Recurrence properties for linear dynamical systems: An approach via invariant
measures

by

Sophie Grivaux & Antoni López-Martínez††Acknowledgments: 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
author was partially supported by the Spanish Ministerio de Ciencia,
Innovación y Universidades, grant FPU2019/04094, and partially by
MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00.
2020 Mathematics Subject Classification: 47A16, 47A35, 37A05, 37B20.
Keywords and phrases: Linear dynamics, Recurrence, Invariant measures,
Frequent recurrence,
Unimodular eigenvectors, Uniformly recurrent operators.
Journal-ref: Journal de Mathématiques Pures et Appliquées, Volume 169, January
2023, Pages 155-188, https://doi.org/10.1016/j.matpur.2022.11.011

###### Abstract

We study different pointwise recurrence notions for linear dynamical systems
from the Ergodic Theory point of view. We show that from any reiteratively
recurrent vector $x_{0}$, for an adjoint operator $T$ on a separable dual
Banach space $X$, one can construct a $T$-invariant probability measure which
contains $x_{0}$ in its support. This allows us to establish some
equivalences, for these operators, between some strong pointwise recurrence
notions which in general are completely distinguished. In particular, we show
that (in our framework) reiterative recurrence coincides with frequent
recurrence; for complex Hilbert spaces uniform recurrence coincides with the
property of having a spanning family of unimodular eigenvectors; and the same
happens for power-bounded operators on complex reflexive Banach spaces. These
(surprising) properties are easily generalized to product and inverse
dynamical systems, which implies some relations with the respective
hypercyclicity notions. Finally we study how typical is an operator with a
non-zero reiteratively recurrent vector in the sense of Baire category.

##  1 Introduction and main results

###  1.1 General background

This paper focuses on some aspects of the interplay between topological and
measurable dynamics for linear dynamical systems, our main aim being to
investigate the links in this context between various notions of recurrence.

A (real or complex) linear dynamical system $(X,T)$ is given by the action of
a bounded linear operator $T$ on a (real or complex) separable infinite-
dimensional Banach space $X$, and we will denote by $\mathcal{L}(X)$ the set
of bounded linear operators acting on such a space $X$. A linear dynamical
system is a particular case of a Polish dynamical system (i.e. a system given
by the action of a continuous map on a separable completely metrizable space),
and some of the results obtained in the paper hold in this more general
context. Given a dynamical system $T:X\rightarrow X$ and a point $x\in X$ we
will denote by

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

the $T$-orbit of $x$, where $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. Examples
of topological properties, for linear dynamical systems, which will be of
interest to us in this work are:

  1. (a)

recurrence: the operator $T$ is said to be recurrent if the set

$\textup{Rec}(T):=\left\\{x\in
X:x\in\overline{\operatorname{Orb}(Tx,T)}\right\\},$

is dense in $X$, where each vector $x\in\textup{Rec}(T)$ is called a recurrent
vector for $T$. By the (not so well-known) Costakis-Manoussos-Parissis’
theorem (see [14, Proposition 2.1]), this notion is equivalent to that of
topological recurrence, i.e. for each non-empty open subset $U$ of $X$ one can
find $n\in\mathbb{N}$ such that $T^{n}(U)\cap U\neq\varnothing$; and in this
case, the set $\textup{Rec}(T)$ of recurrent vectors for $T$ is a dense
$G_{\delta}$ subset of $X$;

  2. (b)

hypercyclicity: the operator $T$ is said to be hypercyclic if there exists a
vector $x\in X$, called a hypercyclic vector for $T$, whose orbit
$\operatorname{Orb}(x,T)$ is dense in $X$. By the Birkhoff’s Transitivity
theorem (see [29, Theorem 1.16]), this notion is equivalent to that of
topological transitivity, i.e. for each pair $U,V$ of non-empty open subsets
of $X$ one can find $n\in\mathbb{N}_{0}$ such that $T^{n}(U)\cap
V\neq\varnothing$; and in this case, the set of hypercyclic vectors for $T$,
denoted by $\textup{HC}(T)$, is a dense $G_{\delta}$ subset of $X$.

If given a point $x\in X$ and a set $A\subset X$ we denote the return set from
$x$ to $A$ as

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

which will be denoted by $N(x,A)$ if no confusion with the map arises, we can
reformulate the above notions in the following terms: a vector $x\in X$ is
recurrent if and only if $N(x,U)$ is an infinite set for every neighbourhood
$U$ of $x$; and a vector $x\in X$ is hypercyclic if and only if $N(x,U)$ is an
infinite set for every non-empty open subset $U$ of $X$. Historically,
hypercyclicity and its generalizations have been the most studied notions in
linear dynamics while the systematic study of the linear dynamical recurrence-
kind properties started recently in 2014 with [14], in spite of the great non-
linear dynamical knowledge already existing in this area (see for instance
[22]).

Direct relations between these properties and Ergodic Theory arise when we are
able to consider a probability (or a positive finite) Borel measure $\mu$ on
$X$ (i.e. defined on $\mathscr{B}(X)$, the $\sigma$-algebra of Borel sets of
$X$), which will sometimes be required to have full support (i.e. $\mu(U)>0$
for every non-empty open subset $U$ of $X$). We will only consider Borel
measures in this work, and the word “Borel” will sometimes be omitted. If such
a measure $\mu$ exists, we can study the dynamical system
$(X,\mathscr{B}(X),\mu,T)$ from the point of view of Ergodic Theory and
relevant properties are:

  1. (a)

invariance: the operator $T$ is said to be $\mu$-invariant, or equivalently,
the measure $\mu$ is called $T$-invariant, if for each $A\in\mathscr{B}(X)$
the equality $\mu(T^{-1}(A))=\mu(A)$ holds. By the Poincaré’s Recurrence
theorem (see [36, Theorem 1.4]), this notion implies that for every
$A\in\mathscr{B}(X)$ with $\mu(A)>0$ there is $n\in\mathbb{N}$ such that
$T^{n}(A)\cap A\neq\varnothing$. The Dirac mass $\delta_{0}$ at $0$ is always
an invariant measure for any operator $T$, and we will say that a
$T$-invariant probability measure $\mu$ is non-trivial if it is different from
$\delta_{0}$.

  2. (b)

ergodicity: the operator $T$ is said to be ergodic with respect to $\mu$,
provided that the measure $\mu$ is $T$-invariant, and for each
$A\in\mathscr{B}(X)$ with $T^{-1}(A)=A$ we have that $\mu(A)\in\\{0,1\\}$. It
is well known that the last statement is equivalent to the fact that, for each
pair of sets $A,B\in\mathscr{B}(X)$ with $\mu(A),\mu(B)>0$ there is
$n\in\mathbb{N}_{0}$ such that $\mu\big{(}T^{-n}(A)\cap B\big{)}>0$ (see [36,
Theorem 1.5]).

When $T$ is ergodic with respect to a measure with full support, it follows
from the Birkhoff’s Pointwise Ergodic theorem that $T$ is not only
hypercyclic, but even frequently hypercyclic: there exists a vector $x\in X$
such that for each non-empty open subset $U$ of $X$ the return set $N(x,U)$
has positive lower density; in other words:

$\underline{\operatorname{dens}}(N(x,U))=\liminf_{N\to\infty}\dfrac{\operatorname{card}(N(x,U)\cap[0,N])}{N+1}>0.$

Such a vector $x$ is said to be a frequently hypercyclic vector for $T$, and
the set of all frequently hypercyclic vectors is denoted by
$\operatorname{FHC}(T)$. See [3, Corollary 5.5] for the details of this
argument, and for more on frequent hypercyclicity.

When $T$ is only supposed to admit an invariant measure $\mu$, it follows
easily from the Poincaré’s Recurrence theorem that $\mu$-almost every $x\in X$
is a recurrent point for $T$ (see [22, Theorem 3.3]). Our main line of thought
in this work will be to connect various (stronger) notions of recurrence via
invariant measures, proceeding essentially in two steps:

  1. -

if $T$ admits vectors with a certain (rather weak) recurrence property, prove
that it admits a non-trivial invariant measure, perhaps with full support (see
Theorem 2.3);

  2. -

if $T$ admits a non-trivial invariant measure (perhaps with full support),
prove that it admits vectors with a certain strong recurrence property (see
Lemmas 3.1 and 4.4).

This approach in the context of linear dynamical systems comes from the paper
[24], which extends to the linear setting some well-known results in the
context of compact dynamical systems (see [22, Chapter 3 and Lemma 3.17]). The
various recurrence notions which we will consider were introduced and studied
in the work [10], but the initial study of recurrence in linear dynamics
started in [14]. In the next subsection, we recall the relevant definitions
and present the first main result of this paper.

###  1.2 Furstenberg families: recurrence and hypercyclicity notions

The Banach spaces $X$ considered in this subsection can be either real or
complex. Let us first recall the following definitions from [10]:

######  Definition 1.1.

Given a non-empty collection of sets
$\mathcal{F}\subset\mathscr{P}(\mathbb{N}_{0})$ we say that it is a
Furstenberg family if for each $A\in\mathcal{F}$ we have

  1. (i)

$A$ is infinite;

  2. (ii)

if $A\subset B\subset\mathbb{N}_{0}$ then $B\in\mathcal{F}$.

The dual family of $\mathcal{F}$ is defined as the collection of sets

$\mathcal{F}^{*}:=\\{A\subset\mathbb{N}_{0}\text{ infinite}:A\cap
B\neq\varnothing\text{ for all }B\in\mathcal{F}\\}.$

######  Definition 1.2.

Let $(X,T)$ be a linear dynamical system and let $\mathcal{F}$ be a
Furstenberg family. A point $x\in X$ is said to be $\mathcal{F}$-recurrent
(resp. $\mathcal{F}$-hypercyclic) if $N(x,U)\in\mathcal{F}$ for every
neighbourhood $U$ of $x$ (resp. for every non-empty open subset $U$ of $X$).
We will denote by $\mathcal{F}\textup{Rec}(T)$ (resp.
$\mathcal{F}\textup{HC}(T)$) the set of such points and we say that $T$ is
$\mathcal{F}$-recurrent (resp. $\mathcal{F}$-hypercyclic) if
$\mathcal{F}\textup{Rec}(T)$ is dense in $X$ (resp. if
$\mathcal{F}\textup{HC}(T)\neq\varnothing$).

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 since having an orbit distributed around the whole space is much more
complicated than having it just around the initial point of the orbit.
Furstenberg families associated just to recurrence will be used in the
following subsection, but in the present one we focus on the most known cases
of families for which both notions exist. In particular, a point $x\in X$ is
said to be

  1. (a)

frequently recurrent (resp. frequently hypercyclic) if
$\underline{\operatorname{dens}}(N(x,U))>0$ for every neighbourhood $U$ of $x$
(resp. for every non-empty open subset $U$ of $X$). We will denote by
$\operatorname{FRec}(T)$ (resp. $\operatorname{FHC}(T)$) the set of such
points, and we say that $T$ is frequently recurrent (resp. frequently
hypercyclic) if $\operatorname{FRec}(T)$ is dense in $X$ (resp. if
$\operatorname{FHC}(T)\neq\varnothing$);

  2. (b)

$\mathcal{U}$-frequently recurrent (resp. $\mathcal{U}$-frequently
hypercyclic) if $\overline{\operatorname{dens}}(N(x,U))>0$ for every
neighbourhood $U$ of $x$ (resp. for every non-empty open subset $U$ of $X$).
We will denote by $\operatorname{UFRec}(T)$ (resp. $\operatorname{UFHC}(T)$)
the set of such points, and we say that $T$ is $\mathcal{U}$-frequently
recurrent (resp. $\mathcal{U}$-frequently hypercyclic) if
$\operatorname{UFRec}(T)$ is dense in $X$ (resp. if
$\operatorname{UFHC}(T)\neq\varnothing$);

  3. (c)

reiteratively recurrent (resp. reiteratively hypercyclic) if
$\overline{\operatorname{Bd}}(N(x,U))>0$ for every neighbourhood $U$ of $x$
(resp. for every non-empty open subset $U$ of $X$). We will denote by
$\operatorname{RRec}(T)$ (resp. $\operatorname{RHC}(T)$) the set of such
points and we say that $T$ is reiteratively recurrent (resp. reiteratively
hypercyclic) if $\operatorname{RRec}(T)$ is dense in $X$ (resp. if
$\operatorname{RHC}(T)\neq\varnothing$);

where for any $A\subset\mathbb{N}_{0}$ its:

  1. (a)

lower density is
$\displaystyle\underline{\operatorname{dens}}(A):=\liminf_{N\to\infty}\dfrac{\operatorname{card}(A\cap[0,N])}{N+1}$;

  2. (b)

upper density is
$\displaystyle\overline{\operatorname{dens}}(A):=\limsup_{N\to\infty}\dfrac{\operatorname{card}(A\cap[0,N])}{N+1}$;

  3. (c)

upper Banach density is
$\displaystyle\overline{\operatorname{Bd}}(A):=\limsup_{N\to\infty}\left(\max_{m\geq
0}\dfrac{\operatorname{card}(A\cap[m,m+N])}{N+1}\right)$.

The introduced notions follow Definition 1.2 applied to the respective
families of positive (lower, upper and upper Banach) density sets, and in
fact, the inequalities between the respective densities imply the inclusions
$\operatorname{FRec}(T)\subset\operatorname{UFRec}(T)\subset\operatorname{RRec}(T)\subset\textup{Rec}(T)$
and
$\operatorname{FHC}(T)\subset\operatorname{UFHC}(T)\subset\operatorname{RHC}(T)\subset\textup{HC}(T)$.
In particular, frequent, $\mathcal{U}$-frequent and reiterative recurrence are
clearly stronger notions than “usual” recurrence as defined in Subsection 1.1,
and frequent recurrence is a stronger notion than $\mathcal{U}$-frequent
recurrence, which is in its turn stronger than reiterative recurrence.

We point out that all these notions are not specific to the linear setting; we
will actually use them in the context of Polish dynamical systems in Sections
2, 3, 5 and 6. However, since we are focused on linear dynamical systems, our
first main result connects all of them in the framework of adjoint operators
on separable dual Banach spaces:

######  Theorem 1.3.

Let $T:X\rightarrow X$ be an adjoint operator on a (real or complex) separable
dual Banach space $X$. Then we have the equality

$\overline{\operatorname{FRec}(T)}=\overline{\operatorname{RRec}(T)}.$

Moreover:

  1. (a)

The following statements are equivalent:

    1. (i)

$\operatorname{FRec}(T)\setminus\\{0\\}\neq\varnothing$;

    2. (ii)

$\operatorname{UFRec}(T)\setminus\\{0\\}\neq\varnothing$;

    3. (iii)

$\operatorname{RRec}(T)\setminus\\{0\\}\neq\varnothing$;

    4. (iv)

$T$ admits a non-trivial invariant probability measure.

  2. (b)

The following statements are equivalent:

    1. (i)

$T$ is frequently recurrent;

    2. (ii)

$T$ is $\mathcal{U}$-frequently recurrent;

    3. (iii)

$T$ is reiteratively recurrent;

    4. (iv)

$T$ admits an invariant probability measure with full support.

In particular, these results hold whenever $T$ is an operator on a (real or
complex) separable reflexive Banach space $X$.

The above theorem is in spirit similar to [24, Theorem 1.3], where it is
proved that every ($\mathcal{U}$-)frequently hypercyclic operator on a
separable reflexive space admits an invariant measure with full support. It is
observed in [24, Remark 3.5] that the arguments extend to every adjoint
operator acting on a separable dual Banach space. It is also proved in [24,
Proposition 2.11] that, in this same setting, operators admitting an invariant
measure with full support are exactly those which are frequently recurrent.
However, the notion of frequent recurrence introduced in [24, Section 2.5] is
rather different from the one given in Definition 1.2 since in [24],
$T\in\mathcal{L}(X)$ is called frequently recurrent if for every non-empty
open subset $U$ of $X$ there exists a vector $x_{U}\in U$ for which just the
positive lower density of the return set $N(x_{U},U)$ is required. This notion
is (at least formally) weaker than the one used here (see Remark 2.5).

The proof of Theorem 1.3 relies on some modifications of the arguments of [24,
Section 2], which will be presented in Sections 2 and 3. We mention that it
cannot be extended to all operators acting on separable (infinite-dimensional)
Banach spaces. Indeed, it is shown in [10, Theorem 5.7 and Corollary 5.8] that
there even exist reiteratively hypercyclic operators on the space
$c_{0}(\mathbb{N})$ which do not admit any non-zero $\mathcal{U}$-frequently
recurrent vector.

###  1.3 Uniform, $\mathcal{IP}^{*}$, $\Delta^{*}$-recurrence and unimodular
eigenvectors

In this subsection the underlying Banach spaces $X$ are assumed to be complex.
A vector $x\in X$ is a unimodular eigenvector for $T$ provided $x\neq 0$ and
$Tx=\lambda x$ for some unimodular complex number
$\lambda\in\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$. The set of unimodular
eigenvectors of $T$ will be denoted by $\mathcal{E}(T)$, and

$\mathcal{E}(T)=\bigcup_{\lambda\in\mathbb{T}}\ker(\lambda-T)\setminus\\{0\\}.$

Unimodular eigenvectors are clearly frequently recurrent vectors for $T$, but
they enjoy some stronger recurrence properties like uniform,
$\mathcal{IP}^{*}$ and even $\Delta^{*}$-recurrence (see Definition 1.5
below). Our general aim in this paper is to investigate some contexts in which
these strong forms of recurrence actually imply the existence of unimodular
eigenvectors. We will see that it is indeed the case in (at least) the
following two situations:

  1. -

when $T$ is an operator on a complex Hilbert space (see Theorem 1.7 below);

  2. -

when $T$ is a power-bounded operator on a complex reflexive Banach space
(Theorem 1.9).

Let us now introduce these stronger recurrence notions which are defined by
considering Furstenberg families only relevant for the notion of recurrence,
and, contrary to those used in Subsection 1.2, having no hypercyclicity
analogue.

######  Definition 1.4.

Let $A\subset\mathbb{N}_{0}$. We say that $A$ is a

  1. (a)

syndetic set, if there is $m\in\mathbb{N}$ such that for every
$x\in\mathbb{N}_{0}$ we have $[x,x+m]\cap A\neq\varnothing$. We will denote by

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

the Furstenberg family of syndetic sets.

  2. (b)

IP-set, if there is an increasing sequence
$(x_{n})_{n=1}^{\infty}\in\mathbb{N}_{0}^{\mathbb{N}}$ such that

$\left\\{\sum_{n\in F}x_{n}:F\subset\mathbb{N}\text{ finite}\right\\}\subset
A.$

We will denote by

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

the Furstenberg family of IP-sets.

  3. (c)

$\Delta$-set, if there is an infinite set $B\subset\mathbb{N}_{0}$ such that
$(B-B)\cap\mathbb{N}\subset A$. We will denote by

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

the Furstenberg family of $\Delta$-sets.

From Definition 1.2 and the dual families notation we have:

######  Definition 1.5.

Let $(X,T)$ be a linear dynamical system. A point $x\in X$ is said to be

  1. (a)

uniformly recurrent if $N(x,U)\in\mathcal{S}$ for every neighbourhood $U$ of
$x$. We will denote by $\operatorname{URec}(T)$ the set of such points and $T$
is uniformly recurrent if $\operatorname{URec}(T)$ is dense in $X$;

  2. (b)

$\mathcal{IP}^{*}$-recurrent if $N(x,U)\in\mathcal{IP}^{*}$ for every
neighbourhood $U$ of $x$. We will denote by $\mathcal{IP}^{*}\textup{Rec}(T)$
the set of such points, and $T$ is $\mathcal{IP}^{*}$-recurrent if
$\mathcal{IP}^{*}\textup{Rec}(T)$ is dense in $X$;

  3. (c)

$\Delta^{*}$-recurrent if $N(x,U)\in\Delta^{*}$ for every neighbourhood $U$ of
$x$. We will denote by $\Delta^{*}\textup{Rec}(T)$ the set of such points, and
$T$ is $\Delta^{*}$-recurrent if $\Delta^{*}\textup{Rec}(T)$ is dense in $X$.

It is shown in [6, Proposition 2] that the above Furstenberg families do not
admit a respective hypercyclicity counterpart. As in the previous subsection
these recurrence notions could be defined for (non-linear) Polish dynamical
systems, but since the eigenvectors will play a fundamental role in the
connection between those concepts we will directly work with complex linear
maps. The relation $\Delta^{*}\subset\mathcal{IP}^{*}\subset\mathcal{S}$
between the families (see [8]), Proposition 4.1 and [10] imply the inclusions

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

From there the following question was proposed in [10]:

######  Question 1.6 ([10, Question 6.3]).

Does there exist an operator which is uniformly recurrent but not
$\mathcal{IP}^{*}$-recurrent?

The uniformly recurrent operators considered in [10] were also
$\mathcal{IP}^{*}$-recurrent, and in fact a partial negative answer to
Question 1.6 was already given in [10, Theorem 6.2] for the particular case of
power-bounded operators, condition which implies the equality of the two sets
$\mathcal{IP}^{*}\textup{Rec}(T)$ and $\operatorname{URec}(T)$. The second
main result of this paper provides a negative answer to Question 1.6 for
operators acting on a complex separable Hilbert space $H$, by showing the
following stronger statement: any uniformly recurrent operator
$T\in\mathcal{L}(H)$ has a spanning set of unimodular eigenvectors. More
precisely, define the sets

$\displaystyle\operatorname{FRec}^{bo}(T)$ $\displaystyle:=$
$\displaystyle\operatorname{FRec}(T)\cap\\{x\in H\textup{ with bounded
$T$-orbit}\\};$ $\displaystyle\operatorname{UFRec}^{bo}(T)$ $\displaystyle:=$
$\displaystyle\operatorname{UFRec}(T)\cap\\{x\in H\textup{ with bounded
$T$-orbit}\\};$ $\displaystyle\operatorname{RRec}^{bo}(T)$ $\displaystyle:=$
$\displaystyle\operatorname{RRec}(T)\cap\\{x\in H\textup{ with bounded
$T$-orbit}\\}.$

Since uniformly recurrent vectors have bounded orbit, we have
$\operatorname{URec}(T)\subset\operatorname{FRec}^{bo}(T)\subset\operatorname{UFRec}^{bo}(T)\subset\operatorname{RRec}^{bo}(T)$
and hence:

######  Theorem 1.7.

Let $T\in\mathcal{L}(H)$ where $H$ is a complex separable Hilbert space. Then
we have the equalities

$\overline{\operatorname{span}(\mathcal{E}(T))}=\overline{\operatorname{URec}(T)}=\overline{\operatorname{RRec}^{bo}(T)}.$

Moreover:

  1. (a)

The following statements are equivalent:

    1. (i)

$\mathcal{E}(T)\neq\varnothing$;

    2. (ii)

$\Delta^{*}\textup{Rec}(T)\setminus\\{0\\}\neq\varnothing$;

    3. (iii)

$\mathcal{IP}^{*}\textup{Rec}(T)\setminus\\{0\\}\neq\varnothing$;

    4. (iv)

$\operatorname{URec}(T)\setminus\\{0\\}\neq\varnothing$;

    5. (v)

$\operatorname{FRec}^{bo}(T)\setminus\\{0\\}\neq\varnothing$;

    6. (vi)

$\operatorname{UFRec}^{bo}(T)\setminus\\{0\\}\neq\varnothing$;

    7. (vii)

$\operatorname{RRec}^{bo}(T)\setminus\\{0\\}\neq\varnothing$;

    8. (viii)

$T$ admits a non-trivial invariant probability measure $\mu$ with
$\int_{H}\|z\|^{2}d\mu(z)<\infty$.

  2. (b)

The following statements are equivalent:

    1. (i)

the set $\operatorname{span}(\mathcal{E}(T))$ is dense in $H$;

    2. (ii)

$T$ is $\Delta^{*}$-recurrent;

    3. (iii)

$T$ is $\mathcal{IP}^{*}$-recurrent;

    4. (iv)

$T$ is uniformly recurrent;

    5. (v)

the set $\operatorname{FRec}^{bo}(T)$ is dense in $H$;

    6. (vi)

the set $\operatorname{UFRec}^{bo}(T)$ is dense in $H$;

    7. (vii)

the set $\operatorname{RRec}^{bo}(T)$ is dense in $H$;

    8. (viii)

$T$ admits an invariant probability measure $\mu$ with full support and
$\int_{H}\|z\|^{2}d\mu(z)\leavevmode\nobreak\ <\leavevmode\nobreak\ \infty$.

The proof of Theorem 1.7 is really specific to the Hilbertian setting in a
somewhat roundabout way. It relies on the following three main arguments:

  1. -

the existence of a non-trivial invariant measure with a finite second-order
moment, under the assumption of the existence of a reiteratively recurrent
vector with bounded orbit; this argument is the same as the one employed in
the proof of Theorem 1.3 above;

  2. -

the fact that any operator on a space of type 2, admitting an invariant
measure with a finite second-order moment, admits in fact a Gaussian invariant
measure whose support contains that of the initial measure (see Remark 4.5);

  3. -

and lastly, the fact that on spaces of cotype 2, the existence of a Gaussian
invariant measure for an operator $T$ implies that the unimodular eigenvectors
of $T$ span a dense subspace of the support of the measure (see Step 3 of
Lemma 4.4).

These last two “facts” are far from being trivial, and we refer the reader to
[3, Chapter 5] for a proof, as well as for an introduction to the role of
Gaussian measures in linear dynamics. Since the only spaces which are both of
type 2 and of cotype 2 are those which are isomorphic to a Hilbert space, our
proof of Theorem 1.7 does not seem to admit any possible extension to a non-
Hilbertian setting. The following question remains widely open:

######  Question 1.8.

Let $X$ be a complex Banach space and let $T:X\rightarrow X$ be a uniformly
recurrent operator. Is $\operatorname{span}(\mathcal{E}(T))$ a dense set in
$X$? What about the cases where $T$ is an adjoint operator on a separable dual
Banach space or where $X$ is a reflexive Banach space?

A partial (but not completely satisfactory) answer is our third and last main
result, which only concerns the power-bounded operators on complex reflexive
Banach spaces. It extends [10, Theorem 6.2] by showing that such an operator
$T\in\mathcal{L}(X)$ is again uniformly recurrent if and only if has a
spanning set of unimodular eigenvectors. More precisely, we have:

######  Theorem 1.9.

Let $T:X\rightarrow X$ be a power-bounded operator on a complex reflexive
Banach space $X$. Then we have the equality

$\overline{\operatorname{span}(\mathcal{E}(T))}=\overline{\operatorname{URec}(T)}.$

In particular:

  1. (a)

The following statements are equivalent:

    1. (i)

$\mathcal{E}(T)\neq\varnothing$;

    2. (ii)

$\Delta^{*}\textup{Rec}(T)\setminus\\{0\\}\neq\varnothing$;

    3. (iii)

$\mathcal{IP}^{*}\textup{Rec}(T)\setminus\\{0\\}\neq\varnothing$;

    4. (iv)

$\operatorname{URec}(T)\setminus\\{0\\}\neq\varnothing$.

  2. (b)

The following statements are equivalent:

    1. (i)

the set $\operatorname{span}(\mathcal{E}(T))$ is dense in $X$;

    2. (ii)

$T$ is $\Delta^{*}$-recurrent;

    3. (iii)

$T$ is $\mathcal{IP}^{*}$-recurrent;

    4. (iv)

$T$ is uniformly recurrent.

The proof of Theorem 1.9 relies on the splitting theorem of Jacobs-Deleeuw-
Glicksberg (see [31, Section 2.4]). Here the unimodular eigenvectors are
obtained in a very different way than in the proof of Theorem 1.7 (via
characters on a certain compact abelian group).

Even though the arguments used in the proofs of the two theorems above still
hold for complex finite-dimensional spaces, in this situation one can use
directly the canonical Jordan decomposition (see [14, Theorem 4.1] and [10,
Theorem 7.3]) to obtain a spanning set of unimodular eigenvectors even from
“usual” recurrence as defined in Subsection 1.1.

###  1.4 Organization of the paper

Section 2 is devoted to the statement and proof of a purely non-linear result
(Theorem 2.3) which allows to construct invariant measures from reiteratively
recurrent points for a rather general class of Polish dynamical systems (which
includes the compact ones). Theorem 2.3 is a modest improvement of [24,
Theorem 1.5, Remarks 2.6 and 2.12] and its proof is based on a modification of
the construction given in [24, Section 2]. In Section 3, we prove some results
where frequent recurrence is deduced from reiterative recurrence, in
particular Theorem 1.3. Theorems 1.7 and 1.9, which provide links between
strong forms of recurrence and the existence of unimodular eigenvectors, are
proved in Section 4. Sections 5 and 6 present some applications of the above
results in terms of product and inverse dynamical systems respectively, while
we study in Section 7 the “typicality”, in the Baire Category sense, of some
recurrence properties. Lastly, we gather in Section 8 some possibly
interesting open questions and a few comments related to them.

##  2 Invariant measures from reiterative recurrence

In this section, we present a modification of the construction of [24, Section
2] which allows to construct invariant measures from reiteratively recurrent
points for a rather general class of Polish dynamical systems, including the
compact ones (see Remark 2.4).

###  2.1 Topological assumptions and initial comments

We begin this section with some notation: whenever we consider a space of
functions we will use the symbol ${\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$ to denote the
function constantly equal to $1$, and given a subset $A$ of the domain of the
functions, we will write ${\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{A}$ for the
indicator function of $A$, i.e. ${\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{A}(x)=1$ if $x\in A$
and ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{A}(x)=0$ if $x\notin A$. For instance,
if we consider $\ell^{\infty}=\ell^{\infty}(\mathbb{N})$, the space of all
bounded sequences of real numbers, ${\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\in\ell^{\infty}$ is
the sequence with all its terms equal to $1$, and for every
$A\subset\mathbb{N}$, ${\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{A}\in\ell^{\infty}$
will be the sequence in which the $n$-th coordinate is $1$ if $n\in A$ and $0$
otherwise.

A Banach limit is a continuous functional
$\mathfrak{m}:\ell^{\infty}\rightarrow\mathbb{R}$ such that for every pair of
sequences $\phi=(\phi(n))_{n\geq 1},\psi=(\psi(n))_{n\geq 1}\in\ell^{\infty}$,
every $\alpha,\beta\in\mathbb{R}$ and every $a\in\mathbb{N}$:

  1. (a)

$\mathfrak{m}(\alpha\phi+\beta\psi)=\alpha\mathfrak{m}(\phi)+\beta\mathfrak{m}(\psi)$
(linearity);

  2. (b)

$\phi(n)\geq 0$ for every $n\in\mathbb{N}$ implies $\mathfrak{m}(\phi)\geq 0$
(positivity);

  3. (c)

$\mathfrak{m}((\phi(n+a)_{n\geq 1})=\mathfrak{m}((\phi(n))_{n\geq 1})$ (shift-
invariance);

  4. (d)

if $\phi$ is a convergent sequence then
$\mathfrak{m}(\phi)=\lim_{n\to\infty}\phi(n)$ (which implies
$\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})=1$).

Following [24], each Banach limit $\mathfrak{m}$ should be viewed as a
finitely additive measure on $\mathbb{N}$. In fact we will write the result of
the action of $\mathfrak{m}$ on a “function” $\phi\in\ell^{\infty}$ as the
integral:

$\mathfrak{m}(\phi)=\int_{\mathbb{N}}\phi(i)d\mathfrak{m}(i).$

Given a topological space $(X,\tau)$ we will denote by $\mathscr{B}(X,\tau)$
its $\sigma$-algebra of Borel sets. If there is no confusion with the topology
we will simply write $\mathscr{B}(X)$. All the measures considered in this
section will be non-negative finite Borel measures, i.e. they could be the
null measure, and since they will be defined on Polish spaces the finiteness
condition will imply their regularity (see [13, Proposition 8.1.12]). Given a
(non-negative) finite Borel measure $\mu$ on a topological space $(X,\tau)$ we
will denote its support by

$\operatorname{supp}(\mu):=X\setminus\bigcup_{\underset{\mu(U)=0}{U\in\tau}}U.$

When $\mu$ is positive and regular it is easy to show that
$\operatorname{supp}(\mu)$ is non-empty, and the smallest $\tau$-closed subset
of $X$ with full measure, i.e. $\mu(\operatorname{supp}(\mu))=\mu(X)$, the
later being true even if $\mu$ is not regular but $X$ is second-countable (see
[30, Proposition 2.3]). Moreover, a point $x$ belongs to
$\operatorname{supp}(\mu)$ if and only if $\mu(U)>0$ for every neighbourhood
$U$ of $x$.

Before presenting the “measures’ constructing machine” that will be used in
the rest of this work, we give name to some properties that a Polish dynamical
system $(X,T)$ may have. In particular, let $(X,\tau_{X})$ be the underlying
Polish space, $\tau$ a Hausdorff topology in $X$ and let $\mathcal{K}_{\tau}$
be the set of $\tau$-compact subsets of $X$. The properties that we are going
to consider are the following:

  1. (I)

$T$ is a continuous self-map of $(X,\tau)$ (i.e. $T:X\rightarrow X$ is
$\tau$-continuous);

  2. (II)

$\tau\subset\tau_{X}$ (i.e. $\tau$ is coarser than $\tau_{X}$);

  3. (III)

$\mathscr{B}(X,\tau)=\mathscr{B}(X,\tau_{X})$ (i.e. both topologies have the
same Borel sets);

  4. (IV)

every $\tau$-compact set is $\tau$-metrizable (i.e. every
$K\in\mathcal{K}_{\tau}$ is $\tau$-metrizable);

  5. (III*)

every point of $X$ has a neighbourhood basis for $\tau_{X}$ consisting of
$\tau$-compact sets.

In [24, Fact 2.1] it is shown easily how (II) and (III*) imply conditions
(III) and (IV). For the concrete recurrence results that we obtain, it is
necessary to assume conditions (I), (II) and (III*) in order to use the
reiteratively recurrent points in a successful way. However, without property
(III*) and assuming just conditions (I), (II), (III) and (IV) we can carry out
the “construction” on which everything is based:

######  Lemma 2.1 (Modification of [24, Remarks 2.6 and 2.12]).

Let $(X,T)$ be a Polish dynamical system. Assume that $X$ is endowed with a
Hausdorff topology $\tau$ which fulfills (I), (II), (III) and (IV). Then for
each $x_{0}\in X$ and each Banach limit
$\mathfrak{m}:\ell^{\infty}\rightarrow\mathbb{R}$ one can find a (non-
negative) $T$-invariant finite Borel regular measure $\mu$ on $X$ for which
$\mu(X)\leq 1$ and such that

$\mu(K)\geq\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})\quad\text{ for every }K\in\mathcal{K}_{\tau}.$

Moreover, we have the inclusion

$\operatorname{supp}(\mu)\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.$

######  Remark 2.2.

Lemma 2.1 is a rather technical result which allows us to construct invariant
measures. Note that:

  1. (a)

Assumptions (I), (II), (III) and (IV) are fulfilled by the initial topology
$\tau_{X}$. However, if the $\tau_{X}$-compact sets are too small, given an
arbitrary point $x_{0}\in X$ (even with some kind of recurrence property) we
could have $\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{N(x_{0},K)})=0$ for
every $\tau_{X}$-compact set $K\subset X$ and hence the measure $\mu$ obtained
could be the null measure on $X$. We will consider a strictly coarser topology
$\tau\subsetneq\tau_{X}$ in order to obtain “interesting measures” from Lemma
2.1 (see Theorems 1.3 and 1.7).

  2. (b)

Following the previous comment, even if the $\tau$-compact sets are big
enough, the measure $\mu$ could be the null measure on $X$ if we choose a
point $x_{0}\in X$ for which the return sets $N(x_{0},K)$ are too small and
hence $\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{N(x_{0},K)})=0$ for every
$K\in\mathcal{K}_{\tau}$. We will get “interesting measures” whenever we
combine Lemma 2.1 together with the existence of a point $x_{0}\in X$ and a
Banach limit $\mathfrak{m}$ for which $\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})>0$ for some $\tau$-compact subsets $K$ of $X$.
Those conditions will come from property (III*) together with the existence of
a reiteratively recurrent point $x_{0}\in\operatorname{RRec}(T)$, see Theorem
2.3.

  3. (c)

In the proof of [24, Theorem 1.5] it is shown that under conditions (I), (II)
and (III*), one can change the final statement of Lemma 2.1 into:

then for each $x_{0}\in X$ one can find a $T$-invariant finite Borel measure
$\mu$ on $X$ such that $\mu(K)\geq\underline{\operatorname{dens}}(N(x_{0},K))$
for every $K\in\mathcal{K}_{\tau}$,

simply by choosing a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$
and considering the Banach limit

$\mathfrak{m}(\phi):=\lim_{\mathcal{U}}\frac{1}{n}\sum_{i=1}^{n}\phi(i)\quad\text{
for every }\phi\in\ell^{\infty}.$

Moreover, under the same assumptions it is also stated in [24, Remark 2.12]
that

for each $x_{0}\in X$ and each $K\in\mathcal{K}_{\tau}$ one can find a
$T$-invariant finite Borel measure $\mu$ on $X$ such that
$\mu(K)\geq\overline{\operatorname{dens}}(N(x_{0},K))$.

This just ensures that the above inequality holds true for only one fixed
$\tau$-compact subset $K$ of $X$. We will encounter the same problem when
working with the upper Banach density, and we will have to combine some more
sophisticated Banach limits in order to cope with several $\tau$-compact sets
at the same time, see Subsection 2.3.

Here is the main result of this section:

######  Theorem 2.3.

Let $(X,T)$ be a Polish dynamical system. Assume that $X$ is endowed with a
Hausdorff topology $\tau$ which fulfills (I), (II), and (III*). If $x_{0}\in
X$ is a reiteratively recurrent point for $T$, then one can find a
$T$-invariant probability measure $\mu_{x_{0}}$ on $X$ such that

$x_{0}\in\operatorname{supp}(\mu_{x_{0}})\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.$

Moreover, if $T$ is reiteratively recurrent then one can find a $T$-invariant
probability measure $\mu$ on $X$ with full support.

######  Remark 2.4.

If the Polish dynamical system $T:(X,\tau_{X})\rightarrow(X,\tau_{X})$ is
locally compact, its initial topology $\tau_{X}$ already fulfills properties
(I), (II) and (III*), and hence (III) and (IV). In particular, the later is
true whenever $(X,\tau_{X})$ is a (metrizable) compact space.

###  2.2 Proof of Lemma 2.1

We modify the construction given in [24, Section 2.2]. Let $(X,T)$ be a Polish
dynamical system, denote by $\tau_{X}$ the initial topology of $X$ and assume
that $X$ is endowed with a Hausdorff topology $\tau$ which fulfills (I), (II),
(III) and (IV). Fix $x_{0}\in X$ and let
$\mathfrak{m}:\ell^{\infty}\rightarrow\mathbb{R}$ be a Banach limit. For each
$K\in\mathcal{K}_{\tau}$ denote by $\mathscr{C}(K,\tau)$ the space of all
$\tau$-continuous real-valued functions on $K$.

######  Fact 2.2.1 (Modification of [24, Fact 2.2]).

For every $K\in\mathcal{K}_{\tau}$ there is a unique (non-negative) finite
Borel regular measure $\mu_{K}$ on $K$ such that

$\int_{K}fd\mu_{K}=\int_{\mathbb{N}}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{K}f)(T^{i}x_{0})d\mathfrak{m}(i)\quad\text{ for every
}f\in\mathscr{C}(K,\tau).$

The measure $\mu_{K}$ satisfies $0\leq\mu_{K}(K)=\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})\leq 1$.

###### Proof.

The first part is obvious by the Riesz’s Representation theorem since, as
mentioned in [24, Fact 2.2], the formula

$L(f):=\int_{\mathbb{N}}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{K}f)(T^{i}x_{0})d\mathfrak{m}(i)\quad\text{ for every
}f\in\mathscr{C}(K,\tau),$

defines a (non-negative) linear functional on $\mathscr{C}(K,\tau)$. Moreover,
the measure $\mu_{K}$ satisfies

$0\leq\mu_{K}(K)=\int_{\mathbb{N}}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{K})(T^{i}x_{0})d\mathfrak{m}(i)=\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})\leq\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}})=1.\qed$

By (III) we have the equality $\mathscr{B}(X,\tau)=\mathscr{B}(X,\tau_{X})$
and hence for each $K\in\mathcal{K}_{\tau}$ we can extend the measure
$\mu_{K}$ into a Borel measure on the whole space $X$ (still denoted by
$\mu_{K}$) using the formula

$\mu_{K}(A):=\mu_{K}(K\cap A)\quad\text{ for every Borel set
}A\in\mathscr{B}(X).$

Clearly $\mu_{K}(X)\leq 1$, which implies the regularity of these measures.
However, since the compact sets $K\in\mathcal{K}_{\tau}$ are not necessarily
$T$-invariant and we could have $T^{-1}(K)\cap K=\varnothing$, the measures
$\mu_{K}$ are not necessarily $T$-invariant. As in [24] we will define the
$T$-invariant measure we are looking for by taking the supremum of the
measures $\mu_{K}$, and this is possible due to the following fact:

######  Fact 2.2.2 ([24, Fact 2.3]).

If $K,F\in\mathcal{K}_{\tau}$ and if $K\subset F$, then $\mu_{K}\leq\mu_{F}$.

###### Proof.

The proof is exactly the same than that of [24, Fact 2.3] and it uses
essentially conditions (II), (IV) and the positivity of $\mathfrak{m}$. ∎

Since a finite union of $\tau$-compact subsets of $X$ is still an element of
$\mathcal{K}_{\tau}$, from Fact 2.2.2 we deduce that the family
$(\mu_{K})_{K\in\mathcal{K}_{\tau}}$ has the following property: for any pair
$K_{1},K_{2}\in\mathcal{K}_{\tau}$ one can find $F\in\mathcal{K}_{\tau}$ such
that $\mu_{F}\geq\max\\{\mu_{K_{1}},\mu_{K_{2}}\\}$. Hence we can continue the
construction given in [24, Section 2.2]:

######  Fact 2.2.3 (Modification of [24, Fact 2.4]).

If we set

$\mu(A):=\sup_{K\in\mathcal{K}_{\tau}}\mu_{K}(A)\quad\text{ for every Borel
set }A\in\mathscr{B}(X),$

then $\mu$ is a (non-negative) Borel measure on $X$ such that $\mu(X)\leq 1$.

###### Proof.

The proof is exactly the same than that of [24, Fact 2.4] but with $\mu$
having the possibility of being the null measure. ∎

######  Fact 2.2.4 (Modification of [24, Fact 2.5]).

The measure $\mu$ is $T$-invariant and we have the inequality

$\mu(K)\geq\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})\quad\text{ for every }K\in\mathcal{K}_{\tau}.$

###### Proof.

The first part of the proof is exactly the same than that of [24, Fact 2.5]
and it uses essentially conditions (I), (IV), the positivity of $\mathfrak{m}$
and the fact that it is shift-invariant. By Fact 2.2.1, for each
$K\in\mathcal{K}_{\tau}$ we have that

$\mu(K)\geq\mu_{K}(K)=\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{N(x_{0},K)}).\qed$

To finish the proof of Lemma 2.1 we include a property, not shown in [24,
Section 2.2], about the support of the measure constructed:

######  Fact 2.2.5.

We have the inclusion
$\operatorname{supp}(\mu)\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}$.

###### Proof.

Write $O(x_{0}):=\overline{\operatorname{Orb}(x_{0},T)}^{\tau}$. First we show
that for each $K\in\mathcal{K}_{\tau}$ we have the inclusion
$\operatorname{supp}(\mu_{K})\subset K\cap O(x_{0})$: indeed, for any
$K\in\mathcal{K}_{\tau}$ and any point $x\in K\setminus O(x_{0})$, by
compactness, there exists a positive function $f\in\mathscr{C}(K,\tau)$ such
that

$f=0\text{ on }K\cap O(x_{0})\quad\text{ and }\quad f(x)=1.$

If we suppose that $x\in\operatorname{supp}(\mu_{K})$ and if we take a
$\tau$-neighbourhood $U$ of $x$ in $(K,\tau)$ such that
$f(U)\subset[1/2,\infty[$ then we have

$0<\frac{1}{2}\mu_{K}(U)\leq\int_{K}fd\mu_{K}=\int_{\mathbb{N}}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{K}f)(T^{i}x_{0})d\mathfrak{m}(i)=\mathfrak{m}(\boldsymbol{0})=0,$

since $f(T^{i}x_{0})=0$ for every $i\in\mathbb{N}$, which is a contradiction.
Hence $x\notin\operatorname{supp}(\mu_{K})$.

Finally, given $x\notin O(x_{0})$ there is an $\tau$-open neighbourhood $U$ of
$x$ in $(X,\tau)$, which by (II) is also a $\tau_{X}$-neighbourhood of $x$ in
$(X,\tau_{X})$, such that $U\cap O(x_{0})=\varnothing$. Hence, since
$\operatorname{supp}(\mu_{K})\subset O(x_{0})$ for every
$K\in\mathcal{K}_{\tau}$, we deduce that $\mu_{K}(U)=\mu_{K}(K\cap U)=0$ for
every $K\in\mathcal{K}_{\tau}$. By the definition of $\mu$ we get $\mu(U)=0$
and hence that $x\notin\operatorname{supp}(\mu)$. ∎

###  2.3 Proof of Theorem 2.3

Let $(X,T)$ be a Polish dynamical system, denote by $\tau_{X}$ the initial
topology of $X$ and assume that $X$ is endowed with a Hausdorff topology
$\tau$ which fulfills (I), (II) and (III*).

######  Fact 2.3.1.

Given $x_{0}\in X$ and $U\in\mathcal{K}_{\tau}$ with
$\overline{\operatorname{Bd}}(N(x_{0},U))>0$, there exists a $T$-invariant
probability measure $\mu$ on $X$ such that $\mu(U)>0$. Moreover, we have the
inclusion

$\operatorname{supp}(\mu)\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.$

Fact 2.3.1 allow us to (slightly) extend Theorem 2.3 in terms of the
recurrence notion introduced in [24, Section 2.5] (see Remark 2.5 below for
the explicit statement) which at the end turns out to be equivalent to the
recurrence notion used here (see Theorem 3.3).

###### Proof of Fact 2.3.1.

Since

$\overline{\operatorname{Bd}}(N(x_{0},U)):=\limsup_{N\to\infty}\left(\max_{m\geq
0}\dfrac{\operatorname{card}(N(x_{0},U)\cap[m,m+N])}{N+1}\right)>0,$

there exists an increasing sequence of natural numbers
$(N_{k})_{k\in\mathbb{N}}\in\mathbb{N}^{\mathbb{N}}$ and a sequence of
intervals $I_{k}=[i_{k}+1,i_{k}+N_{k}]\subset\mathbb{N}$ such that

$\overline{\operatorname{Bd}}(N(x_{0},U))=\lim_{k\to\infty}\dfrac{\operatorname{card}(N(x_{0},U)\cap
I_{k})}{N_{k}}.$ (1)

Then we fix the Banach limit $\mathfrak{m}:\ell^{\infty}\rightarrow\mathbb{R}$
defined as

$\mathfrak{m}(\phi):=\lim_{\mathcal{U}}\frac{1}{N_{k}}\sum_{n\in
I_{k}}\phi(n)\quad\text{ for every }\phi\in\ell^{\infty},$

for some fixed non-principal ultrafilter
$\mathcal{U}\subset\mathscr{P}(\mathbb{N})$ on $\mathbb{N}$. By (1) we have

$\mathfrak{m}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},U)})=\overline{\operatorname{Bd}}(N(x_{0},U))>0.$

Since $\tau$ fulfills (I), (II) and (III*), by [24, Fact 2.1] it also has
properties (III) and (IV) so we can apply Lemma 2.1 to $x_{0}$ and
$\mathfrak{m}$ obtaining a (non-negative) $T$-invariant finite Borel measure
$\mu$ on $X$ for which $\mu(K)\geq\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},K)})$ for each $K\in\mathcal{K}_{\tau}$ and such
that

$\operatorname{supp}(\mu)\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.$

In particular we get $\mu(U)\geq\mathfrak{m}({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{N(x_{0},U)})>0$ so $\mu$ is a positive $T$-invariant finite
Borel measure. Normalizing $\mu$ we get the desired measure. ∎

######  Fact 2.3.2.

Given $x_{0}\in\operatorname{RRec}(T)$, there exists a $T$-invariant
probability measure $\mu_{x_{0}}$ on $X$ such that

$x_{0}\in\operatorname{supp}(\mu_{x_{0}})\subset\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.$

###### Proof.

Set $O(x_{0}):=\overline{\operatorname{Orb}(x_{0},T)}^{\tau}$. Using (III*),
let $(U_{n})_{n\in\mathbb{N}}$ be a basis of $\tau_{X}$-neighbourhoods of
$x_{0}$ consisting of $\tau$-compact sets. Applying Fact 2.3.1 to each set
$U_{n}$ we obtain a sequence $(\mu_{n})_{n\in\mathbb{N}}$ of $T$-invariant
probability measures on $X$ for which $\mu_{n}(U_{n})>0$ and such that
$\operatorname{supp}(\mu_{n})\subset O(x_{0})$ for each $n\in\mathbb{N}$. Then
the measure

$\mu_{x_{0}}:=\sum_{n\in\mathbb{N}}\frac{\mu_{n}}{2^{n}}$

is a $T$-invariant probability measure on $X$. Moreover, for any
$\tau_{X}$-neighbourhood $U$ of $x_{0}$ there is an integer $n\in\mathbb{N}$
with $U_{n}\subset U$ and hence

$\mu_{x_{0}}(U)\geq\mu_{x_{0}}(U_{n})\geq\frac{\mu_{n}(U_{n})}{2^{n}}>0.$

This implies that $x_{0}\in\operatorname{supp}(\mu_{x_{0}})$. Also, given
$x\notin O(x_{0})$ there is a $\tau$-neighbourhood $V$ of $x$, which by (II)
is also a $\tau_{X}$-neighbourhood of $x$, such that $V\cap
O(x_{0})=\varnothing$. Since $\operatorname{supp}(\mu_{n})\subset O(x_{0})$
for every $n\in\mathbb{N}$ we deduce that $\mu_{n}(V)=0$ for every
$n\in\mathbb{N}$ and by the definition of $\mu_{x_{0}}$ we get
$\mu_{x_{0}}(V)=0$. This implies that
$x\notin\operatorname{supp}(\mu_{x_{0}})$ and hence

$x_{0}\in\operatorname{supp}(\mu_{x_{0}})\subset
O(x_{0})=\overline{\operatorname{Orb}(x_{0},T)}^{\tau}.\qed$

To complete the proof of Theorem 2.3, let $T$ be reiteratively recurrent.
Since $X$ is separable there is a countable set
$\\{x_{n}:n\in\mathbb{N}\\}\subset\operatorname{RRec}(T)$ which is dense in
$X$. Applying Fact 2.3.2 to each point $x_{n}$ we obtain a sequence
$(\mu_{x_{n}})_{n\in\mathbb{N}}$ of $T$-invariant probability measures on $X$
such that $x_{n}\in\operatorname{supp}(\mu_{x_{n}})$ for each
$n\in\mathbb{N}$. Finally, the measure

$\mu:=\sum_{n\in\mathbb{N}}\frac{\mu_{x_{n}}}{2^{n}}$

is a $T$-invariant probability measure on $X$ with full support.∎

######  Remark 2.5.

Under the initial topological assumptions of Theorem 2.3, and in view of Fact
2.3.1, a generalization in terms of the recurrence notion introduced in [24,
Section 2.5], and following the spirit of [24, Proposition 2.11], can be
shown:

If for each open subset $U$ of $X$ there is a point $x_{U}\in X$ such that
$\overline{\operatorname{Bd}}(N(x_{U},U))>0$, then one can find a
$T$-invariant probability measure $\mu$ on $X$ with full support.

Indeed, one just has to use (III*) to consider an appropriate countable family
of $\tau$-compact sets whose $\tau_{X}$-interiors form a base of the initial
topology $\tau_{X}$, apply Fact 2.3.1 to those $\tau$-compact sets and take an
infinite convex combination of the obtained measures.

##  3 From reiterative to frequent recurrence

Theorem 2.3 allows us to construct invariant measures starting from
reiteratively recurrent points. In this section, we exploit this result in
order to show that reiterative recurrence for adjoint operators on separable
dual Banach spaces actually implies the stronger notion of frequent recurrence
(Theorem 1.3).

###  3.1 A key lemma

An important tool for the proof of Theorem 1.3 is the following lemma:

######  Lemma 3.1 (Frequent Recurrence from Invariant Measures).

Let $T:X\rightarrow X$ be a continuous map on a second-countable space $X$ and
let $\mu$ be a $T$-invariant probability measure on $X$. Then
$\mu(\operatorname{FRec}(T))=1$ and in particular we have the inclusion

$\operatorname{supp}(\mu)\subset\overline{\operatorname{FRec}(T)}.$

The above result is the recurrence version of [3, Corollary 5.5], and since
recurrence is a local property the measure is not required to be with full
support, condition under which the map $T$ would clearly be frequently
recurrent.

###### Proof of Lemma 3.1.

Let $B\in\mathscr{B}(X)$ be an arbitrary but fixed Borel set with $\mu(B)>0$.
By the Ergodic Decomposition theorem (see [23, Theorem 3.42]) there is a
$T$-invariant probability measure $m$ on $X$ for which $T$ is an ergodic map
and such that $m(B)>0$. Let $(U_{n})_{n\in\mathbb{N}}$ be a countable basis of
the topology and apply the Birkhoff’s Pointwise Ergodic theorem (see [23,
Theorem 3.41]) to each of the indicator functions ${\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{U_{n}}$. This yields

$\displaystyle\operatorname{dens}(N(x,U_{n}))$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}\frac{\operatorname{card}(N(x,U_{n})\cap[0,N])}{N+1}=\lim_{N\to\infty}\frac{1}{N+1}\sum_{k=0}^{N}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{U_{n}}(T^{k}x)$ $\displaystyle=$ $\displaystyle\int_{X}\
{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{U_{n}}\ dm=m(U_{n}),$

for $m$-a.e. point $x\in X$, that is, for each $n\in\mathbb{N}$ there is a set
$A_{n}\subset X$ with $m(A_{n})=1$ such that
$\operatorname{dens}(N(x,U_{n}))=m(U_{n})$ for every $x\in A_{n}$. Since a
countable union of null sets is again null, the set

$A:=\operatorname{supp}(m)\cap\left(\bigcap_{n\in\mathbb{N}}A_{n}\right)$

satisfies $m(A)=1$. We claim that $A\subset\operatorname{FRec}(T)$. Indeed,
for every $x\in A$ and every neighbourhood $U$ of $x$ there is an integer
$n\in\mathbb{N}$ such that $x\in U_{n}\subset U$. Since
$A\subset\operatorname{supp}(m)$ we have that
$U_{n}\cap\operatorname{supp}(m)\neq\varnothing$ and hence

$\underline{\operatorname{dens}}(N(x,U))\geq\underline{\operatorname{dens}}(N(x,U_{n}))=m(U_{n})>0.$

The arbitrariness of the neighbourhood $U$ of $x$ implies that
$x\in\operatorname{FRec}(T)$. Now, since $m(A)=1$ and $m(B)>0$ we obtain
$A\cap B\neq\varnothing$ and hence

$\operatorname{FRec}(T)\cap B\neq\varnothing.$

Since this is true for every set $B\in\mathscr{B}(X)$ with $\mu(B)>0$ we
deduce that $\mu(\operatorname{FRec}(T))=1$. Then
$\mu(\overline{\operatorname{FRec}(T)})=1$ and in particular, since
$\operatorname{supp}(\mu)$ is the smallest closed subset of $X$ with full
$\mu$-measure, we get that

$\operatorname{supp}(\mu)\subset\overline{\operatorname{FRec}(T)}.\qed$

######  Remark 3.2.

Lemma 3.1 improves [22, Theorem 3.3] in terms of frequent recurrence by using
the Birkhoff’s Pointwise Ergodic theorem. Indeed, under the assumptions of
Lemma 3.1, [22, Theorem 3.3] shows that $\mu$-a.e. point is recurrent, i.e.
$\mu(\textup{Rec}(T))=1$.

Combining Theorem 2.3 and Lemma 3.1 we deduce the following result:

######  Theorem 3.3 (From Reiterative to Frequent Recurrence).

Let $(X,T)$ be a Polish dynamical system, denote by $\tau_{X}$ the initial
topology of $X$ and assume that $X$ is endowed with a Hausdorff topology
$\tau$ which fulfills (I), (II), and (III*). Then we have the equality

$\overline{\operatorname{FRec}(T)}^{\tau_{X}}=\overline{\operatorname{RRec}(T)}^{\tau_{X}}.$

Moreover:

  1. (a)

The following statements are equivalent:

    1. (i)

$\operatorname{FRec}(T)\neq\varnothing$;

    2. (ii)

$\operatorname{UFRec}(T)\neq\varnothing$;

    3. (iii)

$\operatorname{RRec}(T)\neq\varnothing$;

    4. (iv)

$T$ admits an invariant probability measure.

  2. (b)

The following statements are equivalent:

    1. (i)

$T$ is frequently recurrent;

    2. (ii)

$T$ is $\mathcal{U}$-frequently recurrent;

    3. (iii)

$T$ is reiteratively recurrent;

    4. (iv)

$T$ admits an invariant probability measure with full support.

###### Proof.

By definition we always have
$\operatorname{FRec}(T)\subset\operatorname{UFRec}(T)\subset\operatorname{RRec}(T)$,
so we just have to show that

$\operatorname{RRec}(T)\subset\overline{\operatorname{FRec}(T)}^{\tau_{X}}.$

Suppose that $\operatorname{RRec}(T)\neq\varnothing$. Given any
$x_{0}\in\operatorname{RRec}(T)$, by Theorem 2.3 one can find a $T$-invariant
probability measure $\mu_{x_{0}}$ on $X$ for which
$x_{0}\in\operatorname{supp}(\mu_{x_{0}})$. Since separable and metrizable
spaces are second-countable, Lemma 3.1 implies that
$x_{0}\in\overline{\operatorname{FRec}(T)}^{\tau_{X}}$.

Moreover, in both cases (a) and (b) we have: (i) implies (ii) which implies
(iii) by definition; (iii) implies (iv) by Theorem 2.3; and (iv) implies (i)
by Lemma 3.1. ∎

As we already mentioned in the Introduction, this result is false for general
Polish dynamical systems: there exist even reiteratively hypercyclic operators
on $c_{0}(\mathbb{N})$ without any non-zero $\mathcal{U}$-frequently recurrent
vector (see [10, Theorem 5.7 and Corollary 5.8]). Working with linear
dynamical systems implies a reformulation of the above result, which is
Theorem 1.3.

###  3.2 Proof of Theorem 1.3

Let $T:X\rightarrow X$ be an adjoint operator on a separable dual Banach space
$X$. Denote by $\tau_{\|\cdot\|}$ the norm topology, consider the weak-star
topology $w^{*}$ and note that:

  1. (I)

since $T$ is an adjoint operator, it is a continuous self-map of $(X,w^{*})$;

  2. (II)

by the definition of the topologies, we have $w^{*}\subset\tau_{\|\cdot\|}$;

  3. (III*)

by the Alaoglu-Bourbaki’s theorem, the translation of the family of closed
balls centred at $0$ is a $\tau_{\|\cdot\|}$-neighbourhood basis consisting of
$w^{*}$-compact sets.

If $T:X\rightarrow X$ is an operator on a separable reflexive Banach space $X$
the same conditions hold for the weak topology. From here one can apply the
same arguments as those used in the proof of Theorem 3.3. In particular, if we
consider a point $x_{0}\in\operatorname{RRec}(T)\setminus\\{0\\}$ then the
measure $\mu_{x_{0}}$ obtained by Theorem 2.3 is a non-trivial invariant
probability measure.∎

######  Remark 3.4.

The equality
$\overline{\operatorname{FRec}(T)}=\overline{\operatorname{RRec}(T)}$ and
hence the equivalences (i) $\Leftrightarrow$ (ii) $\Leftrightarrow$ (iii)
established in Theorem 1.3 are still true when the underlying space $X$ is a
non-separable reflexive Banach space. Indeed, given an operator
$T:X\rightarrow X$ on a non-separable reflexive Banach space $X$, and given a
point $x_{0}\in\operatorname{RRec}(T)$ we can consider the separable closed
$T$-invariant subspace

$Z:=\overline{\operatorname{span}(\operatorname{Orb}(x_{0},T))},$

which is again reflexive. Then $T\arrowvert_{Z}:Z\rightarrow Z$ is an operator
on a separable reflexive Banach space. Moreover, recurrence is a local
property, i.e. for each Furstenberg family $\mathcal{F}$ we have the equality:

$\mathcal{F}\textup{Rec}(T\arrowvert_{Z})=\mathcal{F}\textup{Rec}(T)\cap Z.$

Applying Theorem 1.3 to $T\arrowvert_{Z}$ we have

$x_{0}\in\operatorname{RRec}(T\arrowvert_{Z})\text{ and hence
}x_{0}\in\overline{\operatorname{FRec}(T\arrowvert_{Z})}\subset\overline{\operatorname{FRec}(T)}.$

However, we cannot say the same about statement (iv) of Theorem 1.3 since
separability is essential to construct and extend the invariant measures onto
the whole space. The above arguments are also restricted to the reflexive case
because closed subspaces of a dual Banach space are not necessarily dual
Banach spaces (consider $c_{0}(\mathbb{N})\subset\ell^{\infty}(\mathbb{N})$).

##  4 From uniform recurrence to unimodular eigenvectors

Our aim in this section is to connect some recurrence properties (stronger
than those considered in Sections 2 and 3), for linear dynamical systems on
complex Banach spaces, to the existence of unimodular eigenvectors. This
investigation is motivated by the fact that, given a complex linear map
$T:X\rightarrow X$ on a complex topological vector space $X$, the linear span
of its unimodular eigenvectors $\mathcal{E}(T)$ consists of
$\Delta^{*}$-recurrent vectors. It is shown in [10, Lemma 7.1 and Corollary
7.2] that they are $\mathcal{IP}^{*}$-recurrent, and in fact, the same
arguments hold by using [22, Proposition 9.8] applied to the Kronecker system
consisting of the compact group $\mathbb{T}^{k}$ and the (left) multiplication
$(z_{1},...,z_{k})\mapsto(\lambda_{1}z_{1},...,\lambda_{k}z_{k})$ for a fixed
$k$-tuple $(\lambda_{1},...,\lambda_{k})\in\mathbb{T}^{k}$. We give an
alternative proof via invariant measures:

######  Proposition 4.1.

Let $T:X\rightarrow X$ be a complex linear map on a complex topological vector
space $X$. A linear combination of unimodular eigenvectors $\mathcal{E}(T)$ is
a $\Delta^{*}$-recurrent vector, i.e.
$\operatorname{span}(\mathcal{E}(T))\subset\Delta^{*}\textup{Rec}(T)$.

###### Proof.

Given $\lambda\in\mathbb{T}$ let $R_{\lambda}:\mathbb{T}\rightarrow\mathbb{T}$
be the $\lambda$-rotation map where $z\mapsto\lambda z$. Then given
$\varepsilon>0$, since the Haar measure on $\mathbb{T}$ is a
$R_{\lambda}$-invariant measure with full support, by the Poincaré’s
Recurrence theorem (see [22, Theorem 3.2 and Page 177]) there is
$A\in\Delta^{*}$ such that for the set
$B(1,\varepsilon/2):=\\{z\in\mathbb{T}:|1-z|<\varepsilon/2\\}$ we have

$R_{\lambda}^{n}(B(1,\varepsilon/2))\cap
B(1,\varepsilon/2)\neq\varnothing\text{ for every }n\in A.$

By the triangular inequality we get $|\lambda^{n}-1|<\varepsilon$ for each
$n\in A$ and hence

$\Delta^{*}\ni A\subset\\{n\in\mathbb{N}:|\lambda^{n}-1|<\varepsilon\\}\text{
so }\\{n\in\mathbb{N}:|\lambda^{n}-1|<\varepsilon\\}\in\Delta^{*}.$

Since the Furstenberg family $\Delta^{*}$ is a filter (see [8]) and
$\lambda\in\mathbb{T}$ and $\varepsilon>0$ were chosen arbitrarily the proof
is finished. ∎

Hence, given a complex linear dynamical system $T:X\rightarrow X$ we will
always have:

$\operatorname{span}(\mathcal{E}(T))\subset\Delta^{*}\textup{Rec}(T)\subset\mathcal{IP}^{*}\textup{Rec}(T)\subset\operatorname{URec}(T)\subset\operatorname{RRec}^{bo}(T).$

Our goal is now to prove Theorem 1.7, which states that for any operator
acting on a complex Hilbert space, the existence of a non-zero reiteratively
recurrent vector with bounded orbit, and in particular the existence of a
uniformly recurrent vector, implies the existence of a unimodular eigenvector.
The proof of Theorem 1.7 relies heavily on the machinery of Gaussian measures
on (complex separable) Hilbert spaces. We begin by recalling some basic facts
concerning these Gaussian measures, as well as some deeper results pertaining
to the Ergodic Theory of Gaussian linear dynamical systems. We refer the
reader to one of the references [12] or [15] for more about Gaussian measures
on Banach spaces, and to [3] and [4] for more on their role in linear
dynamics.

###  4.1 Ergodic Theory for linear dynamical systems and Gaussian measures

The study of Ergodic Theory in the framework of linear dynamics started with
the pioneering work of Flytzanis (see [20, 21]), and was then further
developed in the papers [1], [2] and [4], among others, focusing on the
existence of invariant Gaussian measures satisfying some further dynamical
properties such as weak/strong mixing.

######  Definition 4.2.

A Borel probability measure $m$ on a complex Banach space $X$ is said to be a
Gaussian measure if every continuous linear functional $x^{*}\in X^{*}$ has a
complex Gaussian distribution when considered as a random variable on
$(X,\mathscr{B}(X),m)$.

It is now well understood that the dynamics of a linear dynamical system
$(X,T)$ are closely related to the properties of the unimodular eigenvectors
of $T$. The situation is especially well understood in the Hilbertian setting,
since the existence of an invariant Gaussian measure (with full support, or
with respect to which $T$ is ergodic or weakly/strongly mixing) can be fully
characterized in terms of the properties of the set $\mathcal{E}(T)$. See [1]
and [3] for details. These characterizations do not hold true, in general, in
the Banach space setting, but still many results are preserved allowing for a
rather through understanding of Ergodic Theory of linear dynamical systems in
this Gaussian framework. See [2], [3] and [4] for details. Even though
Gaussian measures are an essential tool for our proof of Theorem 1.7 (see
Lemma 4.4 below), the properties that such measures (may) have are properties
that arbitrary probability measures can have too. We introduce these
properties following [12]:

######  Definition 4.3.

Let $\mu$ be a probability measure on a Banach space $X$:

  1. (a)

suppose that there exists an element $x\in X$ such that

$\int_{X}\langle x^{*},z\rangle d\mu(z)=\langle x^{*},x\rangle\quad\text{ for
every }x^{*}\in X^{*},$

then $x$ is called the expectation of the measure $\mu$, and in this case we
will write

$\int_{X}zd\mu(z):=x;$

  2. (b)

we say that $\mu$ is centered if its expectation exists and it is equal to
$0\in X$;

  3. (c)

we say that $\mu$ has a finite second-order moment, if

$\int_{X}\|z\|^{2}d\mu(z)<\infty.$

If $\mu$ has a finite second-order moment then its expectation (called the
Pettis integral of $\mu$) exists (see [16, Page 55]). Given a centered
probability measure $\mu$ on $X$ with a finite second-order moment, following
[12, Page 169] and [3, Theorem 5.9], we can define the covariance operator of
such a measure $\mu$ as the bounded linear operator $R:X^{*}\rightarrow X$
satisfying

$\langle y^{*},Rx^{*}\rangle=\int_{X}\langle y^{*},z\rangle\langle
x^{*},z\rangle d\mu(z)$

for every pair of elements $x^{*}$ and $y^{*}$ of $X^{*}$. In other words,

$Rx^{*}:=\int_{X}\langle x^{*},z\rangle zd\mu(z)\quad\text{ for every
}x^{*}\in X^{*}.$ (2)

Any Gaussian measure $m$ on $X$ has a finite second-order moment (see [3,
Exercise 5.5]), and since we will consider in this work only centered Gaussian
measures, we will always have an associated covariance operator for such a
measure $m$.

When $H$ is a complex separable Hilbert space, the covariance operator of a
centered probability measure $\mu$ on $H$ with a finite second-order moment is
usually defined, in a slightly different way, as the bounded linear operator
$S:H\rightarrow H$ for which

$\langle Sx,y\rangle=\int_{H}\langle x,z\rangle\overline{\langle
y,z\rangle}d\mu(z)\quad\text{ for every }x,y\in H,$

i.e.

$Sx:=\int_{H}\langle x,z\rangle zd\mu(z)\quad\text{ for every }x\in H.$ (3)

Observe that, contrary to (2), in this case $\langle
Sx,\cdot\rangle:H\rightarrow\mathbb{C}$ is an anti-linear functional acting on
$H$. Also, $S$ is a self-adjoint positive trace-class operator on $H$. It is a
standard result (see for instance [3, Corollary 5.15]) that the Gaussian
covariance operators on $H$ are exactly the positive trace-class operators on
$H$, i.e. for such an operator $S$ there exists a Gaussian measure $m$ on $H$
for which we also have that

$\langle Sx,y\rangle=\int_{H}\langle x,z\rangle\overline{\langle
y,z\rangle}dm(z)\quad\text{ for every }x,y\in H.$

The possibility of constructing a Gaussian measure $m$ with the same
covariance operator as $\mu$, together with the fact that the support of the
Gaussian measure $m$ is exactly $\overline{S(H)}$ (i.e. the closed linear span
of the range of its covariance operator, see [3, Proposition 5.18]) is the key
to prove the following lemma, inspired from the pioneering work [21] of
Flytzanis. This lemma is crucial for the proof of Theorem 1.7.

######  Lemma 4.4 (Unimodular Eigenvectors from Invariant Measures).

Let $T\in\mathcal{L}(H)$, where $H$ is a complex separable Hilbert space, and
let $\mu$ be a (non-trivial) $T$-invariant probability measure on $H$ such
that $\int_{H}\|z\|^{2}d\mu(z)<\infty$. Then we have the inclusions

$\operatorname{supp}(\mu)\subset\overline{\operatorname{span}(\operatorname{supp}(\mu))}\subset\overline{\operatorname{span}(\mathcal{E}(T))}.$

###### Proof.

Suppose first that $\mu$ is a centered measure on $H$. Then, since $H$ is a
Hilbert space, the covariance operator $S$ of $\mu$ defined as in (3)
satisfies

$\langle Sx,y\rangle=\int_{H}\langle x,z\rangle\overline{\langle
y,z\rangle}d\mu(z)=\int_{\operatorname{supp}(\mu)}\langle
x,z\rangle\overline{\langle y,z\rangle}d\mu(z)\quad\text{ for every }x,y\in
H,$

and by [3, Corollary 5.15], it is also the covariance operator of a certain
Gaussian measure $m$ on $H$. From now on we split the proof in three steps:

  1. Step 1.

The Gaussian measure $m$ is $T$-invariant:

Given $x,y\in H$ we have

$\displaystyle\langle TST^{*}x,y\rangle$ $\displaystyle=$
$\displaystyle\langle ST^{*}x,T^{*}y\rangle=\int_{H}\langle
T^{*}x,z\rangle\overline{\langle T^{*}y,z\rangle}d\mu(z)=\int_{H}\langle
x,Tz\rangle\overline{\langle y,Tz\rangle}d\mu(z)$ $\displaystyle=$
$\displaystyle\int_{H}\langle x,z\rangle\overline{\langle
y,z\rangle}d(\mu\circ T^{-1})(z)=\int_{H}\langle x,z\rangle\overline{\langle
y,z\rangle}d\mu(z)=\langle Sx,y\rangle,$

since $\mu$ is $T$-invariant. By [3, Proposition 5.22] we deduce that $m$ is
$T$-invariant.

  2. Step 2.

We have the equality
$\overline{\operatorname{span}(\operatorname{supp}(\mu))}=\operatorname{supp}(m)$:

By [3, Proposition 5.18] we know that
$\operatorname{supp}(m)=\ker(S)^{\perp}=\overline{S(H)}$. Moreover, the
subspace $\overline{\operatorname{span}(\operatorname{supp}(\mu))}^{\perp}$ is
included in the set

$\left\\{y\in H:\langle Sx,y\rangle=\int_{\operatorname{supp}(\mu)}\langle
x,z\rangle\overline{\langle y,z\rangle}d\mu(z)=0\text{ for every }x\in
H\right\\}=\\\ $ $\displaystyle=$
$\displaystyle\overline{S(H)}^{\perp}\subset\left\\{y\in
H:\int_{\operatorname{supp}(\mu)}|\langle y,z\rangle|^{2}d\mu(z)=0\right\\}$
$\displaystyle=$ $\displaystyle\left\\{y\in H:\langle y,z\rangle=0\text{ for
}\mu\text{-a.e. }z\in H\right\\}$ $\displaystyle\overset{(*)}{=}$
$\displaystyle\left\\{y\in H:\langle y,z\rangle=0\text{ for every
}z\in\operatorname{supp}(\mu)\right\\}=\overline{\operatorname{span}(\operatorname{supp}(\mu))}^{\perp},$

where the equality $(*)$ follows from the continuity of the maps $\langle
y,\cdot\rangle:H\rightarrow\mathbb{C}$.

  3. Step 3.

We have the inclusion
$\operatorname{supp}(\mu)\subset\overline{\operatorname{span}(\mathcal{E}(T))}$:

In [3, Theorem 5.46] it is stated that if a Banach space $X$ has cotype 2,
then every operator in $\mathcal{L}(X)$ admitting a Gaussian invariant measure
with full support has a spanning set of unimodular eigenvectors. Since the
support of $m$ is a closed linear subspace of $H$ (Step 2) and every Hilbert
space has cotype 2, [3, Theorem 5.46] applied to the $T$-invariant measure $m$
(Step 1) implies that
$\operatorname{supp}(m)\subset\overline{\operatorname{span}(\mathcal{E}(T))}$,
and hence using again Step 2 we get that

$\operatorname{supp}(\mu)\subset\overline{\operatorname{span}(\operatorname{supp}(\mu))}=\operatorname{supp}(m)\subset\overline{\operatorname{span}(\mathcal{E}(T))}.$

Suppose now that $\mu$ is not centered and define the measure

$\nu(A):=\int_{\mathbb{T}}\mu(\lambda A)d\lambda\quad\text{ for every Borel
set }A\in\mathscr{B}(H).$

Then $\nu$ is a (non-trivial) probability measure on $H$ and it is
$T$-invariant since

$\nu(T^{-1}(A))=\int_{\mathbb{T}}\mu(\lambda
T^{-1}(A))d\lambda=\int_{\mathbb{T}}\mu(T^{-1}(\lambda A))d\lambda=\nu(A).$

Using the density of the simple functions in $L^{1}(H,\mathscr{B}(H),\nu)$ one
can show that $\nu$ is centered since

$\int_{H}zd\nu(z)=\int_{\mathbb{T}}\left(\int_{H}\overline{\lambda}zd\mu(z)\right)d\lambda=\int_{\mathbb{T}}\overline{\lambda}\left(\int_{H}zd\mu(z)\right)d\lambda=0,$

and also that $\nu$ has a finite second-order moment since

$\int_{H}\|z\|^{2}d\nu(z)=\int_{\mathbb{T}}\left(\int_{H}\|\overline{\lambda}z\|^{2}d\mu(z)\right)d\lambda=\int_{H}\|z\|^{2}d\mu(z)<\infty.$

The first part of the proof implies that
$\operatorname{supp}(\nu)\subset\overline{\operatorname{span}(\operatorname{supp}(\nu))}\subset\overline{\operatorname{span}(\mathcal{E}(T))}$
and hence we only have to show that
$\operatorname{supp}(\mu)\subset\operatorname{supp}(\nu)$. In order to see
this, pick $x_{0}\in\operatorname{supp}(\mu)$ and $\varepsilon>0$. Then let
$\delta:=\mu(B(x_{0},\varepsilon/2))>0$, where $B(x_{0},\varepsilon/2)$
denotes the open ball of $X$ centred at $x_{0}$ and of radius $\varepsilon/2$,
and note that $B(x_{0},\varepsilon/2)\subset\lambda B(x_{0},\varepsilon)$ for
any $\lambda\in\mathbb{T}$ with

$|\lambda-1|<\frac{\varepsilon}{2(\|x_{0}\|+1)}.$

Indeed, given $x\in B(x_{0},\varepsilon/2)$ we have that $\|\lambda
x_{0}-x\|\leq\|(\lambda-1)x_{0}\|+\|x_{0}-x\|<\varepsilon$. Then for such a
$\lambda\in\mathbb{T}$ we have that $\mu(\lambda
B(x_{0},\varepsilon))\geq\delta$ and hence $\nu(B(x_{0},\varepsilon))>0$. The
arbitrariness of $\varepsilon>0$ implies that
$x_{0}\in\operatorname{supp}(\nu)$. ∎

######  Remark 4.5.

If we start the proof of Lemma 4.4 with the underlying space being a Banach
space $X$ which has type 2, then there exists a Gaussian measure $m$ on $X$
whose covariance operator is $R$, as defined in (2). Indeed, since $R$ is a
symmetric and positive operator it admits a square root: there exist some
separable Hilbert space $H$ and an operator $K:H\rightarrow X$ such that
$R=KK^{*}$ (see [3, Page 101]). Moreover, by the finite second-order moment
condition of $\mu$, the operator $K^{*}$ is an absolutely 2-summing operator
and hence such a Gaussian measure $m$ on $X$ exists by [3, Corollary 5.20].
However, in the Step 3 of the proof above the underlying space needs to have
cotype 2. Since the only spaces which are both of type 2 and of cotype 2 are
those which are isomorphic to a Hilbert space, the proof of Lemma 4.4 does not
extend outside of the Hilbertian setting.

We are now ready to prove Theorem 1.7.

###  4.2 Proof of Theorem 1.7

Let $T:H\rightarrow H$ be an operator on a complex separable Hilbert space
$H$. We already know that
$\operatorname{span}(\mathcal{E}(T))\subset\operatorname{URec}(T)\subset\operatorname{RRec}^{bo}(T)$,
so we just have to prove that

$\operatorname{RRec}^{bo}(T)\subset\overline{\operatorname{span}(\mathcal{E}(T))}.$

To see this, let $x_{0}\in\operatorname{RRec}^{bo}(T)\setminus\\{0\\}$ and let
$M>0$ be such that $\operatorname{Orb}(x_{0},T)$ is contained in $MB_{H}$, the
$\|\cdot\|$-closed ball of radius $M$ centred at $0$. If we denote by $w$ the
weak topology of $H$, we have the inclusion

$\overline{\operatorname{Orb}(x_{0},T)}^{w}\subset MB_{H}.$

By Theorem 2.3 there is a (non-trivial, because $x_{0}\neq 0$) $T$-invariant
probability measure $\mu_{x_{0}}$ on $X$ such that

$x_{0}\in\operatorname{supp}(\mu_{x_{0}})\subset\overline{\operatorname{Orb}(x_{0},T)}^{w},$

and hence

$\int_{H}\|z\|^{2}d\mu_{x_{0}}(z)=\int_{\operatorname{supp}(\mu_{x_{0}})}\|z\|^{2}d\mu_{x_{0}}(z)\leq
M^{2}<\infty.$

By Lemma 4.4 we get that
$x_{0}\in\operatorname{supp}(\mu_{x_{0}})\subset\overline{\operatorname{span}(\mathcal{E}(T))}$
as we wanted to show.

Suppose now that there is a countable set
$\\{x_{n}:n\in\mathbb{N}\\}\subset\operatorname{RRec}^{bo}(T)$ which is dense
in $H$. For each $n\in\mathbb{N}$ pick $M_{n}>0$ and $k_{n}\in\mathbb{N}$ such
that

$\overline{\operatorname{Orb}(x_{n},T)}^{w}\subset M_{n}B_{H}\quad\text{ and
}\quad 2^{n}M_{n}^{2}\leq 2^{k_{n}}.$

Applying Theorem 2.3 to each vector $x_{n}$ we obtain a sequence
$(\mu_{x_{n}})_{n\in\mathbb{N}}$ of $T$-invariant probability measures on $H$
such that
$x_{n}\in\operatorname{supp}(\mu_{x_{n}})\subset\overline{\operatorname{Orb}(x_{n},T)}^{w}$
for each $n\in\mathbb{N}$. Consider the measure

$\mu:=\sum_{n\in\mathbb{N}}\frac{\mu_{x_{n}}}{2^{k_{n}}},$

which is a (positive) $T$-invariant finite Borel measure on $H$ with full
support such that

$\int_{H}\|z\|^{2}d\mu(z)=\sum_{n\in\mathbb{N}}\frac{1}{2^{k_{n}}}\int_{H}\|z\|^{2}d\mu_{x_{n}}(z)\leq\sum_{n\in\mathbb{N}}\frac{M_{n}^{2}}{2^{k_{n}}}\leq
1.$

Normalizing $\mu$ we get a $T$-invariant probability measure with full support
and finite second-order moment.

Moreover, in both cases (a) and (b) we have: (i) $\Rightarrow$ (ii)
$\Rightarrow$ (iii) $\Rightarrow$ (iv) $\Rightarrow$ (v) $\Rightarrow$ (vi)
$\Rightarrow$ (vii) by definition; (vii) implies (viii) using Theorem 2.3 as
in the above arguments; and (viii) implies (i) by Lemma 4.4.∎

######  Remark 4.6.

The equalities
$\overline{\operatorname{span}(\mathcal{E}(T))}=\overline{\operatorname{URec}(T)}=\overline{\operatorname{RRec}^{bo}(T)}$
and hence the equivalences (i) $\Leftrightarrow$ (ii) $\Leftrightarrow$ (iii)
$\Leftrightarrow$ (iv) $\Leftrightarrow$ (v) $\Leftrightarrow$ (vi)
$\Leftrightarrow$ (vii) established in Theorem 1.7 are still true when the
underlying space $H$ is a complex non-separable Hilbert space. Since the
closed subspaces of a Hilbert space are again Hilbert spaces, the same
arguments as those used in Remark 3.4 apply. We loose again the measures
equivalences, i.e. statement (viii).

As mentioned in the Introduction and in Remark 4.5, the proof of Lemma 4.4 and
hence that of Theorem 1.7 do not extend outside of the Hilbertian setting.

We finish this section with the proof of Theorem 1.9, which concerns the
power-bounded operators on complex reflexive Banach spaces $X$. The proof
relies on the splitting theorem of Jacobs-Deleeuw-Glicksberg, and is really
specific to the setting of power-bounded operators. We follow the presentation
and notation of [31, Section 2.4]: if $\mathscr{S}$ is a semigroup of
$\mathcal{L}(X)$, we say that $\mathscr{S}$ is weakly almost periodic if for
any $x\in X$ the set $\mathscr{S}x=\\{Sx:S\in\mathscr{S}\\}$ has a $w$-compact
closure.

###  4.3 Proof of Theorem 1.9

Given a power-bounded operator $T:X\rightarrow X$ on a complex reflexive
Banach space $X$, we already know that
$\operatorname{span}(\mathcal{E}(T))\subset\operatorname{URec}(T)$ so we just
have to show the inclusion
$\operatorname{URec}(T)\subset\overline{\operatorname{span}(\mathcal{E}(T))}$.
We set $O(x):=\overline{\operatorname{Orb}(x,T)}^{w}$ for each $x\in X$.

Since $T$ is power-bounded, every $T$-orbit is bounded and has a $w$-compact
closure. Hence, by the Jacobs-Deleeuw-Glicksberg theorem [31, Section 2.4,
Theorem 4.4] applied to the (weakly almost periodic) abelian semigroup of
operators $\\{T^{n}:n\in\mathbb{N}_{0}\\}\subset\mathcal{L}(X)$, we obtain
that $X=X_{rev}\oplus X_{fl}$ where

$X_{rev}:=\\{x\in X:y\in O(x)\Rightarrow x\in O(y)\\}\quad\text{ and }\quad
X_{fl}:=\\{x\in X:0\in O(x)\\}.$

Moreover, by the second part of this same theorem [31, Section 2.4, Theorem
4.5] we also get that

$X_{rev}=\overline{\operatorname{span}(\mathcal{E}(T))}.$

Let us now show that $\operatorname{URec}(T)\subset X_{rev}$. Indeed, given
$x\in\operatorname{URec}(T)\setminus\\{0\\}$ we can consider the map
$T\arrowvert_{O(x)}:(O(x),w)\rightarrow(O(x),w)$ which is a $w$-compact
dynamical system. Since the weak topology is coarser than the norm topology we
have $x\in\operatorname{URec}(T\arrowvert_{O(x)})$ and hence by [22, Theorem
1.17] the system $T\arrowvert_{O(x)}$ is minimal so every
$T\arrowvert_{O(x)}$-orbit is dense in $O(x)$. Finally, given $y\in O(x)$ we
have

$O(y)=\overline{\operatorname{Orb}(y,T)}^{w}=\overline{\operatorname{Orb}(y,T\arrowvert_{O(x)})}^{w}=O(x)$

which implies that $x\in O(y)$. The arbitrariness of $y\in O(x)$ shows that
$x\in X_{rev}$.∎

##  5 Product dynamical systems

Given a property of a dynamical system $T:X\rightarrow X$, it is usual to ask
whether the product dynamical system $T\times T:X\times X\rightarrow X\times
X$ has the same property. Studied cases in linear dynamics are transitivity or
hypercyclicity (which gives us the concept of topological weak mixing), and in
general $\mathcal{F}$-transitivity or $\mathcal{F}$-hypercyclicity (see [7]
and [18]). Here we show that the above theorems still work for the product
systems.

######  Theorem 5.1 (From Reiterative to $\boldsymbol{N}$-Dimensional Frequent
Recurrence).

Let $N\in\mathbb{N}$ and suppose that for each $1\leq i\leq N$ there is a
Polish dynamical system $(X_{i},T_{i})$ such that $(X_{i},\tau_{X_{i}})$ can
be endowed with a Hausdorff topology $\tau_{i}$ which fulfills (I), (II), and
(III*) with respect to the map $T_{i}$ and the topology $\tau_{X_{i}}$. Then
for the product dynamical system $T:(X,\tau_{X})\rightarrow(X,\tau_{X})$,
where $\tau_{X}$ is the product topology of the $N$-th $\tau_{X_{i}}$
topologies, we have the equality

$\overline{\operatorname{FRec}(T)}^{\tau_{X}}=\prod_{i=1}^{N}\overline{\operatorname{RRec}(T_{i})}^{\tau_{X_{i}}}.$

In particular:

  1. (a)

The following statements are equivalent:

    1. (i)

$\operatorname{FRec}(T)\neq\varnothing$;

    2. (ii)

$\operatorname{UFRec}(T)\neq\varnothing$;

    3. (iii)

$\operatorname{RRec}(T)\neq\varnothing$;

    4. (iv)

$\operatorname{RRec}(T_{i})\neq\varnothing$ for every $1\leq i\leq N$.

  2. (b)

The following statements are equivalent:

    1. (i)

$T$ is frequently recurrent;

    2. (ii)

$T$ is $\mathcal{U}$-frequently recurrent;

    3. (iii)

$T$ is reiteratively recurrent;

    4. (iv)

$T_{i}$ is reiteratively recurrent for every $1\leq i\leq N$.

###### Proof.

We clearly have the inclusion

$\operatorname{FRec}(T)\subset\prod_{i=1}^{N}\operatorname{RRec}(T_{i}).$

Now given $\boldsymbol{x}_{0}=(x_{1},...,x_{N})\in X$ such that
$x_{i}\in\operatorname{RRec}(T_{i})$ for each $1\leq i\leq N$, let us show
that $\boldsymbol{x}_{0}\in\overline{\operatorname{FRec}(T)}^{\tau_{X}}$.
Applying Theorem 2.3 we obtain a $T_{i}$-invariant measure $\mu_{x_{i}}$ on
$X_{i}$ such that $x_{i}\in\operatorname{supp}(\mu_{x_{i}})$ for each $1\leq
i\leq N$. Since

$\mathscr{B}(X,\tau_{X})=\prod_{i=1}^{N}\mathscr{B}(X,\tau_{X_{i}}),$

we can consider the product measure
$\mu_{\boldsymbol{x}_{0}}:=\prod_{i=1}^{N}\mu_{x_{i}}$ on the product space
$X$, which is a $T$-invariant measure (see [36, Theorem 1.1 and Definition
1.2]) for which
$\boldsymbol{x}_{0}\in\operatorname{supp}(\mu_{\boldsymbol{x}_{0}})$. Applying
now Lemma 3.1 we deduce that
$\boldsymbol{x}_{0}\in\overline{\operatorname{FRec}(T)}^{\tau_{X}}$. ∎

The following immediate corollaries yield a product version of Theorem 1.3:

######  Corollary 5.2.

Let $N\in\mathbb{N}$ and consider for each $1\leq i\leq N$ an adjoint operator
$T_{i}:\leavevmode\nobreak\ X_{i}\rightarrow\leavevmode\nobreak\ X_{i}$ on a
separable dual Banach space $X_{i}$. Then, for the direct sum operator
$T=T_{1}\oplus\cdots\oplus T_{N}:X\rightarrow X$ on the direct sum space
$X=X_{1}\oplus\cdots\oplus X_{N}$, we have the equality

$\overline{\operatorname{FRec}(T)}=\prod_{i=1}^{N}\overline{\operatorname{RRec}(T_{i})}.$

In particular, the following statements are equivalent:

  1. (i)

$T$ is frequently recurrent;

  2. (ii)

$T$ is $\mathcal{U}$-frequently recurrent;

  3. (iii)

$T$ is reiteratively recurrent;

  4. (iv)

$T_{i}$ is reiteratively recurrent for every $1\leq i\leq N$.

Moreover, the result holds whenever some of the $T_{i}$ are operators defined
on some reflexive Banach spaces $X_{i}$.

In the statement above, and whenever we consider a direct sum space
$X_{1}\oplus\cdots\oplus X_{N}$, one can use any norm defining the usual
product topology on $X_{1}\oplus\cdots\oplus X_{N}$ (see Theorem 5.8).

######  Definition 5.3.

Let $(X,T)$ be a linear dynamical system and let $n\in\mathbb{N}$. We will
denote by $T_{n}:X^{n}\rightarrow X^{n}$ the $n$-fold direct sum of $T$ with
itself, i.e. the dynamical system

$T_{n}:=\underbrace{T\oplus\cdots\oplus T}_{n}:\underbrace{X\oplus\cdots\oplus
X}_{n}\longrightarrow\underbrace{X\oplus\cdots\oplus X}_{n},$

where $X^{n}:=\underbrace{X\oplus\cdots\oplus X}_{n}$ is the $n$-fold direct
sum of $X$ with itself.

######  Corollary 5.4.

Let $T:X\rightarrow X$ be an adjoint operator on a separable dual Banach space
$X$. Then the following statements are equivalent:

  1. (i)

for every $n\in\mathbb{N}$, $T_{n}$ is frequently recurrent;

  2. (ii)

for every $n\in\mathbb{N}$, $T_{n}$ is $\mathcal{U}$-frequently recurrent;

  3. (iii)

for every $n\in\mathbb{N}$, $T_{n}$ is reiteratively recurrent;

  4. (iv)

$T$ is reiteratively recurrent.

In particular, the result holds whenever $T$ is an operator on a reflexive
Banach space $X$.

As a consequence of the above fact we can prove some results related with
hypercyclicity. We start with an independent proof of [18, Theorem 2.5 and
Corollary 2.6] for the particular case of the reiteratively hypercyclic
(adjoint) operators:

######  Theorem 5.5.

Let $T:X\rightarrow X$ be a reiteratively hypercyclic adjoint operator on a
separable dual Banach space $X$. Then for every $n\in\mathbb{N}$ the operator
$T_{n}$ is reiteratively hypercyclic and frequently recurrent. In particular,
the result holds whenever $T$ is an operator on a separable reflexive Banach
space $X$.

###### Proof.

Let $n\in\mathbb{N}$. Since $T$ is reiteratively hypercyclic we know that:

  1. (a)

$T$ is topologically weakly mixing (see [6, Page 548]), and hence $T_{n}$ is
topologically transitive, and in particular hypercyclic;

  2. (b)

$T$ is reiteratively recurrent, and by the above results $T_{n}$ is frequently
recurrent, in particular reiteratively recurrent.

By [10, Theorem 2.1], reiterative recurrence plus hypercyclicity imply
reiterative hypercyclicity. We deduce that $T_{n}$ is reiteratively
hypercyclic and frequently recurrent. ∎

If we start just with reiterative recurrence, having a dense set of orbits
converging to $0$ implies a strong notion of hypercyclicity:

######  Theorem 5.6.

Let $T:X\rightarrow X$ be an adjoint operator on a separable dual Banach space
$X$. Suppose that there is a dense set $X_{0}\subset X$ such that $T^{k}x\to
0$ as $k\to\infty$ for each $x\in X_{0}$. The following statements are
equivalent:

  1. (i)

for every $n\in\mathbb{N}$, $T_{n}$ is $\mathcal{U}$-frequently hypercyclic
and frequently recurrent;

  2. (ii)

$T$ is reiteratively recurrent.

In particular, the result holds if $T$ is an operator on a separable reflexive
Banach space $X$.

###### Proof.

Clearly (i) implies (ii) even if $T:X\rightarrow X$ is not a linear map. If we
suppose (ii) and we fix $n\in\mathbb{N}$, by the above results we get that
$T_{n}$ is frequently recurrent and in particular $\mathcal{U}$-frequently
recurrent. Let $Y_{0}:=X_{0}\oplus\cdots\oplus X_{0}$ the $n$-direct sum of
the set $X_{0}$. Then $Y_{0}$ is a dense subset of the $n$-fold direct sum
$X^{n}$ and every orbit of a point of $Y_{0}$ converges to $(0,...,0)\in
X^{n}$. By [10, Theorem 2.12], the existence of $Y_{0}$ and the
$\mathcal{U}$-frequent recurrence imply that $T_{n}$ is
$\mathcal{U}$-frequently hypercyclic. ∎

It would be interesting to change the assumption of $\mathcal{U}$-frequent
hypercyclicity in the above statement into that of frequent hypercyclicity.
However, as exposed in [10, Question 2.13], the following is an open problem:

######  Question 5.7 ([10, Question 2.13]).

Let $T$ be a frequently recurrent operator admitting a dense set of vectors
with orbit convergent to $0$. Is $T$ is frequently hypercyclic?

If now we focus on Theorems 1.7 and 1.9, their generalizations for product
linear dynamical systems follow in a much easier way, since any $N$-tuple
formed by unimodular eigenvectors is a linear combination of such vectors for
the direct sum map:

######  Theorem 5.8.

Let $N\in\mathbb{N}$ and suppose that for each $1\leq i\leq N$:

  1. (a)

we have an operator $T_{i}:H_{i}\rightarrow H_{i}$ on a complex Hilbert space
$H_{i}$. Then, for the direct sum operator $T=T_{1}\oplus\cdots\oplus
T_{N}:H\rightarrow H$ on the direct sum Hilbert space
$H=H_{1}\oplus\cdots\oplus H_{N}$, we have the equality:

$\overline{\operatorname{span}(\mathcal{E}(T))}=\displaystyle\prod_{i=1}^{N}\overline{\operatorname{RRec}^{bo}(T_{i})},$

In particular, the following statements are equivalent:

    1. (i)

the set $\operatorname{span}(\mathcal{E}(T))$ is dense in $H$;

    2. (ii)

the set $\operatorname{RRec}^{bo}(T_{i})$ is dense in $H_{i}$ for every $1\leq
i\leq N$.

  2. (b)

we have a power-bounded operator $T_{i}:X_{i}\rightarrow X_{i}$ on a complex
reflexive Banach space $X_{i}$. Then, for the direct sum operator
$T=T_{1}\oplus\cdots\oplus T_{N}:X\rightarrow X$ on the direct sum space
$X=X_{1}\oplus\cdots\oplus X_{N}$, we have the equality

$\overline{\operatorname{span}(\mathcal{E}(T))}=\prod_{i=1}^{N}\overline{\operatorname{URec}(T_{i})}.$

In particular, the following statements are equivalent:

    1. (i)

the set $\operatorname{span}(\mathcal{E}(T))$ is dense in $X$;

    2. (ii)

the set $\operatorname{URec}(T_{i})$ is dense in $X_{i}$ for every $1\leq
i\leq N$.

###### Proof.

Since the vector $(0,...,0,x_{i},0,...,0)$ belongs to $\mathcal{E}(T)$
whenever $x_{i}\in\mathcal{E}(T_{i})$ for each $1\leq i\leq N$, it is enough
to apply Theorems 1.7 and 1.9 to each operator $T_{i}$. ∎

Finally we get the desired generalization of Theorems 1.7 and 1.9:

######  Corollary 5.9.

Let $T\in\mathcal{L}(H)$ where $H$ is a complex Hilbert space. The following
statements are equivalent:

  1. (i)

for every $n\in\mathbb{N}$, the set $\operatorname{span}(\mathcal{E}(T_{n}))$
is dense in $H^{n}$;

  2. (ii)

for every $n\in\mathbb{N}$, $T_{n}$ is $\Delta^{*}$-recurrent;

  3. (iii)

for every $n\in\mathbb{N}$, $T_{n}$ is $\mathcal{IP}^{*}$-recurrent;

  4. (iv)

for every $n\in\mathbb{N}$, $T_{n}$ is uniformly recurrent;

  5. (v)

for every $n\in\mathbb{N}$, the set $\operatorname{FRec}^{bo}(T_{n})$ is dense
in $H^{n}$;

  6. (vi)

for every $n\in\mathbb{N}$, the set $\operatorname{UFRec}^{bo}(T_{n})$ is
dense in $H^{n}$;

  7. (vii)

for every $n\in\mathbb{N}$, the set $\operatorname{RRec}^{bo}(T_{n})$ is dense
in $H^{n}$;

  8. (viii)

the set $\operatorname{RRec}^{bo}(T)$ is dense in $H$.

######  Corollary 5.10.

Let $T:X\rightarrow X$ be a power-bounded operator on a complex reflexive
Banach space $X$. The following statements are equivalent:

  1. (i)

for every $n\in\mathbb{N}$, the set $\operatorname{span}(\mathcal{E}(T_{n}))$
is dense in $X^{n}$;

  2. (ii)

for every $n\in\mathbb{N}$, $T_{n}$ is $\Delta^{*}$-recurrent;

  3. (iii)

for every $n\in\mathbb{N}$, $T_{n}$ is $\mathcal{IP}^{*}$-recurrent;

  4. (iv)

for every $n\in\mathbb{N}$, $T_{n}$ is uniformly recurrent;

  5. (v)

$T$ is uniformly recurrent.

##  6 Inverse dynamical systems

As in the case of products, given a (topological) dynamical system
$T:X\rightarrow X$ with some property, it is natural to ask whether the
inverse dynamical system $T^{-1}:X\rightarrow X$ (if it exists and is
continuous) has the same property. This is true for hypercyclicity and
reiterative hypercyclicity (see [9]), but it fails for $\mathcal{U}$-frequent
hypercyclicity (see [33]) and frequent hypercyclicity (see [34]). It is also
known that the inverse of a frequently hypercyclic operator is
$\mathcal{U}$-frequently hypercyclic (see [5, Proposition 20]).

If we focus on recurrence properties, the inverse of a recurrent operator is
again recurrent as [14, Proposition 2.6] shows. A simpler proof (in a
transitive style) of that fact would be:

######  Proposition 6.1 ([14, Proposition 2.6]).

Let $T:X\rightarrow X$ be an invertible operator. Then $T$ is recurrent if and
only if so is $T^{-1}$.

###### Proof.

By [14, Proposition 2.1] the result follows from the equivalence

$T^{n}(U)\cap U\neq\varnothing\text{ if and only if }U\cap
T^{-n}(U)\neq\varnothing,$

valid for any non-empty open subset $U$ of $X$. ∎

However, it is also shown in [14, Remark 2.7] that the sets $\textup{Rec}(T)$
and $\textup{Rec}(T^{-1})$ may not be equal in spite of the fact that their
closures coincide. For general $\mathcal{F}$-recurrence notions the following
problem was proposed in [10]:

######  Question 6.2 ([10, Question 2.14]).

Let $T$ be an invertible operator. If $T$ is reiteratively (resp.
$\mathcal{U}$-frequently, frequently or uniformly) recurrent, does $T^{-1}$
has the same property?

In fact, we could ask the same question for $\mathcal{IP}^{*}$,
$\Delta^{*}$-recurrence and unimodular eigenvalues. However, for the latest
the linearity is enough to show it since if

$Tx=\lambda x\text{ for some }\lambda\in\mathbb{T}\text{, then
}T^{-1}x=\overline{\lambda}(T^{-1}\lambda x)=\overline{\lambda}x,$

so clearly
$\operatorname{span}(\mathcal{E}(T))=\operatorname{span}(\mathcal{E}(T^{-1}))$.
In order to answer Question 6.2 in our dual/reflexive or Hilbertian setting we
just have to recall the following trivial fact: given a homeomorphism
$T:X\rightarrow X$ on a Polish space $X$ and a Borel measure $\mu$ on $X$,
$\mu$ is $T$-invariant if and only if it is $T^{-1}$-invariant.

######  Theorem 6.3 (From Reiterative to Inverse Frequent Recurrence).

Let $T:X\rightarrow X$ be a homeomorphism of the Polish space $(X,\tau_{X})$,
and assume that $X$ is endowed with a Hausdorff topology $\tau_{1}$ which
fulfills (I) for $T$, (II), and (III*). Then we have

$\overline{\operatorname{FRec}(T)}^{\tau_{X}}=\overline{\operatorname{RRec}(T)}^{\tau_{X}}\subset\overline{\operatorname{FRec}(T^{-1})}^{\tau_{X}}\subset\overline{\operatorname{RRec}(T^{-1})}^{\tau_{X}}.$

Moreover:

  1. (a)

If $T$ is reiteratively recurrent then $T^{-1}$ is frequently recurrent.

  2. (b)

If $\operatorname{RRec}(T)\neq\varnothing$ then
$\operatorname{FRec}(T)\cap\operatorname{FRec}(T^{-1})\neq\varnothing$.

  3. (c)

If $X$ can be endowed with a Hausdorff topology $\tau_{2}$ which fulfills (I)
for $T^{-1}$, (II), and (III*), then the above inclusions are equalities and
$T$ is reiteratively (and hence frequently) recurrent if and only if so is
$T^{-1}$.

###### Proof.

The equality is shown in Theorem 3.3. The first inclusion follows from Lemma
3.1 applied to the measures constructed with Theorem 2.3 for each point of
$\operatorname{RRec}(T)$, using the fact that they are $T^{-1}$-invariant. The
second inclusion follows by definition. Moreover, if there exists
$x_{0}\in\operatorname{RRec}(T)$ and if we take the invariant probability
measure $\mu_{x_{0}}$ on $X$ constructed with Theorem 2.3, then by Lemma 3.1
we have

$\mu_{x_{0}}(\operatorname{FRec}(T))=1=\mu_{x_{0}}(\operatorname{FRec}(T^{-1}))$

which implies that
$\operatorname{FRec}(T)\cap\operatorname{FRec}(T^{-1})\neq\varnothing$.
Finally, if such a topology $\tau_{2}$ exists we can apply the first part of
the result to $T^{-1}$ obtaining
$\overline{\operatorname{RRec}(T^{-1})}^{\tau_{X}}\subset\overline{\operatorname{FRec}(T)}^{\tau_{X}}$.
∎

As a corollary of the above theorem, and using the arguments from Theorem 1.3
we have:

######  Corollary 6.4.

Let $T:X\rightarrow X$ be an invertible adjoint operator on a separable dual
Banach space $X$. Then we have the equalities

$\overline{\operatorname{RRec}(T)}=\overline{\operatorname{FRec}(T)}=\overline{\operatorname{FRec}(T^{-1})}=\overline{\operatorname{RRec}(T^{-1})}.$

Moreover:

  1. (a)

$T$ is reiteratively (and hence frequently) recurrent if and only if so is
$T^{-1}$.

  2. (b)

If $\operatorname{RRec}(T)\setminus\\{0\\}\neq\varnothing$ then
$[\operatorname{FRec}(T)\cap\operatorname{FRec}(T^{-1})]\setminus\\{0\\}\neq\varnothing$.

In particular, the result holds whenever $T$ is an operator on a reflexive
Banach space $X$.

###### Proof.

Let $S:Y\rightarrow Y$ be an operator on a Banach space $Y$ such that
$Y^{*}=X$ and $S^{*}=T$. It is a known fact that $T$ is invertible if and only
if $S$ is invertible, and in this case, $T^{-1}=(S^{-1})^{*}$, so $T^{-1}$ is
also an adjoint operator on the separable Banach space $X$ and hence it is
$w^{*}$-continuous. The result follows from the above theorem applied to
$T:X\rightarrow X$, $(X,\|\cdot\|)$ and the topology $w^{*}$. ∎

With the above fact we give an alternative prove of [9, Theorem 3.6] for
adjoint operators:

######  Theorem 6.5.

Let $T:X\rightarrow X$ be an invertible adjoint operator on a separable dual
Banach space. If $T$ is reiteratively hypercyclic (and hence frequently
recurrent) then so is $T^{-1}$. In particular, the result holds whenever $T$
is an operator on a separable reflexive Banach space $X$.

###### Proof.

By the above theorem $T^{-1}$ is frequently recurrent and in particular
reiteratively recurrent. Since hypercyclicity (or transitivity) is also
preserved by taking the inverse system, [10, Theorem 2.1] implies that
$T^{-1}$ is reiteratively hypercyclic. ∎

We cannot change the assumption of reiterative hypercyclicity in the statement
of Theorem 6.5 above into the assumption of $\mathcal{U}$-frequent
hypercyclicity since there are invertible $\mathcal{U}$-frequently hypercyclic
operators on $\ell^{p}(\mathbb{N})$ ($1\leq p<\infty$) whose inverse is not
$\mathcal{U}$-frequently hypercyclic (see [33]). However, it would be
interesting to know whether it is possible to change the assumption of
reiterative hypercyclicity into that of frequent hypercyclicity: even though
it is known that there are invertible frequently hypercyclic operators on
$\ell^{1}(\mathbb{N})$ whose inverse is not frequently hypercyclic (see [34]),
one can check that these are not adjoint operators and moreover by [5,
Proposition 20] the inverse of a frequently hypercyclic operator is always
$\mathcal{U}$-frequently hypercyclic.

All the counterexamples mentioned here are C-type operators, which were
introduced for the first time in [32] and further developed in [26, 33, 34],
so a possible counterexample for the frequent hypercyclicity case could arise
from those operators. If, on the other hand, one wishes to prove an analogue
of Theorem 6.5 for the frequent hypercyclicity case in our dual/reflexive
framework, one cannot take a similar approach since there are chaotic
operators, which are in particular frequently recurrent and hypercyclic, but
not $\mathcal{U}$-frequently hypercyclic (see [32] and [26]) and hence not
frequently hypercyclic.

If we now focus on uniform, $\mathcal{IP}^{*}$ and $\Delta^{*}$-recurrence,
Theorems 1.7 and 1.9 combined with the equality
$\operatorname{span}(\mathcal{E}(T))=\operatorname{span}(\mathcal{E}(T^{-1}))$
give us the following:

######  Corollary 6.6.

Let $T:H\rightarrow H$ be an invertible operator on a complex Hilbert space
$H$. Then we have the equalities

$\overline{\operatorname{RRec}^{bo}(T)}=\overline{\operatorname{span}(\mathcal{E}(T))}=\overline{\operatorname{span}(\mathcal{E}(T^{-1}))}=\overline{\operatorname{RRec}^{bo}(T^{-1})}.$

In particular, $T$ is uniformly (and hence $\mathcal{IP}^{*}$ and
$\Delta^{*}$) recurrent if and only if so is $T^{-1}$.

######  Corollary 6.7.

Let $T:X\rightarrow X$ be an invertible operator on a complex reflexive space
$X$. If $T$ is power-bounded, then we have

$\overline{\operatorname{URec}(T)}=\overline{\operatorname{span}(\mathcal{E}(T))}=\overline{\operatorname{span}(\mathcal{E}(T^{-1}))}\subset\overline{\operatorname{URec}(T^{-1})}.$

In particular, if $T$ is uniformly recurrent then
$\operatorname{span}(\mathcal{E}(T^{-1}))$ is a dense set in $X$. Moreover, if
$T^{-1}$ is also power-bounded then the above inclusion is an equality and $T$
is uniformly (and hence $\mathcal{IP}^{*}$ and $\Delta^{*}$) recurrent if and
only if so is $T^{-1}$.

##  7 How typical is a reiteratively recurrent operator?

Let $H$ be a complex separable Hilbert space. For any $M>0$, denote by
$\mathcal{L}_{M}(H)$ the set of bounded operators $T\in\mathcal{L}(H)$ such
that $\|T\|\leq M$. Our aim in this short section is to present a result
pertaining to the typicality of reiteratively recurrent operators of
$\mathcal{L}_{M}(H)$, with $M>1$, for one of the two (Polish) topologies SOT
and SOT∗. The framework that we use here is presented in detail in [26,
Chapters 2 and 3], so we will be rather brief in our presentation and refer
the readers to the works [25], [26] or [27] for more on typical properties of
operators on Hilbert or Banach spaces.

We recall that the Strong Operator Topology (SOT) on $\mathcal{L}(H)$ is
defined as follows: any $T_{0}\in\mathcal{L}(H)$ has a SOT-neighbourhood basis
consisting of sets of the form

$U_{T_{0},x_{1},...,x_{s},\varepsilon}:=\\{T\in\mathcal{L}(H):\|(T-T_{0})x_{i}\|<\varepsilon\text{
for }i=1,...,s\\},$

where $x_{1},...,x_{s}\in H$ and $\varepsilon>0$.

The Strong∗ Operator Topology (SOT∗) is the “self-adjoint” version of SOT: a
basis of SOT∗-neighbourhoods of $T_{0}\in\mathcal{L}(H)$ is provided by the
sets of the form

$V_{T_{0},x_{1},...,x_{s},\varepsilon}:=\\{T\in\mathcal{L}(H):\|(T-T_{0})x_{i}\|<\varepsilon\text{
and }\|(T-T_{0})^{*}x_{i}\|<\varepsilon\text{ for }i=1,...,s\\},$

where $x_{1},...,x_{s}\in H$ and $\varepsilon>0$.

It is easily shown that $(\mathcal{L}_{M}(H),\text{SOT})$ and
$(\mathcal{L}_{M}(H),\text{SOT}^{*})$ are Polish spaces for any $M>0$ (see
[35, Section 4.6.2]), and hence, a property of operators
$T\in\mathcal{L}_{M}(H)$ will be called typical if the set of operators
fulfilling it is co-meager (i.e. contains a dense $G_{\delta}$ set), and
atypical if its negation is typical. Following the notation used in [26] we
can write

$\displaystyle\textup{HC}(H)$ $\displaystyle:=$
$\displaystyle\\{T\in\mathcal{L}(H)\text{ hypercyclic}\\};$
$\displaystyle\text{INV}(H)$ $\displaystyle:=$
$\displaystyle\\{T\in\mathcal{L}(H)\text{ admitting a non-trivial invariant
measure}\\};$

and for each $M>1$ the set $\textup{HC}_{M}(H)$ is defined as
$\textup{HC}(H)\cap\mathcal{L}_{M}(H)$. Following the spirit of this study we
introduce the following notation:

$\displaystyle\operatorname{RHC}(H)$ $\displaystyle:=$
$\displaystyle\\{T\in\mathcal{L}(H)\text{ reiteratively hypercyclic}\\};$
$\displaystyle\operatorname{RRec}(H)$ $\displaystyle:=$
$\displaystyle\\{T\in\mathcal{L}(H)\text{ reiteratively recurrent}\\};$
$\displaystyle\operatorname{RRec}^{\neq\varnothing}(H)$ $\displaystyle:=$
$\displaystyle\\{T\in\mathcal{L}(H):\operatorname{RRec}(T)\setminus\\{0\\}\neq\varnothing\\};$

and as it was done previously for the set $\textup{HC}(H)$, for each $M>1$, we
will denote the respective bounded versions of these sets of operators by
$\operatorname{RHC}_{M}(H)$, $\operatorname{RRec}_{M}(H)$ and
$\operatorname{RRec}_{M}^{\neq\varnothing}(H)$.

Let us first recall that a SOT-typical operator in $\mathcal{L}_{M}(H)$, for
$M>1$, has any form of recurrence one can wish for: by [17] it is known that a
typical $T\in\mathcal{L}_{M}(H)$ is unitarily similar to $MB_{\infty}$, where
$B_{\infty}$ denotes the backward shift of infinite multiplicity on
$\ell^{2}(\mathbb{N},\ell^{2}(\mathbb{N}))$, and $MB_{\infty}$ is such that
the linear span of its unimodular eigenvectors is dense in
$\ell^{2}(\mathbb{N},\ell^{2}(\mathbb{N}))$, see [17, Theorem 5.2].

With respect to the topology SOT∗, it is proved in [26, Theorem 2.29] that for
every $M>1$ the set $\textup{HC}_{M}(H)\setminus\textup{INV}(H)$ is co-meager
in the space $(\textup{HC}_{M}(H),\textup{SOT}^{*})$. In other words, a
SOT∗-typical hypercyclic operator on $H$ admits no non-trivial invariant
measure. Combining Theorem 2.3 of the present work with [26, Theorem 2.29] we
obtain:

######  Corollary 7.1.

For every $M>1$, the set $\operatorname{RRec}^{\neq\varnothing}_{M}(H)$ is
meager in $(\mathcal{L}_{M}(H),\textup{SOT}^{*})$. In other words, a
SOT∗-typical operator on $H$ has no non-zero reiteratively recurrent point.

Moreover, since [10, Theorem 2.1] implies that reiterative recurrence plus
hypercyclicity is equivalent to reiterative hypercyclicity, we have that
$\operatorname{RHC}(H)=\operatorname{RRec}(H)\cap\textup{HC}(H)$. Using
Corollary 7.1 we can improve [26, Corollary 2.36] in terms of reiterative
recurrence:

######  Corollary 7.2.

For every $M>1$, the set
$\operatorname{RRec}^{\neq\varnothing}_{M}(H)\cap\textup{HC}_{M}(H)$ is meager
in the space $(\textup{HC}_{M}(H),\textup{SOT}^{*})$. In particular, the set
$\operatorname{RHC}_{M}(H)$ is meager in
$(\textup{HC}_{M}(H),\textup{SOT}^{*})$. In other words, a SOT∗-typical
hypercyclic operator on $H$ does not admit any non-zero reiteratively
recurrent point, and, in particular, is not reiteratively hypercyclic.

##  8 Open problems

In this section we gather some possibly interesting open questions and a few
comments related to them. We start by Questions 1.6 and 1.8, already stated in
Subsection 1.3, which we recall here with some extra generality:

######  Question 8.1 (Question 1.8).

Let $T$ be a uniformly recurrent operator on a complex Fréchet space $X$. Is
$\operatorname{span}(\mathcal{E}(T))$ a dense set in $X$? What about the cases
where $T$ is an adjoint operator on a separable dual Banach space or where $X$
is a reflexive Banach space?

######  Question 8.2 (From [10, Question 6.3] and Question 1.6).

Does there exist an operator (possibly on a Fréchet space) which is uniformly
recurrent but not $\Delta^{*}$-recurrent? What about distinguishing uniform
recurrence from $\mathcal{IP}^{*}$-recurrence?

Note that these two questions make sense in the more general context of
complex Fréchet spaces, and in fact both questions are still unsolved for that
rather general class of spaces. It is clear that, in every possible complex
context, a positive answer to Question 8.1 implies a negative one to Question
8.2. Moreover, it would even imply a negative answer for the real case of
Question 8.2: given any uniformly recurrent real linear dynamical system we
could consider its complexification, and by the product-arguments used for
Theorem 5.8 we would get unimodular eigenvectors and hence
$\Delta^{*}$-recurrence; the initial real dynamical system could possibly not
contain the obtained unimodular eigenvectors, but the real and complex parts
of such vectors would clearly be $\Delta^{*}$-recurrent for the original real
system. It is worth mentioning that uniform and $\mathcal{IP}^{*}$-recurrence
are completely distinguished for compact dynamical systems (see the
construction from [19], its properties in [11] and use [22, Theorems 1.15 and
9.12]), so the question is if the linearity avoids that distinction.

The technique used in the proof of Theorem 1.7 (via Gaussian measures) is very
different from the one used in Theorem 1.9 (via the Jacobs-Deleeuw-Glicksberg
theorem). Indeed we loose the contact with measures and the unimodular
eigenvectors are obtained from a totally different construction (see [31,
Section 2.4]). It seems to us that a more general “eigenvectors’ constructing
machine”, not restricted to the measures or power-bounded assumptions, should
be developed in order to provide a better answer to Question 8.1. What we know
for the moment, leaving apart the power-bounded case which seems very
specific, is the following:

######  Proposition 8.3.

Let $T\in\mathcal{L}(H)$, where $H$ is a complex separable Hilbert space.
Given a $T$-invariant $w$-compact subset $K$ of $H$ for which $0\notin K$, we
have

$\overline{\operatorname{span}(\mathcal{E}(T))}\cap
K\neq\varnothing\quad\text{ and in particular
}\quad\mathcal{E}(T)\neq\varnothing.$

###### Proof.

We have that $T\arrowvert_{K}:(K,w)\rightarrow(K,w)$ is a $w$-compact
dynamical system, so it admits a $T\arrowvert_{K}$-invariant probability
measure $\mu$ on $K$ (see [22, Page 62]). Since the norm topology and the weak
topology on $H$ have the same Borel sets, we can extend the measure $\mu$ into
a Borel probability measure on the whole space $H$ (still denoted by $\mu$)
using the formula

$\mu(A):=\mu(K\cap A)\quad\text{ for every Borel set }A\in\mathscr{B}(H).$

Note that $\mu$ is $T$-invariant. We deduce that:

  1. (a)

$\mu$ is non-trivial, since $0\notin K$;

  2. (b)

$\mu$ has a finite second-order moment, since $\operatorname{supp}(\mu)\subset
K$.

Lemma 4.4 implies that $\overline{\operatorname{span}(\mathcal{E}(T))}\cap
K\neq\varnothing$ and in particular $\mathcal{E}(T)\neq\varnothing$. ∎

######  Remark 8.4.

Let $H$ be a complex separable Hilbert space. Since the set
$\overline{\operatorname{Orb}(x,T)}^{w}$ is a $T$-invariant $w$-compact subset
of $H$ for any point $x\in H$ with bounded $T$-orbit, the arguments of the
above proposition imply that for any $M>1$, a SOT∗-typical operator
$T\in\mathcal{L}_{M}(H)$ has the property that every bounded orbit of $T$
contains $0$ in its weak closure.

######  Proposition 8.5.

Let $T$ be an adjoint operator on a complex separable dual Banach space $X$.
Let $n\in\mathbb{N}$ and $\lambda\in\mathbb{T}$. Given a $[\lambda
T]^{n}$-invariant $w^{*}$-compact and convex subset $K$ of $H$ for which
$0\notin K$, we have

$\mathcal{E}(T)\cap\operatorname{span}(\operatorname{Orb}(x,T))\neq\varnothing\quad\text{
for some }x\in K,$

and in particular $\mathcal{E}(T)\neq\varnothing$.

###### Proof.

By the Schauder’s Fixed-Point theorem there is $x\in K$ for which the identity
$[\lambda T]^{n}x=x$ holds. Taking $\alpha=\lambda^{-n}\in\mathbb{T}$ we get
that $(\alpha-T^{n})x=0$. If we split the polynomial
$(\alpha-z^{n})\in\mathbb{C}[z]$ we have

$(\alpha-z^{n})=\prod_{i=1}^{n}(\alpha_{i}-z),$

where the $\alpha_{i}$’s are distinct $n$-th roots of $\alpha$ in
$\mathbb{T}$. Considering the vectors

$y_{0}:=x\quad\text{ and }\quad
y_{j}:=(\alpha_{j}-T)y_{j-1}=\prod_{i=1}^{j}(\alpha_{i}-T)x\quad\text{ for
each }1\leq j\leq n,$

we have $y_{0}\neq 0$ since $0\notin K$, but $y_{n}=(\alpha-T^{n})x=0$. Then
for some $0\leq k\leq n-1$ we have that
$y_{k}\in\mathcal{E}(T)\cap\operatorname{span}(\operatorname{Orb}(x,T))$. In
particular $\mathcal{E}(T)\neq\varnothing$. ∎

Another natural question concerning Theorem 1.7 is the relevance, in
assertions (v) to (vii) of both parts (a) and (b), of the assumption that the
vectors under consideration have bounded orbit. This fact is used in order to
ensure that the invariant measures, which by Theorem 2.3 can be constructed
from each reiteratively recurrent vector, have a finite second-order moment.
To omit this boundedness assumption (or weak versions of it) seems to require
new ideas. We recall here the following open problem from [26]:

######  Question 8.6 ([26, Question 8.3]).

Does there exist an operator on a complex separable Hilbert space admitting a
non-trivial invariant probability measure but no eigenvalues?

The following product and inverse questions also remain open:

######  Question 8.7.

Let $T$ be an operator on a Fréchet space $X$. If $T$ is reiteratively (resp.
$\mathcal{U}$-frequently, frequently, uniformly) recurrent, does the $n$-fold
direct sum $T\oplus\cdots\oplus T$, $n\geq 2$, have the same property?

######  Question 8.8 (From [10, Question 2.14] and Question 6.2).

Let $T$ be an invertible operator on a Fréchet space $X$. If $T$ is
reiteratively (resp. $\mathcal{U}$-frequently, frequently, uniformly,
$\mathcal{IP}^{*}$ or $\Delta^{*}$) recurrent, does $T^{-1}$ have the same
property?

It is worth mentioning that Question 8.7 is also open for usual recurrence, as
defined in Subsection 1.1, and it seems to be a non-trivial question. For the
$\mathcal{IP}^{*}$ and $\Delta^{*}$ cases, the fact that such families are
filters (see [8]) implies a positive answer.

As mentioned in [10], the set of periodic points $\operatorname{Per}(T)$ of an
operator $T$ has the property that $\operatorname{Per}(T)$ is either equal to
$X$ or is a meager set (by the Baire Category theorem, either
$\operatorname{Per}(T)$ is of first category or else $T^{n}=I$ for some
$n\in\mathbb{N}$). The same phenomenon happens (at least when $X$ is a Banach
space) with the set of uniformly recurrent vectors $\operatorname{URec}(T)$,
since by [10, Corollary 3.2] if $\operatorname{URec}(T)$ is co-meager in $X$
then $T$ is power-bounded and $\operatorname{URec}(T)=X$. This motivates the
following question:

######  Question 8.9 ([10, Question 2.9]).

Let $T$ be an operator on a Fréchet space $X$. Do we always have that either
$\operatorname{FRec}(T)=X$ or $\operatorname{FRec}(T)$ is a meager set?

This seems to be a non-trivial question even in the dual/reflexive setting,
since the frequently recurrent points obtained in our construction form a
“big” set with respect to a certain invariant measure, and usually this has
nothing to do with the “bigness” from the topological point of view (i.e. in
the Baire Category sense). In fact, given any chaotic operator $T:X\rightarrow
X$ (i.e. hypercyclic with a dense set of periodic points), it admits an
invariant probability measure $\mu$ on $X$ with full support (see [28,
Corollary 3.6]) and hence $\mu(\operatorname{FRec}(T))=1$ by Lemma 3.1.
However, since $T$ is hypercyclic we have that $\operatorname{FRec}(T)$ is a
meager set, otherwise by [10, Theorem 2.7] the set
$\operatorname{FHC}(T)=\operatorname{FRec}(T)\cap\textup{HC}(T)$ would be co-
meager contradicting [5, Corollary 19].

## References

  * [1] F. Bayart and S. Grivaux, Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083–5117.

  * [2] F. Bayart and S. Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94 (2007), 181–210.

  * [3] F. Bayart and É. Matheron, Dynamics of linear operators. Cambridge University Press, 2009.

  * [4] F. Bayart and É. Matheron, Mixing operators and small subsets of the circle. J. für die Reine und Angew. Math., vol. 2016, no. 715, 2016, pp. 75–123.

  * [5] F. Bayart and I. Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergod. Theory Dyn. Syst.. 35 (2015), 691–709.

  * [6] J. Bès, Q. Menet, A. Peris and Y. Puig, Recurrence properties of hypercyclic operators. Math. Ann., 366, no. 1-2, 545–572, 2016.

  * [7] J. Bès, Q. Menet, A. Peris and Y. Puig, Strong transitivity properties for operators. J. Differ. Equ., 266, no. 2-3, 1313–1337, 2019.

  * [8] 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.

  * [9] A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, Rev. Mat. Complut. 31 (2018), 673–711.

  * [10] 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).

  * [11] Z. Chen, G. Liao and L. Wang, The complexity of a minimal sub-shift on symbolic spaces, _J. Math. Anal. Appl._ , 317 (2006) 136–145.

  * [12] S. A. Chobanyan, V. I. Tarieladze and N. N. Vakhania, Probability distributions on Banach spaces, volume 14 of Mathematics and its Applications. Reidel, 1987.

  * [13] D.L. Cohn, Measure Theory, Second edition. Birkhauser, 2013.

  * [14] G. Costakis, A. Manoussos and I. Parissis, Recurrent linear operators, Complex Anal. Oper. Theory 8 (2014), 1601–1643.

  * [15] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press, 1995.

  * [16] J. Diestel and J.J. Uhl, Jr, Vector Measures. Mathematical Surveys and Monographs. Volume: 15, 1977.

  * [17] T. Eisner, T. Mátrai, On typical properties of Hilbert space operators. Isr. J. Math. 195 (2013), 247–281.

  * [18] R. Ernst, C. Esser and Q. Menet, $\mathcal{U}$-Frequent hypercyclicity notions and related weighted densities, Isr. J. Math. 241 (2021), p. 817–848.

  * [19] Q. Fan and G. Liao, Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy, _Sci. China Math._ 41, 33 (1998).

  * [20] E. Flytzanis, Mixing properties of linear operators in Hilbert spaces, Séminaire d’Initiation à l’Analyse 34ème année (1994/1995), no. Exposé no. 6.

  * [21] E. Flytzanis, Unimodular eigenvalues and linear chaos in Hilbert spaces. Geom. Funct. Anal. 5 (1995), 1–13.

  * [22] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, New Jersey 1981.

  * [23] E. Glasner, Ergodic Theory via Joinings. American Mathematical Society, 2003.

  * [24] S. Grivaux and É. Matheron, Invariant measures for frequently hypercyclic operators, Adv. Math., 265 (2014), p. 371–427.

  * [25] S. Grivaux and É. Matheron, Local spectral properties of typical contractions on $\ell_{p}$-spaces, Anal. Math., 48, 755–778 (2022).

  * [26] S. Grivaux, É. Matheron and Q. Menet, Linear dynamical systems on Hilbert spaces: Typical properties and explicit examples, Memoirs of the American Mathematical Society, vol. 269, 2021.

  * [27] S. Grivaux, É. Matheron and Q. Menet, Does a typical $\ell_{p}$-space contraction have a non-trivial invariant subspace? Trans. Amer. Math. Soc. 374 (2021), no. 10, 7359–7410.

  * [28] K.-G. Grosse-Erdmann, Dynamics of operators, Topics in Complex Analysis and Operator Theory, Proceedings of the Winter School held in Antequera, Málaga, Spain (February 5-9, 2006), pages 41–84.

  * [29] K.-G. Grosse-Erdmann and A. Peris, Linear Chaos. Springer, London 2011.

  * [30] P. A. Kozarzewski, On the existence of the support of a Borel measure. Demonstr. Math. 2018, 51, 76–84.

  * [31] U. Krengel, Ergodic Theorems. De Gruyter, 1985.

  * [32] Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., 369 (2017), 4977–4994.

  * [33] Q. Menet, Inverse of $\mathcal{U}$-frequently hypercyclic operators, J. Funct. Anal., Vol 279, Issue 4, 1 September 2020, 108543.

  * [34] Q. Menet, Inverse of frequently hypercyclic operators, J. Inst. Math. Jussieu., (2021), pp. 1–20.

  * [35] G. Pedersen, Analysis now. Springer, 1989.

  * [36] P. Walters, An Introduction to Ergodic Theory. Springer New York, Graduate Texts in Mathematics, 1982.

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, Acces F, 4a planta, 46022 València, Spain
E-mail<EMAIL_ADDRESS>