
contradiction:

$\mathcal{A}^{\prime}:=\\{A\cap[n,\infty[:A\in\mathcal{A}\\}\subset\mathcal{F}\quad\text{
with }\quad\\#\mathcal{A}^{\prime}<\infty\quad\text{ but
}\quad\bigcap_{A\in\mathcal{A}^{\prime}}A=\varnothing.$

  2. (b)

Let $\mathcal{F}$ be a Furstenberg family. Then, $\mathcal{F}$ is a filter if
and only if it is a free filter: note that given any $B$ from a filter
$\mathcal{F}$ which is also a Furstenberg family we have that

$\bigcap_{A\in\mathcal{F}}A\subset\bigcap_{n\in\mathbb{N}}B\cap[n,\infty[=\varnothing,$

by the definition of Furstenberg family used in this paper.

We are now ready to characterize quasi-rigidity in terms of
$\mathcal{F}$-recurrence and free filters:

######  Proposition 4.5.

Let $(X,T)$ be a dynamical system. The following are equivalent:

  1. (i)

$T$ is quasi-rigid;

  2. (ii)

$T$ is $\mathcal{F}(A)$-recurrent for some infinite subset
$A\subset\mathbb{N}_{0}$;

  3. (iii)

$T$ is $\mathcal{F}$-recurrent with respect to a free filter $\mathcal{F}$
with a countable base.

Moreover, if $X$ is a second-countable space, the previous statements are
equivalent to:

  1. (iv)

$T$ is $\mathcal{F}$-recurrent for a family $\mathcal{F}$ with the finite
intersection property.

###### Proof.

For (i) $\Rightarrow$ (ii), if $T$ is quasi-rigid with respect to
$(n_{k})_{k\in\mathbb{N}}$ it is enough to take
$A:=\\{n_{k}:k\in\mathbb{N}\\}$. The implication (ii) $\Rightarrow$ (iii) is
obvious since $\mathcal{F}(A)$ has a countable base. To see (iii)
$\Rightarrow$ (i) assume that the map $T$ is $\mathcal{F}$-recurrent and let
$(A_{k})_{k\in\mathbb{N}}$ be a decreasing base of the free filter
$\mathcal{F}$. Setting $n_{k}:=\min(A_{k})$ for each $k\in\mathbb{N}$, and
taking a subsequence if necessary, we get an increasing sequence
$(n_{k})_{k\in\mathbb{N}}$ with respect to which $T$ is quasi-rigid.

We always have that (i), (ii) and (iii) $\Rightarrow$ (iv) even if the space
$X$ is not second-countable. Assume now that $X$ is second-countable and that
$T$ is $\mathcal{F}$-recurrent for a family $\mathcal{F}$ with the finite
intersection property. Let
$\\{x_{s}:s\in\mathbb{N}\\}\subset\mathcal{F}\textup{Rec}(T)$ be a dense set
in $X$ and, for each $s\in\mathbb{N}$, let $(U_{s,k})_{k\in\mathbb{N}}$ be a
decreasing neighbourhood basis of $x_{s}$. We can now recursively construct an
increasing sequence of positive integers $(n_{k})_{k\in\mathbb{N}}$ such that

$T^{n_{k}}x_{s}\in U_{s,k}\quad\text{ for every }s,k\in\mathbb{N}\text{ with
}1\leq s\leq k,$

by taking a sufficiently large integer
$n_{k}\in\bigcap_{s=1}^{k}N_{T}(x_{s},U_{s,k})\in\mathcal{I}$. It is easily
seen that $T$ is quasi-rigid with respect to $(n_{k})_{k\in\mathbb{N}}$, which
shows that (iv) $\Rightarrow$ (i), (ii) and (iii). ∎

###  4.2 Appropriate Furstenberg families for $\mathcal{F}$-recurrence

We have just shown that quasi-rigidity is indeed a particular case of
$\mathcal{F}$-recurrence for a Furstenberg family not listed in Example 4.2.
It is then natural to ask the following:

######  Question 4.6.

Are the families of Example 4.2, i.e.
$\mathcal{F}=\mathcal{I},\mathcal{AP},\overline{\mathcal{BD}},\overline{\mathcal{D}},\underline{\mathcal{D}},\mathcal{S},\mathcal{IP}^{*},\Delta^{*}$,
and the free filters $\mathcal{F}(A)$ the only ones for which
$\mathcal{F}$-recurrence should be considered?

This may seem a very open query. Our objective in this part of the paper is to
look in detail into two very different classes of Furstenberg families: those
for which $\mathcal{F}$-recurrence is the weakest possible recurrence notion,
i.e. families $\mathcal{F}\subsetneq\mathcal{I}$ such that
$\mathcal{F}\textup{Rec}(T)=\textup{Rec}(T)$; and those for which
$\mathcal{F}$-recurrence is the strongest possible recurrence notion, i.e.
periodicity.

Starting with the first of these two classes: for any dynamical system $(X,T)$
every recurrent point $x\in\textup{Rec}(T)$ is an $\mathcal{IP}$-recurrent
point (this is shown in [24, Theorem 2.17] when $X$ is a metric space, but the
proof easily extends for arbitrary dynamical systems). Hence
$\textup{Rec}(T)\subset\mathcal{IP}\textup{Rec}(T)\subset\Delta\textup{Rec}(T)\subset\mathcal{I}\textup{Rec}(T)$.
Since $\mathcal{I}\textup{Rec}(T)=\textup{Rec}(T)$ we always have

$\mathcal{IP}\textup{Rec}(T)=\Delta\textup{Rec}(T)=\textup{Rec}(T).$

This relation allows us to give an alternative (and much simpler) proof for
the so-called Ansari and León-Müller recurrence theorems, which assert that
powers and unimodular multiples of an operator share the same set of recurrent
vectors (see [19, Proposition 2.3] for the original, different for each of the
$T^{p}$ and $\lambda T$ cases, and rather long proof):

######  Proposition 4.7 ([19, Proposition 2.3]).

Let $(X,T)$ be a dynamical system. Then:

  1. (a)

For every positive integer $p\in\mathbb{N}$ we have that
$\textup{Rec}(T)=\textup{Rec}(T^{p})$.

  2. (b)

If $(X,T)$ is linear, then for every $\lambda\in\mathbb{T}$ we have that
$\textup{Rec}(T)=\textup{Rec}(\lambda T)$.

###### Proof.

Let $x\in\textup{Rec}(T)$, i.e. $N_{T}(x,V)$ belongs to $\Delta$ for every
neighbourhood $V$ of $x$. Fixed any $p\in\mathbb{N}$ and any neighbourhood $U$
of $x$, since the set $(p\cdot\mathbb{N}_{0})$ belongs to $\Delta^{*}$ we have
that

$\varnothing\neq N_{T}(x,U)\cap\left(p\cdot\mathbb{N}_{0}\right)\subset
N_{T^{p}}(x,U),$

which implies that $x\in\textup{Rec}(T^{p})$, proving (a). To check (b) assume
moreover that $(X,T)$ is a linear dynamical system and fix any
$\lambda\in\mathbb{T}$. Let $\varepsilon>0$ and let $U_{0}\subset U$ be a
neighbourhood of $x$ such that $\mu\cdot U_{0}\subset U$ for all
$|\mu-1|<\varepsilon$. Since the set
$\\{n\in\mathbb{N}:|\lambda^{n}-1|<\varepsilon\\}$ belongs to $\Delta^{*}$
(see for instance [29, Proposition 4.1]) we have that

$\varnothing\neq
N_{T}(x,U_{0})\cap\\{n\in\mathbb{N}:|\lambda^{n}-1|<\varepsilon\\}\subset
N_{\lambda T}(x,U),$

which implies that $x\in\textup{Rec}(\lambda T)$. ∎

It is worth mentioning that, since the families $\Delta^{*}$ and
$\mathcal{IP}^{*}$ are filters (see [11]), then the proof of Proposition 4.7
can be adapted to show the equalities

$\Delta^{*}\textup{Rec}(T^{p})=\Delta^{*}\textup{Rec}(T)=\Delta^{*}\textup{Rec}(\lambda
T),$

and also

$\mathcal{IP}^{*}\textup{Rec}(T^{p})=\mathcal{IP}^{*}\textup{Rec}(T)=\mathcal{IP}^{*}\textup{Rec}(\lambda
T),$

which were discussed in [13, Section 6] in a different way via product
recurrent points.

Let us now look at the strongest possible recurrence notion by using families
formed by very big sets of natural numbers that have been also used in Linear
Dynamics:

######  Example 4.8.

Consider the Furstenberg families formed by:

  1. (a)

the cofinite sets $\mathcal{I}^{*}:=\\{A\subset\mathbb{N}_{0}:A\text{ is
cofinite}\\}$, which is the dual family of that formed by the infinite sets
$\mathcal{I}$, and which is used to define the notion of (topological) mixing:
a dynamical system $(X,T)$ is called mixing if the set
$\\{n\in\mathbb{N}_{0}:T^{n}(U)\cap V\neq\varnothing\\}$ belongs to
$\mathcal{I}^{*}$ for every pair of non-empty open subset $U,V\subset X$ (see
[31, Definition 1.38]).

  2. (b)

the thick sets $\mathcal{T}:=\\{A\subset\mathbb{N}_{0}:A\text{ is thick}\\}$,
i.e. $A\in\mathcal{T}$ if $\forall m\in\mathbb{N},\ \exists a_{m}\in A$ with
$[a_{m},a_{m}+m]\subset A$. This family is known to characterize the weak-
mixing property (see [30, Theorem 3] or [31, Theorem 1.54]): for every
dynamical system $(X,T)$ the following statements are equivalent:

    1. (i)

$(X,T)$ is weakly mixing;

    2. (ii)

the set $\\{n\in\mathbb{N}_{0}:T^{n}(U)\cap V\neq\varnothing\\}$ belongs to
$\mathcal{T}$ for every pair of non-empty open subsets $U,V\subset X$.

  3. (c)

the thickly syndetic sets $\mathcal{TS}:=\\{A\subset\mathbb{N}_{0}:A\text{ is
thickly syndetic}\\}$, i.e. $A\in\mathcal{TS}$ if $\forall m\in\mathbb{N},\
\exists A_{m}$ syndetic such that $A_{m}+[0,m]\subset A$. This family
characterizes the topological ergodicity (i.e. the property that
$\\{n\in\mathbb{N}_{0}:T^{n}(U)\cap V\neq\varnothing\\}$ is syndetic for every
pair of non-empty open subsets $U,V\subset X$) for the particular case of
linear dynamical systems (see [31, Exercise 2.5.4] and [9]): for every linear
dynamical system $(X,T)$ the following statements are equivalent:

    1. (i)

$(X,T)$ is topologically ergodic;

    2. (ii)

the set $\\{n\in\mathbb{N}_{0}:T^{n}(U)\cap V\neq\varnothing\\}$ belongs to
$\mathcal{TS}$ for every pair of non-empty open subsets $U,V\subset X$.

  4. (d)

the sets of density greater than $\delta>0$, which for each $0<\delta\leq 1$
are the families

$\displaystyle\underline{\mathcal{BD}}_{\delta}:=\\{A\subset\mathbb{N}_{0}:\underline{\textup{Bd}}(A)\geq\delta\\},$
$\displaystyle\overline{\mathcal{BD}}_{\delta}:=\\{A\subset\mathbb{N}_{0}:\overline{\operatorname{Bd}}(A)\geq\delta\\},$
$\displaystyle\underline{\mathcal{D}}_{\delta}:=\\{A\subset\mathbb{N}_{0}:\underline{\operatorname{dens}}(A)\geq\delta\\},$
$\displaystyle\overline{\mathcal{D}}_{\delta}:=\\{A\subset\mathbb{N}_{0}:\overline{\operatorname{dens}}(A)\geq\delta\\}.$

The density families have also been studied in Linear Dynamics, and we refer
the reader to [8, 9] for more about them.

We are about to show that the recurrence notions associated to the Furstenberg
families introduced in Example 4.8 above imply different periodicity notions
(see Proposition 4.11). We replicate the arguments from [8, Proposition 3],
which have been used in an independent way in [17, Lemma 2.14 and Corollary
2.15]. Our contribution here is to rewrite these results in their “pointwise
$\mathcal{F}$-recurrence” version. We start by proving two key lemmas:

######  Lemma 4.9.

Let $(X,T)$ be a dynamical system. Given $x\in X$ and $N\in\mathbb{N}$ the
following statements are equivalent:

  1. (i)

$x\in\operatorname{Per}(T)$ and its period is strictly lower than $N$;

  2. (ii)

for every neighbourhood $U$ of $x$ there exists $n_{U}\in\mathbb{N}_{0}$ such
that

$\\#\ N_{T}(x,U)\cap[n_{U}+1,n_{U}+N]\geq 2;$

  3. (iii)

for every neighbourhood $U$ of $x$ we have that $T^{p}(U)\cap
U\neq\varnothing$ for some $1\leq p<N$.

###### Proof.

For (i) $\Rightarrow$ (ii) let $n_{U}$ be the period of $x$ minus one. For
(ii) $\Rightarrow$ (iii) recall that

$\text{given }n_{1}<n_{2}\in N_{T}(x,U)\quad\text{ we have that }\quad
T^{n_{2}-n_{1}}(U)\cap U\neq\varnothing.$

Conversely, if we suppose that $T^{p}x\neq x$ for $1\leq p<N$, by continuity
we can find an open neighbourhood $U$ of $x$ such that $T^{p}(U)\cap
U=\varnothing$ for every $1\leq p<N$, so (iii) $\Rightarrow$ (i). ∎

######  Lemma 4.10.

Let $0<\delta\leq 1$ and $N\in\mathbb{N}$ with $\frac{1}{N}<\delta$. Then, for
every $A\in\overline{\mathcal{BD}}_{\delta}$ there is $n_{A}\in\mathbb{N}_{0}$
such that

$\\#\ A\cap[n_{A}+1,n_{A}+N]\geq 2.$

###### Proof.

Otherwise we would have that $\\#\ A\cap[n+1,n+N]\leq 1$ for every
$n\in\mathbb{N}_{0}$ obtaining the contradiction

$\overline{\operatorname{Bd}}(A)=\lim_{K\to\infty}\left(\max_{n\geq
0}\dfrac{\\#\ A\cap[n+1,n+(N\cdot K)]}{N\cdot
K}\right)\leq\lim_{K\to\infty}\dfrac{K}{N\cdot K}=\frac{1}{N}<\delta.\qed$

Finally we get the desired result, in which we classify the periodic points of
a dynamical system in terms of the density of the return sets:

######  Proposition 4.11.

Let $(X,T)$ be a dynamical system and let $x\in X$. Then:

  1. (a)

Given any $0<\delta\leq 1$, the following statements are equivalent:

    1. (i)

$x\in\operatorname{Per}(T)$ and its period is lower or equal to
$\left\lfloor\frac{1}{\delta}\right\rfloor$;

    2. (ii)

$x\in\underline{\mathcal{BD}}_{\delta}\textup{Rec}(T)$;

    3. (iii)

$x\in\underline{\mathcal{D}}_{\delta}\textup{Rec}(T)$;

    4. (iv)

$x\in\overline{\mathcal{D}}_{\delta}\textup{Rec}(T)$;

    5. (v)

$x\in\overline{\mathcal{BD}}_{\delta}\textup{Rec}(T)$.

In particular, if $T$ is $\overline{\mathcal{BD}}_{\delta}$-recurrent then
$T^{N}=I$ for $N=1\cdot
2\cdots\left(\left\lfloor\frac{1}{\delta}\right\rfloor-1\right)\cdot\left\lfloor\frac{1}{\delta}\right\rfloor$.

  2. (b)

The following statements are equivalent:

    1. (i)

$x$ is a fixed point, i.e. $Tx=x$;

    2. (ii)

$x\in\mathcal{I}^{*}\textup{Rec}(T)$;

    3. (iii)

$x\in\mathcal{TS}\textup{Rec}(T)$;

    4. (iv)

$x\in\mathcal{T}\textup{Rec}(T)$;

    5. (v)

$x\in\overline{\mathcal{BD}}_{\delta}\textup{Rec}(T)$ for some
$\delta>\frac{1}{2}$;

    6. (vi)

for every neighbourhood $U$ of $x$ the set $N(x,U)$ contains two consecutive
integers.

###### Proof.

(a): Let us first show that (i) $\Rightarrow$ (ii): let
$p\in\left\\{1,2,...,\left\lfloor\frac{1}{\delta}\right\rfloor\right\\}$ be
such that $T^{p}x=x$. Then

$\underline{\textup{Bd}}(N_{T}(x,U))\geq\underline{\textup{Bd}}(p\cdot\mathbb{N}_{0})=\frac{1}{p}\geq\delta$

for every neighbourhood $U$ of $x$, so
$N(x,U)\in\overline{\mathcal{BD}}_{\delta}$. For (ii) $\Rightarrow$ (iii)
$\Rightarrow$ (iv) $\Rightarrow$ (v) recall that
$\underline{\mathcal{BD}}_{\delta}\subset\underline{\mathcal{D}}_{\delta}\subset\overline{\mathcal{D}}_{\delta}\subset\overline{\mathcal{BD}}_{\delta}$.
For (v) $\Rightarrow$ (i) let
$N:=\left\lfloor\frac{1}{\delta}\right\rfloor+1$. By Lemma 4.10, for every
neighbourhood $U$ of $x$ there is $n_{U}\in\mathbb{N}$ such that

$\\#\ N_{T}(x,U)\cap[n_{U}+1,n_{U}+N]\geq 2,$

so Lemma 4.9 finishes the work.

(b): The implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Rightarrow$
(iv) $\Rightarrow$ (v) follow from the well-known relations
$\mathbb{N}_{0}\in\mathcal{I}^{*}\subset\mathcal{TS}\subset\mathcal{T}=\overline{\mathcal{BD}}_{1}\subset\overline{\mathcal{BD}}_{\delta}$
for every $0<\delta\leq 1$ (see [11, 34]). To prove (v) $\Rightarrow$ (vi) use
Lemma 4.10 applied to $N=2$. Finally, (vi) $\Leftrightarrow$ (i) follows from
Lemma 4.9. ∎

The characterization obtained for periodic points in part (a) of Proposition
4.11 tells us which are the Furstenberg families $\mathcal{F}$ whose
respective $\mathcal{F}$-recurrence notion is trivial, in the sense that it
coincides with periodicity. However, it is still interesting to study all the
recurrence notions introduced in Example 4.2 since clearly
$\operatorname{Per}(T)\subset\Delta^{*}\textup{Rec}(T)$ for every dynamical
system $(X,T)$. We have also reproved [8, Proposition 3]:

######  Corollary 4.12.

There is no $\overline{\mathcal{BD}}_{\delta}$-hypercyclic operator for any
$0<\delta\leq 1$.

###  4.3 Quasi-conjugacies and commutants in $\mathcal{F}$-recurrence

We end Section 4 by presenting some basic but useful tools regarding quasi-
conjugacies and (non-linear) commutants which will be used in Section 5 to
study the lineability and dense lineability properties of the set of
$\mathcal{F}$-recurrent vectors. Following [24, 31] we define:

######  Definition 4.13.

Given two dynamical systems $(X,T)$ and $(Y,S)$, we say that a map
$\phi:Y\longrightarrow X$ is an homomorphism of dynamical systems if $\phi$ is
continuous and the diagram

$\begin{CD}Y@>{S}>{}>Y\\\ @V{\phi}V{}V@V{}V{\phi}V\\\ X@>{T}>{}>X\end{CD}$

commutes, i.e. $\phi\circ S=T\circ\phi$. Moreover, we say that the map $\phi$
is a:

  1. –

quasi-conjugacy, if $\phi$ has dense range, and hence that $(X,T)$ is quasi-
conjugate to $(Y,S)$;

  2. –

conjugacy, if $\phi$ is an homeomorphism between $X$ and $Y$ (see Remark 5.2).

Many dynamical properties are preserved by quasi-conjugacy (see [31, Chapter
1]): if $(Y,S)$ admits a dense orbit, is topologically transitive, weakly-
mixing or even (Devaney) chaotic, then so is $(X,T)$. On the other hand, to
preserve the recurrence of a point it is enough to have an homomorphism. This
is a really well-known fact (see [24, Proposition 1.3]), and indeed similar
arguments were already used in [24, Proposition 9.9] for
$\mathcal{F}$-recurrence with respect to the Furstenberg families $\Delta^{*}$
and $\mathcal{IP}^{*}$, or in [17, Proposition 2.7] for general Furstenberg
families. We include here the general argument regarding return sets:

######  Lemma 4.14.

Let $(X,T)$ and $(Y,S)$ be dynamical systems and let $\phi:Y\longrightarrow X$
be an homomorphism between them. Given $y\in Y$ and any neighbourhood
$U\subset X$ of $\phi(y)$, the set $V:=\phi^{-1}(U)\subset Y$ is a
neighbourhood of $y$ for which $N_{S}(y,V)=N_{T}(\phi(y),U)$. In particular,
for any Furstenberg family $\mathcal{F}$ we have that:

  1. (a)

If $y\in Y$ is $\mathcal{F}$-recurrent for $S$ then $\phi(y)$ is
$\mathcal{F}$-recurrent for $T$.

  2. (b)

If $\phi$ is a quasi-conjugacy and $S$ is $\mathcal{F}$-recurrent then so is
$T$.

###### Proof.

By continuity of $\phi$ the set $V=\phi^{-1}(U)$ is a neighbourhood of $y$.
Since $\phi\circ S=T\circ\phi$ we have that $\phi(S^{n}y)=T^{n}\phi(y)$ for
every $n\in\mathbb{N}_{0}$. It follows that

$N_{T}(\phi(y),U)=\\{n\in\mathbb{N}_{0}:T^{n}\phi(y)\in
U\\}=\\{n\in\mathbb{N}_{0}:S^{n}y\in\phi^{-1}(U)\\}=N_{S}(y,V).$

In particular, if $y$ is $\mathcal{F}$-recurrent then the return set
$N_{T}(\phi(y),U)=N_{S}(y,\phi^{-1}(U))$ belongs to $\mathcal{F}$ for every
neighbourhood $U$ of $\phi(y)$. Statement (b) follows immediately from (a). ∎

Considering $Y:=X$, $S:=T$ and $\phi:=S$ we obtain the following:

######  Corollary 4.15.

Let $(X,T)$ be a dynamical system and let $S:X\longrightarrow X$ be a
continuous map commuting with $T$, that is $S\circ T=T\circ S$. Given $x\in X$
and any neighbourhood $U$ of $Sx$, the set $V:=S^{-1}(U)$ is a neighbourhood
of $x$ for which $N_{T}(x,V)=N_{T}(Sx,U)$. In particular, for any Furstenberg
family $\mathcal{F}$ we have that
$S(\mathcal{F}\textup{Rec}(T))\subset\mathcal{F}\textup{Rec}(T)$.

Corollary 4.15 unifies some of the well-known arguments already used in [1,
Proposition 2.5], [19, Proof of Theorem 9.1] and [24, Propositions 1.3 and
9.9], generalizing them to arbitrary Furstenberg families and (not necessarily
linear) maps acting on arbitrary topological spaces. From now on we will write

$\mathscr{C}(X):=\left\\{S:X\longrightarrow X\text{ continuous map}\right\\},$

i.e. the set of continuous (and not necessarily linear) self-maps of the
Hausdorff topological space $X$, and following the notation of [35] we define:

######  Definition 4.16.

Let $(X,T)$ be a dynamical system. The (non-linear) commutant of $T$ is the
set

$\mathcal{C}_{T}:=\left\\{S\in\mathscr{C}(X):S\circ T=T\circ S\right\\}.$

Given $x\in X$ we define the $\mathcal{C}_{T}$-orbit of $x$ as
$\mathcal{C}_{T}(x):=\\{Sx:S\in\mathcal{C}_{T}\\}$. We say that a subset
$Y\subset X$ is $\mathcal{C}_{T}$-invariant if $S(Y)\subset Y$ for every
$S\in\mathcal{C}_{T}$.

######  Remark 4.17.

Since $T\in\mathcal{C}_{T}$ we always have that:

  1. –

every $\mathcal{C}_{T}$-invariant set is also $T$-invariant;

  2. –

if $X$ is a linear space, then $\\{p(T):p\text{
polynomial}\\}\subset\mathcal{C}_{T}$ and for every $x\in X$ we get that

$\textup{span}(\textup{Orb}(x,T))=\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}=\\{p(T)x:p\text{
polynomial}\\}\subset\mathcal{C}_{T}(x),$

so that every cyclic vector for an operator $T\in\mathcal{L}(X)$ has a dense
$\mathcal{C}_{T}$-orbit. Moreover:

######  Lemma 4.18.

Let $(X,T)$ be a linear dynamical system. Then $(\mathcal{C}_{T},+,\circ)$ is
a subring of the ring of continuous maps $(\mathscr{C}(X),+,\circ)$. In
particular, given any $x\in X$ the set $\mathcal{C}_{T}(x)$ is a vector
subspace of $X$ and the smallest $\mathcal{C}_{T}$-invariant subset of $X$
containing the point $x$.

###### Proof.

Given $S,R\in\mathcal{C}_{T}$ and $\alpha,\beta\in\mathbb{K}$ we have that
$(\alpha S+\beta R)\in\mathcal{C}_{T}$ and $(S\circ R)\in\mathcal{C}_{T}$. In
particular, given any $x\in X$ the set $\mathcal{C}_{T}(x)$ is a vector
subspace of $X$. Moreover, if $Y\subset X$ is a $\mathcal{C}_{T}$-invariant
subset of $X$ and $x\in Y$ then
$\mathcal{C}_{T}(x)=\left\\{Sx:S\in\mathcal{C}_{T}\right\\}\subset\bigcup_{S\in\mathcal{C}_{T}}S(Y)\subset
Y$. ∎

When $(X,T)$ is a non-linear system then $\mathcal{C}_{T}(x)$ is still the
smallest $\mathcal{C}_{T}$-invariant subset of $X$ containing the point $x\in
X$. Let us now generalize Corollary 4.15 to direct products by using the
following notation: given a system $(X,T)$ and a subset $Y\subset X$ we will
denote by

$\mathcal{N}_{pr}(Y):=\\{A\subset\mathbb{N}_{0}:N(x,U)\subset A\text{ for some
}x\in Y\text{ and some neighbourhood }U\text{ of }x\\},$

the family of pointwise-recurrent return sets of the points of $Y$. Note that
$\mathcal{N}_{pr}(\\{x\\})$ is a filter if and only if $x\in\textup{Rec}(T)$:
indeed, if $x\notin\textup{Rec}(T)$ then
$\mathcal{N}_{pr}(\\{x\\})=\mathscr{P}(\mathbb{N}_{0})$, which is not a
filter; conversely, for any pair $U$ and $V$ of neighbourhoods for a recurrent
vector $x\in\textup{Rec}(T)$ then we have that $N(x,U\cap V)\subset N(x,U)\cap
N(x,V)$. Note also that a dynamical system $(X,T)$ is $\mathcal{F}$-recurrent
if there exits a dense set $Y\subset X$ such that
$\mathcal{N}_{pr}(Y)\subset\mathcal{F}$.

######  Theorem 4.19.

Let $(X,T)$ be a (linear) dynamical system and let $\mathcal{F}$ be a
Furstenberg family. Given $x\in\mathcal{F}\textup{Rec}(T)$ we have that

$\mathcal{N}_{pr}(\mathcal{C}_{T}(x))\ =\ \mathcal{N}_{pr}(\\{x\\}),$

and hence $\mathcal{C}_{T}(x)$ is a $\mathcal{C}_{T}$-invariant (vector)
subspace of $X$ with the property that

$\mathcal{C}_{T}(x)^{N}\ \subset\ \mathcal{F}\textup{Rec}(T_{(N)})\quad\text{
for every }N\in\mathbb{N}.$

###### Proof.

Obviously
$\mathcal{N}_{pr}(\\{x\\})\subset\mathcal{N}_{pr}(\mathcal{C}_{T}(x))$. The
inclusion
$\mathcal{N}_{pr}(\mathcal{C}_{T}(x))\subset\mathcal{N}_{pr}(\\{x\\})$ follows
from Corollary 4.15: given $y\in\mathcal{C}_{T}(x)$ and any neighbourhood $U$
of $y$ there is a neighbourhood $V$ of $x$ such that

$N_{T}(y,U)=N_{T}(x,V)\in\mathcal{N}_{pr}(\\{x\\}).$

The rest of the result follows from the filter condition satisfied by the
family $\mathcal{N}_{pr}(\\{x\\})$: indeed, given any $N\in\mathbb{N}$ and a
set $\\{x_{1},x_{2},...,x_{N}\\}\subset\mathcal{C}_{T}(x)$ write

$z:=(x_{1},x_{2},...,x_{N})\in X^{N}.$

Then, given any neighbourhood $U\subset X^{N}$ of $z$ we can find
neighbourhoods $U_{i}\subset X$ of $x_{i}$, for $1\leq i\leq N$, such that
$U_{1}\times\cdots\times U_{N}\subset U$, and hence

$N_{T_{(N)}}(z,U)\supset\bigcap_{i=1}^{N}N_{T}(x_{i},U_{i})\in\mathcal{N}_{pr}(\\{x\\})\subset\mathcal{F}.$

The arbitrariness of $U$ implies that $z\in\mathcal{F}\textup{Rec}(T_{(N)})$.
Finally, if $(X,T)$ is a linear dynamical system, then the set
$\mathcal{C}_{T}(x)$ is a $\mathcal{C}_{T}$-invariant vector subspace of $X$
by Lemma 4.18. ∎

######  Remark 4.20.

Theorem 4.19 is an extension of Corollary 4.15 to every $N$-fold direct
product of a given dynamical system. As we have already mentioned, these kind
of arguments were already used in [19, Theorem 9.1] for usual recurrence and
operators on Banach spaces, and in [17, Proposition 2.7] for arbitrary
Furstenberg families.

##  5 Infinite-dimensional vector spaces in $\mathcal{F}\textup{Rec}(T)$

The objective of this section is to study, for a linear dynamical system
$(X,T)$ and any arbitrary Furstenberg family $\mathcal{F}$, when the set of
$\mathcal{F}$-recurrent vectors is lineable or dense lineable, i.e. to
establish whether it contains a (possibly dense) infinite-dimensional vector
subspace. Theorem 4.19 will be our main tool for this study, which motivation
stems from the following two facts already shown in previous sections:

  1. –

quasi-rigidity coincides with $\mathcal{F}(A)$-recurrence (see Proposition
4.5);

  2. –

quasi-rigidity of an operator $T$ implies that $\textup{Rec}(T)$ is dense
lineable (see Proposition 2.7).

It is then natural to ask whether the set $\mathcal{F}\textup{Rec}(T)$
contains an infinite-dimensional vector space for other Furstenberg families
$\mathcal{F}$. We show that $\mathcal{F}\textup{Rec}(T)$ is lineable as soon
as $T$ is $\mathcal{F}$-recurrent (see Theorem 5.5 and Corollary 5.6 below),
and we obtain some (natural) sufficient conditions implying that
$\mathcal{F}\textup{Rec}(T)$ is dense lineable (see Theorem 5.8), obtaining
then the Herrero-Bourdon theorem for $\mathcal{F}$-hypercyclicity (see
Subsection 5.3).

###  5.1 Lineability

Given a vector space $X$ and a subset of vectors $Y\subset X$ with some
property, we say that $Y$ is lineable if there exists an infinite-dimensional
vector subspace $Z\subset X$ such that $Z\setminus\\{0\\}\subset Y$. In our
case, given a linear dynamical system $(X,T)$ we always have that
$0\in\mathcal{F}\textup{Rec}(T)$ for every Furstenberg family $\mathcal{F}$,
so the set of $\mathcal{F}$-recurrent vectors will be lineable if it admits an
infinite-dimensional vector space (which includes the zero-vector).

In order to prove that $\mathcal{F}\textup{Rec}(T)$ is lineable as soon as
$(X,T)$ is $\mathcal{F}$-recurrent, we will observe that the (span of the)
unimodular eigenvectors are the only recurrent vectors whose orbits have a
finite-dimensional linear span (see Lemma 5.3 below). Recall that:

######  Definition 5.1.

Given a complex-linear dynamical system $T:X\longrightarrow X$, a vector $x\in
X$ is called a unimodular eigenvector for $T$ if $x\neq 0$ and $Tx=\lambda x$
for some unimodular complex number $\lambda\in\mathbb{T}$. We denote by
$\mathcal{E}(T)$ the set of unimodular eigenvectors for $T$, i.e.

$\mathcal{E}(T)=\\{x\in X\setminus\\{0\\}:Tx=\lambda x\text{ for some
}\lambda\in\mathbb{T}\\}.$

Every finite linear combination of unimodular eigenvectors is a
$\Delta^{*}$-recurrent vector (see for instance [29, Proposition 4.1]) and the
following holds (see [31, Proposition 2.33]):

$\operatorname{Per}(T)=\textup{span}\\{x\in X:Tx=e^{\alpha\pi i}x\text{ for
some
}\alpha\in\mathbb{Q}\\}\subset\textup{span}(\mathcal{E}(T))\subset\Delta^{*}\textup{Rec}(T).$

In order to treat both real and complex cases at the same time, we need a set
of (real) vectors having an analogous recurrent-behaviour to that of
unimodular eigenvectors:

######  Remark 5.2 (A well-known conjugacy).

The complexification $(\widetilde{X},\widetilde{T})$ of a real-linear system
$T:X\longrightarrow X$ is defined in the following way (see [41, 40] and [31,
Exercise 2.2.7]):

  1. –

the space $\widetilde{X}:=\left\\{x+iy:x,y\in X\right\\}$, which is
topologically identified with $X\oplus X$, and becomes a complex F-space when
endowed with the multiplication by complex scalars
$(\alpha+i\beta)(x+iy)=(\alpha x-\beta y)+i(\alpha y+\beta x)$ for every
$\alpha,\beta\in\mathbb{R}$ and every $x,y\in X$;

  2. –

and the operator $\widetilde{T}:\widetilde{X}\longrightarrow\widetilde{X}$ is
defined as $\widetilde{T}(x+iy)=Tx+iTy$ for every $x,y\in X$. It is a
continuous complex-linear operator acting on $\widetilde{X}$.

Defining the map $J:X\oplus X\longrightarrow\widetilde{X}$ as
$J(x,y):=x+iy\in\widetilde{X}$ for every $(x,y)\in X\oplus X$, the diagram

$\begin{CD}X\oplus X@>{T\oplus T}>{}>X\oplus X\\\ @V{J}V{}V@V{}V{J}V\\\
\widetilde{X}@>{\widetilde{T}}>{}>\widetilde{X}\\\ \end{CD}$

commutes so $J$ is a conjugacy (see Definition 4.13). In this setting we
define the (real) set of unimodular eigenvectors for $T$ as

$\mathcal{E}(T):=\left\\{x\in X:\text{ there exists }y\in X\text{ such that
}x+iy\in\mathcal{E}(\widetilde{T})\right\\}.$

These real unimodular eigenvectors have the same recurrent-behaviour than the
complex unimodular eigenvectors. In fact, the following properties are easily
checked:

$\textup{span}(\mathcal{E}(T))=\left\\{x\in X:\text{ there exists }y\in
X\text{ such that }x+iy\in\textup{span}(\mathcal{E}(\widetilde{T}))\right\\},$
$\operatorname{Per}(T)\subset\textup{span}(\mathcal{E}(T))\subset\Delta^{*}\textup{Rec}(T).$

######  Lemma 5.3.

Let $(X,T)$ be a (real or complex) linear dynamical system. For each recurrent
vector $x\in\textup{Rec}(T)$ the following statements are equivalent:

  1. (i)

$x\in\textup{span}(\mathcal{E}(T))$;

  2. (ii)

$\dim\left(\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}\right)<\infty$.

In particular, given any Furstenberg family $\mathcal{F}$ for which
$\mathcal{F}\textup{Rec}(T)\setminus\textup{span}(\mathcal{E}(T))\neq\varnothing$,
there exists an infinite-dimensional $T$-invariant vector subspace $Z\subset
X$ with the property that $Z^{N}\subset\mathcal{F}\textup{Rec}(T_{(N)})$ for
every $N\in\mathbb{N}$.

###### Proof.

For (i) $\Rightarrow$ (ii) compute the orbit of a vector from
$\textup{span}(\mathcal{E}(T))$ in both real and complex cases, and observe
that its span is finite-dimensional. For (ii) $\Rightarrow$ (i) let
$x\in\textup{Rec}(T)$ and suppose that the $T$-invariant subspace
$E=\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}$ is finite-dimensional:

  1. (1)

If $(X,T)$ is complex, then $T\arrowvert_{E}:E\longrightarrow E$ is a
recurrent complex-linear operator on a finite-dimensional space. By [19,
Theorem 4.1] there exists a basis of $E$ formed by unimodular eigenvectors for
$T\arrowvert_{E}$ (and hence for $T$) so $x\in
E\subset\textup{span}(\mathcal{E}(T))$.

  2. (2)

If $(X,T)$ is real, then we can identify the complexification
$\widetilde{T\arrowvert_{E}}:\widetilde{E}\longrightarrow\widetilde{E}$ with
the direct sum $T\arrowvert_{E}\oplus T\arrowvert_{E}$ (see Remark 5.2).
Theorem 4.19 implies that

$E\oplus E\subset\textup{Rec}(T\arrowvert_{E}\oplus T\arrowvert_{E}),$

so $\widetilde{T\arrowvert_{E}}$ is a recurrent complex-linear operator on a
finite-dimensional space. As in case (1) above, by [19, Theorem 4.1] there
exists a basis of $\widetilde{E}$ formed by unimodular eigenvectors for
$\widetilde{T}$, so
$x+i0\in\widetilde{E}\subset\textup{span}(\mathcal{E}(\widetilde{T}))$ and
hence $x\in\textup{span}(\mathcal{E}(T))$.

Finally, if $\mathcal{F}$ is a Furstenberg family for which there exists some
$x\in\mathcal{F}\textup{Rec}(T)\setminus\textup{span}(\mathcal{E}(T))$, the
equivalence (i) $\Leftrightarrow$ (ii) implies that
$Z:=\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}$ is an infinite-dimensional
$T$-invariant vector subspace of $X$. Theorem 4.19 then shows

$Z^{N}\subset\mathcal{C}_{T}(x)^{N}\subset\mathcal{F}\textup{Rec}(T_{(N)})\quad\text{
for every }N\in\mathbb{N}.\qed$

The strongest $\mathcal{F}$-recurrence notion considered in this paper and
fulfilling the condition that
$\textup{span}(\mathcal{E}(T))\subset\mathcal{F}\textup{Rec}(T)$ for every
$T\in\mathcal{L}(X)$ is that of $\Delta^{*}$-recurrence. The properties of
“having a spanning set of unimodular eigenvectors” and that of “being
$\Delta^{*}$-recurrent” have been deeply related in the recent work [29] and
they are specially near when one considers power-bounded operators (see [29,
Theorem 1.9]). It is then natural to ask if the equality
$\textup{span}(\mathcal{E}(T))=\Delta^{*}\textup{Rec}(T)$ holds for some
general class of linear dynamical systems $(X,T)$, a natural candidate being
that of power-bounded operators. The answer is negative as we show in the
following trivial example by using Lemma 5.3:

######  Example 5.4.

There exists a linear dynamical system admitting a $\Delta^{*}$-recurrent
vector such that the linear span of its orbit is infinite-dimensional:
consider $X=c_{0}(\mathbb{N})$ or $\ell^{p}(\mathbb{N})$ for some $1\leq
p<\infty$ with their usual norms. Define a linear map
$T:c_{00}(\mathbb{N})\longrightarrow c_{00}(\mathbb{N})$ in the following way:
$Te_{1}=e_{1}$ and for every $k\geq 2$

$Te_{k}=\begin{cases}e_{k+1},&\text{if }2^{m}<k<2^{m+1},\\\
e_{2^{m}+1},&\text{if }k=2^{m+1},\end{cases}$

for each $m\geq 0$, where $e_{k}=\\{\delta_{k,n}\\}_{n=1}^{\infty}$ is the
$k$-th vector of the canonical basis of $X$. Since $\|Tx\|=\|x\|$ for each
$x\in c_{00}(\mathbb{N})$, $T$ extends to a linear isometry on the whole space
$X$. We now show that the vector $x=\sum_{m\geq
0}\frac{1}{2^{m}}e_{2^{m}+1}\in X$ has the required properties:

  1. 1)

$x\in\Delta^{*}\textup{Rec}(T)$: given any $\varepsilon>0$ there is
$m_{\varepsilon}\in\mathbb{N}$ such that
$\left\|\sum_{m>m_{\varepsilon}}\frac{1}{2^{m}}e_{2^{m}+1}\right\|<\frac{\varepsilon}{2}$.
Then, given $n\in 2^{m_{\varepsilon}}\cdot\mathbb{N}_{0}$ we have that

$T^{n}\left(\sum_{m\leq
m_{\varepsilon}}\frac{1}{2^{m}}e_{2^{m}+1}\right)=\sum_{m\leq
m_{\varepsilon}}\frac{1}{2^{m}}e_{2^{m}+1}\quad\text{ so
}\quad\|T^{n}x-x\|\leq
2\left\|\sum_{m>m_{\varepsilon}}\frac{1}{2^{m}}e_{2^{m}+1}\right\|<\varepsilon.$

Hence $2^{m_{\varepsilon}}\cdot\mathbb{N}_{0}\subset
N_{T}(x,B(x,\varepsilon))$, so $x\in\Delta^{*}\textup{Rec}(T)$ since
$2^{m}\cdot\mathbb{N}_{0}\in\Delta^{*}$ for all $m\in\mathbb{N}$.

  2. 2)

$\textup{span}\\{T^{n}x:n\in\mathbb{N}_{0}\\}$ is infinite-dimensional:
otherwise there would exist a polynomial

$p(z)=\sum_{i=1}^{N}a_{i}z^{i}\text{ with }a_{i}\in\mathbb{K}\text{ and
}a_{N}\neq 0\quad\text{ such that }\quad p(T)x=0,$

and taking $m_{0}\in\mathbb{N}$ such that $N<2^{m_{0}}$ we would arrive to the
contradiction

$0=\left[p(T)x\right]_{2^{m_{0}}+1+N}=\left[a_{N}\cdot
T^{N}x\right]_{2^{m_{0}}+1+N}=\frac{a_{N}}{2^{m_{0}}}\neq 0.$

We finally state the desired lineability results:

######  Theorem 5.5 (Lineability).

Let $\mathcal{F}$ be a Furstenberg family such that the inclusion
$\textup{span}(\mathcal{E}(S))\subset\mathcal{F}\textup{Rec}(S)$ holds for
every linear dynamical system $(Y,S)$. Then, for each $\mathcal{F}$-recurrent
linear system $(X,T)$ there exists an infinite-dimensional $T$-invariant
vector subspace $Z\subset X$ with the property that
$Z^{N}\subset\mathcal{F}\textup{Rec}(T_{(N)})$ for every $N\in\mathbb{N}$.

###### Proof.

By assumption $\mathcal{F}\textup{Rec}(T)$ is dense. We distinguish two cases:

  1. (1)

If $\textup{span}(\mathcal{E}(T))$ is finite-dimensional (or simply if it is
not dense in $X$), then we have that
$\mathcal{F}\textup{Rec}(T)\setminus\textup{span}(\mathcal{E}(T))\neq\varnothing$
and Lemma 5.3 yields the desired vector subspace.

  2. (2)

If $\textup{span}(\mathcal{E}(T))$ is infinite-dimensional, then
$(\textup{span}(\mathcal{E}(T)))^{N}=\textup{span}(\mathcal{E}(T_{(N)}))\subset\mathcal{F}\textup{Rec}(T_{(N)})$
for every $N\in\mathbb{N}$, and $Z:=\textup{span}(\mathcal{E}(T))$ is the
required vector subspace.∎

The previous theorem is “optimal” for the families considered in this paper:
we always have the inclusions

$\operatorname{Per}(T)\subset\textup{span}(\mathcal{E}(T))\subset\Delta^{*}\textup{Rec}(T),$

and we observed in Section 4 that the families $\mathcal{F}$ considered in
this paper for which
$\mathcal{F}\textup{Rec}(T)\subset\Delta^{*}\textup{Rec}(T)$ are such that
$\mathcal{F}\textup{Rec}(T)\subset\operatorname{Per}(T)$ for every system
$(X,T)$. We deduce (at least) the following result:

######  Corollary 5.6.

Let $(X,T)$ be a linear dynamical system. If the operator $T$ is recurrent
(resp. $\mathcal{AP}$, reiterative, $\mathcal{U}$-frequent, frequent,
uniformly, $\mathcal{IP}^{*}$ or $\Delta^{*}$-recurrent), then
$\textup{Rec}(T)$ (resp. $\mathcal{AP}\textup{Rec}(T)$,
$\operatorname{RRec}(T)$, $\operatorname{UFRec}(T)$, $\operatorname{FRec}(T)$,
$\operatorname{URec}(T)$, $\mathcal{IP}^{*}\textup{Rec}(T)$ or
$\Delta^{*}\textup{Rec}(T)$) contains an infinite-dimensional $T$-invariant
vector space, and in particular it is lineable.

Theorem 5.5 (and hence Corollary 5.6) is still true with the same proof if we
replace the original assumption that “$T$ is $\mathcal{F}$-recurrent” by the
very much less restrictive hypothesis that “$\mathcal{F}\textup{Rec}(T)$ is
dense in some infinite-dimensional closed subspace $Y\subset X$”, or even by
the formally weaker “$\mathcal{F}\textup{Rec}(T)$ spans an infinite-
dimensional vector subspace”. These are also necessary conditions for the
lineability property as the following example shows:

######  Example 5.7.

Let $X=c_{0}(\mathbb{N})$ or $\ell^{p}(\mathbb{N})$ for some $1\leq p<\infty$
with their usual norm. Consider a bounded sequence
$(\lambda_{n})_{n\in\mathbb{N}}\subset\mathbb{C}$ and the respective
multiplication operator defined as

$T\left((x_{n})_{n\in\mathbb{N}}\right):=(\lambda_{n}\cdot
x_{n})_{n\in\mathbb{N}}\quad\text{ for each }(x_{n})_{n\in\mathbb{N}}\in X.$

For any fixed any $N\in\mathbb{N}$, consider a sequence
$(\lambda_{n})_{n\in\mathbb{N}}$ with the properties that

$\lambda_{n}=1\text{ for }n\leq N\qquad\text{ and
}\qquad\lambda_{n}\notin\mathbb{T}\text{ for }n>N.$

Then it is trivial to check that, for any Furstenberg family $\mathcal{F}$,
the set $\mathcal{F}\textup{Rec}(T)$ is exactly a finite-dimensional vector
space of dimension $N$.

###  5.2 Dense Lineability

Given a topological vector space $X$ and a subset of vectors $Y\subset X$ with
some property, we say that $Y$ is dense lineable if there exists a dense
infinite-dimensional vector subspace $Z\subset X$ such that
$Z\setminus\\{0\\}\subset Y$. As it happens for lineability, given a linear
dynamical system $(X,T)$ we always have that $0\in\mathcal{F}\textup{Rec}(T)$
for every Furstenberg family $\mathcal{F}$, so the set of
$\mathcal{F}$-recurrent vectors will be dense lineable if it admits a dense
infinite-dimensional vector space (which includes the zero-vector).

Given a linear dynamical system $(X,T)$ we provide some sufficient conditions
for the set $\mathcal{F}\textup{Rec}(T)$ to be dense lineable. Filters will
play a fundamental role in our study. The motivation for our main result (see
Theorem 5.8 below) is again the notion of quasi-rigidity:

  1. –

quasi-rigidity implies that $\textup{Rec}(T)=\mathcal{I}\textup{Rec}(T)$ is
dense lineable (see Proposition 2.7);

  2. –

quasi-rigidity coincides with $\mathcal{F}(A)$-recurrence, and
$\mathcal{F}(A)$ is a filter (see Proposition 4.5);

  3. –

i.e. there is a dense set $Y\subset X$ and a filter
$\mathcal{F}^{\prime}=\mathcal{F}(A)$ such that
$\mathcal{N}_{pr}(Y)\subset\mathcal{F}^{\prime}\subset\mathcal{I}$.

We can generalize these ideas to arbitrary Furstenberg families $\mathcal{F}$,
finding some filter $\mathcal{F}^{\prime}$ contained in $\mathcal{F}$. In
particular, we extend [13, Theorem 6.1] where it is shown that the set of
$\mathcal{IP}^{*}$-recurrent vectors is always a vector subspace:

######  Theorem 5.8 (Dense Lineability).

Let $(X,T)$ be a linear dynamical system and let $\mathcal{F}$ be a
Furstenberg family. If there exist a dense subset $Y\subset X$ and a filter
$\mathcal{F}^{\prime}$ such that

$\mathcal{N}_{pr}(Y)\subset\mathcal{F}^{\prime}\subset\mathcal{F},$

then $Z:=\textup{span}(\bigcup_{S\in\mathcal{C}_{T}}S(Y))\subset X$ is a dense
infinite-dimensional $T$-invariant vector subspace with the property that
$Z^{N}\subset\mathcal{F}\textup{Rec}(T_{(N)})$ for every $N\in\mathbb{N}$. In
particular, if $\mathcal{F}$ is a filter itself, then for every
$N\in\mathbb{N}$ we have the equality

$\mathcal{F}\textup{Rec}(T)^{N}=\mathcal{F}\textup{Rec}(T_{(N)}),$

which is a $T$-invariant vector subspace of $X^{N}$, and the following are
equivalent:

  1. (i)

$T$ is $\mathcal{F}$-recurrent;

  2. (ii)

$\mathcal{F}\textup{Rec}(T_{(N)})$ is a dense infinite-dimensional vector
subspace of $X^{N}$ for every $N\in\mathbb{N}$.

###### Proof.

It is clear that $Z\subset X$ is a dense infinite-dimensional $T$-invariant
vector subspace. Let us show that
$\mathcal{N}_{pr}(Z)\subset\mathcal{F}^{\prime}$: given
$y_{1},y_{2},...,y_{N}\in\bigcup_{S\in\mathcal{C}_{T}}S(Y)$ and any
neighbourhood $U\subset X$ of $x:=\sum_{i=1}^{N}y_{i}\in Z$ we can find
neighbourhoods $U_{i}$ of $y_{i}$, for $1\leq i\leq N$, such that

$\sum_{i=1}^{N}U_{i}\subset U\subset X\quad\text{ and hence }\quad
N_{T}(x,U)\supset\bigcap_{i=1}^{N}N_{T}(y_{i},U_{i})\in\mathcal{F}^{\prime}\subset\mathcal{F},$

since $\mathcal{N}_{pr}(\\{y_{i}\\})\subset\mathcal{F}^{\prime}$ for all
$1\leq i\leq N$ by Theorem 4.19. Finally, given any $N\in\mathbb{N}$,
$x_{1},x_{2},...,x_{N}\in Z$ and any neighbourhood $V\subset X^{N}$ of the
$N$-tuple $z:=(x_{1},...,x_{N})\in X^{N}$, we can find neighbourhoods $V_{i}$
of $z_{i}$, for $1\leq i\leq N$, such that

$V_{1}\oplus\cdots\oplus V_{N}\subset V\subset X^{N}\quad\text{ and hence
}\quad
N_{T_{(N)}}(z,V)\supset\bigcap_{i=1}^{N}N_{T}(z_{i},V_{i})\in\mathcal{F}^{\prime}\subset\mathcal{F}.$

The arbitrariness of $V\subset X^{N}$ implies that
$z\in\mathcal{F}\textup{Rec}(T_{(N)})$. ∎

Note that Theorem 5.8 extends the result obtained in Proposition 2.7 to every
$N$-fold direct sum operator. Moreover, in view of Theorems 4.19 and 5.8 we
can generalize Proposition 2.9 establishing, for dense lineability, the
following (natural) sufficient conditions:

######  Proposition 5.9.

Let $(X,T)$ be a linear dynamical system and let $\mathcal{F}$ be a
Furstenberg family. If $T$ admits an $\mathcal{F}$-recurrent vector
$x\in\mathcal{F}\textup{Rec}(T)$ with a dense $\mathcal{C}_{T}$-orbit

$\mathcal{C}_{T}(x)=\\{Sx:S\in\mathcal{C}_{T}\\},$

then $\mathcal{F}\textup{Rec}(T_{(N)})$ is dense lineable in $X^{N}$ for every
$N\in\mathbb{N}$. In particular, the later is true whenever any of the
following holds:

  1. –

$T$ admits an $\mathcal{F}$-recurrent and cyclic vector;

  2. –

$T$ admits an $\mathcal{F}$-hypercyclic vector.

###### Proof.

Given a vector $x\in\mathcal{F}\textup{Rec}(T)$ for which $\mathcal{C}_{T}(x)$
is dense in $X$, Theorem 4.19 implies that
$\mathcal{N}_{pr}(\mathcal{C}_{T}(x))\subset\mathcal{N}_{pr}(\\{x\\})\subset\mathcal{F}$
and the result follows from Theorem 5.8. ∎

The dense lineability of the set $\mathcal{F}\textup{Rec}(T_{(N)})$ implies
that $T_{(N)}$ is $\mathcal{F}$-recurrent. A slightly weaker hypothesis allows
us to keep the $\mathcal{F}$-recurrence of every $N$-fold direct product:

######  Proposition 5.10 ($\mathcal{F}$-recurrence for all $(X^{N},T_{(N)})$).

Let $(X,T)$ be a linear dynamical system and let $\mathcal{F}$ be a
Furstenberg family. If $T$ is $\mathcal{F}$-recurrent and if there exists a
vector $x\in X$ with a dense $\mathcal{C}_{T}$-orbit

$\mathcal{C}_{T}(x)=\\{Sx:S\in\mathcal{C}_{T}\\},$

then $T_{(N)}:X^{N}\longrightarrow X^{N}$ is $\mathcal{F}$-recurrent for every
$N\in\mathbb{N}$. In particular, the later is true whenever any of the
following holds:

  1. –

$T$ is $\mathcal{F}$-recurrent and cyclic;

  2. –

$T$ is $\mathcal{F}$-recurrent and hypercyclic.

This has been recently proved in [17, Proposition 2.7], but we repeat here the
argument for the sake of completeness:

###### Proof.

Fix $N\in\mathbb{N}$. By Theorem 4.19 we have the inclusion

$Y:=\bigcup_{y\in\mathcal{F}\textup{Rec}(T)}\mathcal{C}_{T}(y)^{N}\subset\mathcal{F}\textup{Rec}(T_{(N)}),$

so it is enough to show that $Y$ is dense in $X^{N}$. By assumption the point
$x\in X$ has a dense $\mathcal{C}_{T}$-orbit, so given any non-empty open
subset $U\subset X^{N}$ there are $N$ continuous maps
$S_{1},S_{2},...,S_{N}\in\mathcal{C}_{T}$ such that
$(S_{1}x,S_{2}x,...,S_{N}x)\in U$. By the continuity of
$S_{1},S_{2},...,S_{N}$, and since $T$ is $\mathcal{F}$-recurrent, we can find
$y\in\mathcal{F}\textup{Rec}(T)$ near enough to $x$ such that

$(S_{1}y,S_{2}y,...,S_{N}y)\in\ \mathcal{C}_{T}(y)^{N}\cap U\ \subset\ Y\cap
U.\qed$

######  Remark 5.11.

We include here some comments about the previous results:

  1. (a)

Proposition 5.10 remains valid for non-linear dynamical systems. It is a
slight improvement of [19, Theorem 9.1] in terms of Furstenberg families and
non-linear commuting maps, and it has been independently proved in the recent
work [17, Proposition 2.7].

  2. (b)

Given a Furstenberg family $\mathcal{F}$, the following is a natural open
question:

######  Problem 5.12.

Suppose that a linear dynamical system $(X,T)$ does not admit any dense
$\mathcal{C}_{T}$-orbit. Can every $N$-fold direct sum $(X^{N},T_{(N)})$ be
$\mathcal{F}$-recurrent?

  3. (c)

The assumptions of Proposition 5.9 imply those of Proposition 5.10.
Conversely:

######  Problem 5.13.

Let $\mathcal{F}$ be a Furstenberg family. Can an $\mathcal{F}$-recurrent
linear system $(X,T)$ admit dense $\mathcal{C}_{T}$-orbits but fulfill that
$\mathcal{F}\textup{Rec}(T)\cap\\{x\in
X:\overline{\mathcal{C}_{T}(x)}=X\\}=\varnothing$?

This seems to be a tough question since the commutator of an operator is
usually difficult to describe. If instead of dense $\mathcal{C}_{T}$-orbits we
just consider the set of hypercyclic vectors, then the answer is yes: Menet
constructed a chaotic (and hence with dense periodic vectors) operator $T$,
which is not $\mathcal{U}$-frequently hypercyclic (see [39]), i.e.

    1. –

Such an operator $T$ fulfills the assumptions of Proposition 5.10 for every
Furstenberg family $\mathcal{F}$ satisfying that
$\Delta^{*}\subset\mathcal{F}\subset\overline{\mathcal{D}}$, but we have that
$\mathcal{F}\textup{Rec}(T)\cap\textup{HC}(T)=\varnothing$ so that $T$ has no
vector which is $\mathcal{F}$-recurrent and hypercyclic at the same time.

Considering now the set of cyclic vectors (instead of dense
$\mathcal{C}_{T}$-orbits or hypercyclic vectors), Corollary 2.11 and
Proposition 5.9 imply that:

    1. –

If $(X,T)$ is a recurrent and cyclic linear dynamical system, then the set
$\textup{Rec}(T_{(N)})$ is dense lineable for every $N\in\mathbb{N}$.

In general we cannot change $\textup{Rec}(T_{(N)})$ into
$\mathcal{F}\textup{Rec}(T_{(N)})$ unless we could ensure that
$\mathcal{F}\textup{Rec}(T)$ is a co-meager set. This is the case for
$\mathcal{AP}$-recurrence (see [36, 16]):

    1. –

If $(X,T)$ is an $\mathcal{AP}$-recurrent (i.e. multiple recurrent) and cyclic
linear dynamical system, then the set $\mathcal{AP}\textup{Rec}(T_{(N)})$ is
dense lineable for every $N\in\mathbb{N}$.

In [13, Example 2.4] it is exhibited a $\overline{\mathcal{BD}}$-recurrent
(i.e. reiteratively recurrent) operator $T$ for which the set
$\overline{\mathcal{BD}}\textup{Rec}(T)=\operatorname{RRec}(T)$ is meager.
Such an operator $T$ is not cyclic, and in view of [13, Theorem 2.1] (result
which states that: the set of $\overline{\mathcal{BD}}$-recurrent vectors is
co-meager for every $\overline{\mathcal{BD}}$-recurrent and hypercyclic
operator) the following is a natural open question:

######  Problem 5.14.

Let $(X,T)$ be a $\overline{\mathcal{BD}}$-recurrent (i.e. reiteratively
recurrent) and cyclic linear dynamical system. Is $\operatorname{RRec}(T)$ a
co-meager set?

###  5.3 Dense Lineability for $\mathcal{F}$-hypercyclicity

Under some natural conditions on the Furstenberg family $\mathcal{F}$ we can
apply the $\mathcal{F}$-recurrence theory developed above in order to obtain
some results regarding $\mathcal{F}$-hypercyclicity. We will consider right-
invariant and upper Furstenberg families. Let us recall the definitions:

A family $\mathcal{F}$ is said to be right-invariant if for every
$A\in\mathcal{F}$ and every $n\in\mathbb{N}_{0}$ the set

$A+n=\\{k+n:k\in A\\}$

also belongs to $\mathcal{F}$. For example, the families
$\mathcal{I},\mathcal{AP},\mathcal{S},\mathcal{T},\mathcal{TS},\mathcal{I}^{*}$
and the density ones are easily seen to be right-invariant. However, the
families $\mathcal{IP}$ and $\Delta$ together with their dual families
$\mathcal{IP}^{*}$ and $\Delta^{*}$ are not right-invariant (see [9,
Proposition 5.6]).

It is shown in [13] that given any right-invariant Furstenberg family
$\mathcal{F}$, then every single $\mathcal{F}$-recurrent and hypercyclic
vector is indeed $\mathcal{F}$-hypercyclic. As a consequence we obtain a kind
of Herrero-Bourdon theorem (see [31, Theorem 2.55]) for
$\mathcal{F}$-hypercyclicity:

######  Corollary 5.15.

Let $(X,T)$ be a linear dynamical system and assume that $\mathcal{F}$ is a
right-invariant Furstenberg family. If $x$ is an $\mathcal{F}$-hypercyclic
vector for $T$, then

$\\{p(T)x:p\text{ polynomial}\\}\setminus\\{0\\},$

is a dense set of $\mathcal{F}$-hypercyclic vectors. In particular, any
$\mathcal{F}$-hypercyclic operator admits a dense invariant vector space
consisting, except for zero, of $\mathcal{F}$-hypercyclic vectors.

###### Proof.

Follows from the original Herrero-Bourdon theorem, together with Theorem 4.19
and the equality

$\mathcal{F}\textup{HC}(T)=\mathcal{F}\textup{Rec}(T)\cap\textup{HC}(T),$

showed in [13] for every right-invariant Furstenberg family $\mathcal{F}$. ∎

Apart from right-invariance we may assume the u.f.i. upper condition (defined
in [12]) to obtain even stronger results: a family $\mathcal{F}$ is said to be
upper if it can be written as

$\mathcal{F}=\bigcup_{\delta\in D}\mathcal{F}_{\delta}\quad\text{ with
}\quad\mathcal{F}_{\delta}=\bigcap_{m\in M}\mathcal{F}_{\delta,m}$

for sets $\mathcal{F}_{\delta,m}\subset\mathscr{P}(\mathbb{N}_{0})$ such that
$\delta\in D$ and $m\in M$, where $D$ is arbitrary, $M$ is countable and they
have the following properties:

  1. (i)

for any $\mathcal{F}_{\delta,m}$ and any $A\in\mathcal{F}_{\delta,m}$ there
exists a finite set $F\subset\mathbb{N}_{0}$ such that

$A\cap F\subset B\text{ implies }B\in\mathcal{F}_{\delta,m};$

  2. (ii)

for any $A\in\mathcal{F}$ there exists some $\delta\in D$ such that for all
$n\in\mathbb{N}_{0}$ we have

$(A-n)\cap\mathbb{N}_{0}\in\mathcal{F}_{\delta}.$

We say that an upper family $\mathcal{F}$ as above is uniformly finitely
invariant (u.f.i. for short) if for any $A\in\mathcal{F}$ there is some
$\delta\in D$ such that, for all $n\in\mathbb{N}$,
$A\cap[n,\infty[\in\mathcal{F}_{\delta}$.

The families $\mathcal{I},\mathcal{AP},\overline{\mathcal{BD}}$ and
$\overline{\mathcal{D}}$ are easily checked to be u.f.i. upper, while
$\underline{\mathcal{D}}$ is not even upper (see [12, 16]). As it is shown in
[12], there exists a kind of Birkhoff transitivity theorem for these u.f.i.
upper families, and in particular, the following holds:

######  Corollary 5.16.

Let $(X,T)$ be an $\mathcal{F}$-recurrent linear dynamical system where
$\mathcal{F}$ is a u.f.i. upper Furstenberg family. If there is a dense set
$X_{0}\subset X$ such that $T^{n}x\to 0$ for each $x\in X_{0}$, then the set
$\mathcal{F}\textup{Rec}(T_{(N)})$ is dense lineable for every
$N\in\mathbb{N}$. In particular:

  1. (a)

If $T$ is recurrent then $\textup{Rec}(T_{(N)})$ is dense lineable for every
$N\in\mathbb{N}$;

  2. (b)

If $T$ is $\mathcal{AP}$-recurrent then $\mathcal{AP}\textup{Rec}(T_{(N)})$ is
dense lineable for every $N\in\mathbb{N}$;

  3. (c)

If $T$ is reiteratively recurrent then $\operatorname{RRec}(T_{(N)})$ is dense
lineable for every $N\in\mathbb{N}$;

  4. (d)

If $T$ is $\mathcal{U}$-frequently recurrent then
$\operatorname{UFRec}(T_{(N)})$ is dense lineable for every $N\in\mathbb{N}$.

###### Proof.

By [13, Theorem 8.5], if $T$ is $\mathcal{F}$-recurrent for a u.f.i. upper
family $\mathcal{F}$ and if there exists a dense set of vectors $x\in X$ such
that $T^{n}x\to 0$ as $n\to\infty$, then $T$ is $\mathcal{F}$-hypercyclic. By
Proposition 5.9 the set $\mathcal{F}\textup{Rec}(T_{(N)})$ is then dense
lineable for every $N\in\mathbb{N}$. The particular cases follow from the fact
that $\mathcal{I}$, $\mathcal{AP}$, $\overline{\mathcal{BD}}$ and
$\overline{\mathcal{D}}$ are u.f.i. upper families. ∎

Moreover, if we consider the Furstenberg family $\overline{\mathcal{BD}}$ of
positive upper Banach density sets we can prove a particular case of [23,
Theorem 2.5 and Corollary 2.8] in a much simpler way, which is also a
particular case of [17, Proposition 2.8]:

######  Corollary 5.17 ([23, Theorem 2.5 and Corollary 2.8] and [17,
Proposition 2.8]).

Let $(X,T)$ be a reiteratively hypercyclic linear system. Then the $N$-fold
direct sum operator $T_{(N)}:X^{N}\longrightarrow X^{N}$ is reiteratively
hypercyclic for every $N\in\mathbb{N}$.

###### Proof.

By [8, Proposition 4] we have that $T_{(N)}$ is hypercyclic. By Corollary 5.10
we have that $T_{(N)}$ is reiteratively recurrent. By [13, Theorem 2.1] every
hypercyclic vector of $T_{(N)}$ is also reiteratively hypercyclic, so that
$T_{(N)}$ is reiteratively hypercyclic. ∎

##  6 Open problems

In this section we include three open questions and a few comments related to
them. They essentially ask “what happens if we cannot apply Theorem 5.8”:

######  Problems 6.1.

Let $\mathcal{F}$ be a Furstenberg family and assume that it is not a filter.
Suppose that $T\in\mathcal{L}(X)$ is a continuous linear
$\mathcal{F}$-recurrent operator:

  1. (a)

Is the set $\mathcal{F}\textup{Rec}(T_{(N)})$ necessarily dense lineable for
every $N\in\mathbb{N}$?

  2. (b)

Is the set $\mathcal{F}\textup{Rec}(T)$ necessarily dense lineable?

  3. (c)

Is $T\oplus T$ necessarily an $\mathcal{F}$-recurrent operator?

Note that a positive answer to Problem (a) would imply an affirmative answer
to both Problems (b) and (c). However, for usual recurrence (i.e. for
$\mathcal{I}$-recurrence) we already have a negative answer to Problems (a)
and (c):

  1. –

in Section 3 we construct recurrent operators $T\in\mathcal{L}(X)$ for which
$T\oplus T$ is not recurrent.

Nevertheless, that example is not solving Problem (b) in the negative since
the mentioned operator fulfills that $\textup{Rec}(T)=X$. Thus, finding a
possible recurrent operator $T$ for which the set $\textup{Rec}(T)$ is not
dense lineable demands to solve again the $T\oplus T$-recurrence problem.

It is not hard to check that all the operators constructed in Section 3 have a
dense lineable set of $\mathcal{AP}$-recurrent vectors. In particular:

  1. –

there exist multiple recurrent operators $T\in\mathcal{L}(X)$ for which
$T\oplus T$ is not even recurrent;

and the same comments done for usual recurrence hold for
$\mathcal{AP}$-recurrence. These kind arguments are not valid for the other
recurrence notions considered in this document:

######  Proposition 6.2 (Modification of [8, Proposition 4] and [9,
Proposition 5.6]).

Let $(X,T)$ be a linear dynamical system. If $T$ is reiteratively recurrent,
then $T$ is quasi-rigid, and in particular $\textup{Rec}(T_{(N)})$ is dense
lineable for every $N\in\mathbb{N}$.

###### Proof.

Given a non-empty subset $U\subset X$ we can pick a vector
$x\in\operatorname{RRec}(T)\cap U$ and hence
$\overline{\operatorname{Bd}}(N_{T}(x,U))>0$. Recall that

$\big{(}N_{T}(x,U)-N_{T}(x,U)\big{)}:=\left\\{s_{2}-s_{1}:s_{1}\leq s_{2}\in
N_{T}(x,U)\right\\}\subset\left\\{n\geq 0:T^{n}(U)\cap
U\neq\varnothing\right\\}.$

Now we use [24, Theorem 3.18], which implies that $A-A\in\Delta^{*}$ whenever
$A\in\overline{\mathcal{BD}}$, obtaining that: for every non-empty open subset
$U\subset X$ we have that

$\\{n\geq 0:T^{n}(U)\cap U\neq\varnothing\\}\in\Delta^{*}.$

Since $\Delta^{*}$ is a filter (see [11, 34]) it is easily checked that every
$N$-fold direct sum $(X^{N},T_{(N)})$ is topologically recurrent. By Theorem
2.5 we have that $T$ is quasi-rigid and Proposition 2.7 (or Theorem 5.8)
implies that $\textup{Rec}(T_{(N)})$ is dense lineable for every
$N\in\mathbb{N}$. ∎

This last Proposition 6.2 together with the very recent results [29, Theorems
5.1 and 5.8] show that Problems (a), (b) and (c) may have a positive answer
for Furstenberg families implying stronger recurrence notions than those of
usual and $\mathcal{AP}$-recurrence.

We would like to end the document mentioning that, by now, also the
spaceability property (i.e. the existence of an infinite-dimensional closed
subspace) has been studied for the set of recurrent vectors using the notion
of quasi-rigidity in an essential way. Indeed, the results obtained in [38]
justify again that quasi-rigidity is the analogous notion, for recurrence, to
that of weak-mixing for hypercyclicity.

## Funding

The first author was supported by the project FRONT of the French National
Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI
(ANR-11-LABX-0007-01). The second and third authors were supported by
MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00. The second
author was also supported by the Spanish Ministerio de Ciencia, Innovación y
Universidades, grant FPU2019/04094. The third author was also supported by
Generalitat Valenciana, Project PROMETEU/2021/070.

## Acknowledgments

The authors would like to thank Juan Bès, Antonio Bonilla and Karl Grosse-
Erdmann for suggesting the topic, and also for providing valuable ideas on the
subject studied. We also want to thank Robert Deville for letting us know
about the work [2].

## References

  * [1] M. Amouch and O. Benchiheb, On recurrent sets of operators, arXiv:1907.05930, (2019).
  * [2] J.-M. Augé, Linear Operators with Wild Dynamics, Proc. Am. Math. Soc., vol. 140, no. 6, 2103–2116 (2012).
  * [3] J. Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dyn. Syst., vol. 5, no. 1, 83–92 (1999).
  * [4] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Am. Math. Soc., vol. 358, no. 11, 5083–5117 (2006).
  * [5] F. Bayart and É. Matheron, Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces, J. Funct. Anal., vol. 250, 426–441 (2007).
  * [6] F. Bayart and É. Matheron, (Non-)Weakly mixing operators and hypercyclicity sets, Ann. de l’Institut Fourier, vol. 59, no. 1, 1–35 (2009).
  * [7] F. Bayart and É. Matheron, Dynamics of linear operators. Cambridge University Press, 2009.
  * [8] J. Bès, Q. Menet, A. Peris and Y. Puig, Recurrence properties of hypercyclic operators. Math. Ann., vol. 366, no. 1-2, 545–572 (2016).
  * [9] J. Bès, Q. Menet, A. Peris and Y. Puig, Strong transitivity properties for operators. J. Differ. Equ., vol. 266, no. 2-3, 1313–1337 (2019).
  * [10] J. Bès and A. Peris, Hereditarily hypercyclic operators. J. Funct. Anal., 167, 94–112 (1999).
  * [11] V. Bergelson and T. Downarowicz: Large sets of integers and hierarchy of mixing properties of measure preserving systems. Colloq. Math., 110, no. 1, 117–150 (2008).
  * [12] A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, Rev. Mat. Complut., vol. 31, no. 3, 673–711 (2018).
  * [13] A. Bonilla, K-G. Grosse-Erdmann, A. López-Martínez and A. Peris, Frequently recurrent operators, J. Funct. Anal., vol. 283, issue 12, no. 109713 (2022).
  * [14] N. Bourbaki, General Topology, Chaps. 1-4. Springer, 1989.
  * [15] R. Cardeccia and S. Muro, Arithmetic progressions and chaos in linear dynamics, _Integral Equ. Oper. Theory_ , vol. 94, no. 11, 18 pp (2022).
  * [16] R. Cardeccia and S. Muro, Multiple recurrence and hypercyclicity, Math. Scand., vol. 128, no. 3, 16 pages (2022).
  * [17] R. Cardeccia and S. Muro, Frequently recurrence properties and block families, arXiv:2204.13542, (2022).
  * [18] G. Costakis and I. Parissis, Szemerédi’s theorem, frequent hypercyclicity and multiple recurrence. Math. Scand., 110 (2), (2012).
  * [19] G. Costakis, A. Manoussos and I. Parissis, Recurrent linear operators, Complex Anal. Oper. Theory, vol. 8, 1601–1643 (2014).
  * [20] C.-C. Chen, M. Kostić, D. Velinov, A note on recurrent strongly continuous semigroups of operators, Funct. Anal. Approx. Comput., vol. 13, no. 1, 7–12 (2021).
  * [21] M. De La Rosa and C. Read, A hypercyclic operator whose direct sum $T\oplus T$ is not hypercyclic, J. Oper. Theory, vol. 2, (2009).
  * [22] T. Eisner and S. Grivaux, Hilbertian Jamison sequences and rigid dynamical systems. J. Funct. Anal., vol 261, no. 7, 2013–2052 (2011).
  * [23] R. Ernst, C. Esser and Q. Menet, $\mathcal{U}$-frequent hypercyclicity notions and related weighted densities, Isr. J. Math., vol 241, 817–848 (2021).
  * [24] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, New Jersey 1981.
  * [25] H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ.). Lecture Notes in Math., vol. 1978, 127–132 (1977).
  * [26] V. J. Galán, F. Martínez-Jiménez, A. Peris and P. Oprocha, Product Recurrence for Weighted Backward Shifts. Appl. Math. Inf. Sci., 9(5): 2361–2365 (2015).
  * [27] S. Glasner and D. Maon, Rigidity in topological dynamics, Ergod. Theory Dyn. Syst., vol 9, no. 2, 309–320 (1989).
  * [28] G. Grekos, V. Toma and J. Tomanová, A note on uniform or Banach density, Ann. Math. Blaise Pascal, vol. 17, no. 1, 153–163 (2010).
  * [29] S. Grivaux and A. López-Martínez, Recurrence properties for linear dynamical systems: An approach via invariant measures, J. Math. Pures Appl., vol. 169, 155–188 (2023).
  * [30] K-G. Grosse-Erdmann and A. Peris, Weakly mixing operators on topological vector spaces, RACSAM, 104 (2), 413–426, 2010.
  * [31] K-G. Grosse-Erdmann and A. Peris, Linear Chaos. Springer, London 2011.
  * [32] D. A. Herrero, Possible structures for the set of cyclic vectors, Indiana Univ. Math. J., vol. 28, no. 6, 913–926 (1979).
  * [33] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal., vol. 99, 179–190 (1991).
  * [34] N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification. Second Edition. De Gruyter, 1998.
  * [35] G. Jungck, Commuting Mappings and Fixed Points, Amer. Math. Monthly, vol. 83, no. 4, 261–263 (1976).
  * [36] D. Kwietniak, J. Li, P. Oprocha, and X. Ye, Multi-recurrence and van der waerden systems, Science China Mathematics, vol. 60, no. 1, 59–82 (2017).
  * [37] J. Lindestrauss and L. Tzafriri, Classical Banach spaces I and II. Springer, 1977.
  * [38] A. López-Martínez, Recurrent subspaces in Banach spaces, arXiv.2212.04464, (2022).
  * [39] Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., 369 (2017), 4977–4994.
  * [40] M. S. Moslehian, G. A. Muñoz-Fernández, A. M. Peralta and J. B. Seoane-Sepúlveda, Similarities and differences between real and complex Banach spaces: an overview and recent developments, arXiv:2107.03740v3, (2022).
  * [41] G. Muñoz-Fernández, Y. Sarantopoulos and A. Tonge, Complexifications of real Banach spaces, polynomials and multilinear maps. Studia Math., 134, no. 1, (1999).
  * [42] R.I. Ovsepian and A. Pełczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$, Studia Math., vol. 54, 149–159 (1975).
  * [43] S. Shkarin, On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc., vol. 137, 123–134 (2009).
  * [44] E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Proc. of the International Congress of Math. (Vancouver, 1974), vol. 2, 503–505 (1975).

Sophie Grivaux: CNRS, Université de Lille, UMR 8524 - Laboratoire Paul
Painlevé, F-59000 Lille, France. e-mail<EMAIL_ADDRESS>

Antoni López-Martínez: Universitat Politècnica de València, Institut
Universitari de Matemàtica Pura i Aplicada, Edifici 8E, 4a planta, 46022
València, Spain. e-mail<EMAIL_ADDRESS>

Alfred Peris: Universitat Politècnica de València, Institut Universitari de
Matemàtica Pura i Aplicada, Edifici 8E, 4a planta, 46022 València, Spain.
e-mail<EMAIL_ADDRESS>