A decoherence explanation of the gallium neutrino anomaly

Yasaman<EMAIL_ADDRESS>Thomas<EMAIL_ADDRESS>
aSchool of physics, Institute for Research in Fundamental Sciences (IPM)
P.O.Box 19395-5531, Tehran, Iran
bInstitut für Astroteilchenphysik, Karlsruher Institut für Technologie (KIT),
76021 Karlsruhe, Germany

###### Contents

  1. 1 Introduction
  2. 2 The decoherence model
  3. 3 Numerical analysis for gallium data
    1. 3.1 Discussion of the gallium anomaly
    2. 3.2 Fitting gallium data with the decoherence model
  4. 4 Consistency with oscillation data
    1. 4.1 Solar neutrinos
    2. 4.2 Other oscillation data
  5. 5 Summary

##  1 Introduction

The gallium solar-neutrino detectors GALLEX [1, 2] and SAGE [3, 4] have been
used to measure the neutrino induced event rate from radioactive 51Cr and 37Ar
sources, leading to event rates consistently lower than Standard Model (SM)
expectations. This has been called the gallium anomaly [5, 6, 7]. Recently,
the BEST collaboration has performed a dedicated source experiment using a
51Cr source in the center of a two-volume gallium detector, confirming the
previous hints by observing an event deficit of around 20% compared to the SM
prediction at high statistical significance [8, 9].

Traditionally, the gallium anomaly has been interpreted in terms of sterile
neutrino oscillations, see e.g. [7, 10]. This explanation, however, is in
sever tension with constraints from solar neutrinos as well as short-baseline
reactor experiments [11, 12, 13, 14]. It is unlikely that the anomaly can be
resolved by cross section uncertainties [12, 15, 16]. The authors of ref. [17]
discuss various possible explanations based on conventional or non-standard
physics, with no convincing solution. Therefore, at present the gallium
anomaly remains a puzzle.

Below we are going to present an explanation of the gallium anomaly in terms
of quantum-decoherence of neutrinos. We are assuming a modification of
standard quantum mechanics by some exotic new physics which induces a loss of
coherence in the evolution of the quantum states [18, 19, 20]; for previous
applications to neutrino oscillations see e.g., [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31]. We postulate that in the evolution of the electron neutrinos
produced in the gallium experiments, decoherence is lost already at distances
of order few meters, in order to explain the observed event deficit. In a
similar spirit as in the “soft decoherence” scenario of [24] we assume that
decoherence effects dominate at low energies and are suppressed with
increasing energy by a steep power law. In this way we can explain the gallium
anomaly without impacting other oscillation measurements. As we discuss, this
scenario is also consistent with the solar neutrino data.

Hence, we offer an explanation of the gallium anomaly without affecting the
success of the standard three-flavour explanation of oscillation data. To
explain the LSND hint for $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ transitions [32],
our explanation of the gallium anomaly can be combined with the decoherence
model for LSND proposed in [26]. To simultaneously explain the LSND and
gallium anomalies, we may allow for different decoherence parameters for
neutrinos and antineutrinos or accept that decoherence effects happen only
around neutrino energies of 0.75 and 30 MeV, but not in between or at higher
energies. We do not address the MiniBooNE [33] anomaly, which requires an
alternative explanation to this scenario. Similarly, our model predicts no
non-standard effects at short-baseline reactor experiments (see [34] for a
discussion in view of recent developments related to reactor neutrino flux
predictions).

Our model is based on the decoherence of the three standard-model neutrinos
and does not require an introduction of sterile neutrinos. Recent discussions
of decoherence in oscillations of eV-scale sterile neutrinos can be found in
refs. [35, 36]. Let us stress that the decoherence that we postulate here
requires exotic new physics which modifies the standard quantum mechanical
evolution; conventional decoherence based on particle localisation leads only
to tiny effects which are negligible for all oscillation experiments
considered here [37, 38].

The outline of the rest of the paper is as follows. In section 2, we introduce
the quantum decoherence framework and identify the parameters of our scenario.
Section 3 contains our considerations of the gallium anomaly: we present our
numerical analysis and determine the decoherence parameters which can explain
the gallium data. In section 4, we show that our scenario can be consistent
with the global data on the neutrino oscillations, provided that the
decoherence effects decrease rather quickly with energy in order to be
compatible with the solar and reactor neutrino data. We comment on the
possibility to also explain the LSND results along with the gallium data. We
summarize our findings in section 5.

##  2 The decoherence model

In the decoherence model, the evolution of the density matrix, $\rho$ is
modified as follows

$\frac{d\rho}{dt}=-i[H,\rho]-\mathcal{D}[\rho]\ .$ (1)

While $H$ is the standard Hamilton operator, $\mathcal{D}$ accounts for the
decoherence. To maintain complete positivity, $\mathcal{D}[\rho]$ has to be of
Lindblad form [39, 40]

$\mathcal{D}[\rho]=\sum_{n}[\\{\rho,D_{n}D_{n}^{\dagger}\\}-2D_{n}\rho
D_{n}^{\dagger}]\,.$ (2)

To ensure unitarity, i.e., $d{\rm Tr}(\rho)/dt=0$, we impose the condition
$D_{n}^{\dagger}=D_{n}$. This also guarantees the second law of the
thermodynamics [40]. If we furthermore want the average energy Tr($\rho H$) to
be conserved, $H$ and $D_{n}$ should be simultaneously diagonalized:
$[H,D_{n}]=0$.

From now on, we take a single $D$ matrix. With the properties mentioned above,
we can write the Hamiltonian and the $D$ matrix in the neutrino mass basis as

$H=\frac{1}{2E_{\nu}}{\rm diag}(m_{1}^{2},m_{2}^{2},m_{3}^{2})\,,\qquad D={\rm
diag}(d_{1},d_{2},d_{3})\,,$ (3)

where $m_{i}$ are the neutrino masses and $d_{i}$ are real quantities with
dimension of square-root of mass. The decoherence terms lead to exponential
damping of the off-diagonal elements of the density matrix, see e.g., [24],
with a rate set by the decoherence parameters

$\gamma_{ij}=(d_{i}-d_{j})^{2}\,.$ (4)

For instance, we obtain for the $\nu_{e}$ survival probability

$P_{ee}=\sum_{i=1}^{3}|U_{ei}|^{4}+\sum_{i\neq
j}|U_{ei}|^{2}|U_{ej}|^{2}e^{-\gamma_{ij}L}e^{-i\phi_{ij}}\,,$ (5)

where

$\phi_{ij}=\frac{\Delta m^{2}_{ji}L}{2E_{\nu}}\,.$ (6)

Deviations from the standard oscillation formula are controlled by the
decoherence parameters $\gamma_{ij}$. They have units of inverse length and an
unknown energy dependence. Following the usual practice in the literature, we
will assume here an arbitrary power law dependence for $\gamma_{ij}$ as

$\gamma_{ij}=\frac{1}{\lambda_{ij}}\left(\frac{E_{\rm
ref}}{E_{\nu}}\right)^{r}\,,$ (7)

where $\lambda_{ij}$ is the decoherence length. As a reference energy, we
choose $E_{\rm ref}=0.75$ MeV which is close to the dominant neutrino energies
from a Cr source.

In the phenomenological study below, we will take $\lambda_{12},\lambda_{13}$
and the power index $r$ as the independent parameters; $\lambda_{23}$ is then
determined by using eq. 4, which implies that $\gamma_{23}$ is fixed up to a
sign ambiguity:

$\gamma_{23}=\gamma_{12}+\gamma_{13}\pm 2\sqrt{\gamma_{12}\gamma_{13}}\,.$ (8)

Within the three active neutrino framework, for the gallium experiments the
oscillation phases $\phi_{ij}\ll 1$ and we have $e^{i\phi_{ij}}\approx 1$.

Note that in eq. 3 we have assumed that matter effects are negligible and
adopted the vacuum Hamiltonian. If matter effects are important, $H$ and $D$
will no longer commute. In such a case additional damping effects may appear,
not only damping the off-diagonal elements of $\rho$, but also driving $\rho$
towards a matrix proportional to the identity matrix, see e.g., [23, 25, 29,
28]. We will come back to this in section 4.1, when discussing the solar
neutrinos.

##  3 Numerical analysis for gallium data

###  3.1 Discussion of the gallium anomaly

Gallium experiments consist of detector volumes with typical dimensions of few
meters filled with gallium. In particular, the BEST experiments has two
separated volumes, an inner spherical volume with radius 0.67 m and an outer
cylindrical volume with radius 1.09 m and height 2.35 m [8, 9]. In the center
of these volumes they deploy intense radioactive sources providing a flux of
$\nu_{e}$ from electron-capture decay. For the 51Cr source used in most
measurements, the dominant neutrino energy lines are around 750 keV (430 keV)
with branching ratios around 90% (10%). The 37Ar source used in one
measurement campaign of the SAGE experiment has two dominant lines close to
812 keV, see e.g., [16] for more details.

The gallium source experiments report the ratio of observed to expected events

$R=\frac{N^{\rm obs}}{N^{\rm pred,SM}}\,,$ (9)

where $N^{\rm pred,SM}$ is the number of events predicted in the SM without
any $\nu_{e}$ disappearance, which requires to specify the cross section for
the detection reaction ${}^{71}{\rm Ga}(\nu_{e},e^{-})^{71}{\rm Ge}$ for
neutrinos from 51Cr or 37Ar sources, correspondingly. There has been some
discussion in the literature, to what level uncertainties on the cross section
can affect the anomaly, e.g., [12, 15, 16, 17]. In this work we adopt the
recent detailed consideration of the relevant cross section from Haxton et al.
[16]. They obtain corrections to the ground state transition leading to a
ground state cross section about 2.5% smaller than Bahcall [41]. Then they
provide two independent evaluations of the excited state transition, one based
on $(p,n)$ measurements from Krofcheck et al. (1985) [42] and one using
$(^{3}{\rm He},t)$ data from Frekers et al. (2011) [43]. In the following we
denote the corresponding cross sections by (CS1) and (CS2), respectively.
Including a detailed evaluation of the uncertainties, they obtain the
following two “recommended” cross sections [16]

$\displaystyle\sigma(^{51}{\rm Cr})=5.71^{+0.27}_{-0.10}\,,\qquad$
$\displaystyle\sigma(^{37}{\rm Ar})=6.88^{+0.34}_{-0.13}\,,\qquad\text{(CS1)}$
$\displaystyle\sigma(^{51}{\rm Cr})=5.85^{+0.19}_{-0.13}\,,\qquad$
$\displaystyle\sigma(^{37}{\rm Ar})=7.01^{+0.22}_{-0.16}\,,\qquad\text{(CS2)}$
(10)

in units of $10^{-45}\,{\rm cm}^{2}$ and quoted uncertainties at 68% CL.

| CS1 | CS2
---|---|---
Gallex 1 [2] | $0.970\pm 0.112$ | $0.946\pm 0.109$
Gallex 2 [2] | $0.826\pm 0.102$ | $0.806\pm 0.099$
SAGE (Cr) [3] | $0.967\pm 0.122$ | $0.944\pm 0.119$
SAGE (Ar) [4] | $0.805\pm 0.085$ | $0.790\pm 0.084$
BEST (inner) [8] | $0.805\pm 0.045$ | $0.786\pm 0.044$
BEST (outer) [8] | $0.779\pm 0.046$ | $0.761\pm 0.045$
Table 1: Ratio $R$ of observed and predicted event numbers in the gallium
experiments, assuming the two recomended cross sections CS1 and CS2 from
Haxton et al. [16]. The quoted errors correspond to the $1\sigma$ combined
statistical and uncorrelated systematic experimental uncertainties. The
uncertainty on the cross section is not included.

In table 1 we report the ratios $R$ for the 6 data points, using either the
(CS1) or (CS2) cross sections. The errors in the table include statistical and
uncorrelated experimental systematic uncertainties. In order to evaluate the
significance of the effect, they need to be combined with the correlated
uncertainty due to the cross sections from eq. 10. To test the null-hypothesis
of no neutrino disappearance we define

$\displaystyle\chi^{2}_{\text{null}}={\rm min}_{\xi_{\rm
CS}}\left[\sum_{i}\frac{(1+\delta^{i}_{\rm CS}\xi_{\rm
CS}-R_{i})^{2}}{\sigma_{i}^{2}}+\xi_{\rm CS}^{2}\right]\,,$ (11)

with $R_{i}$ and $\sigma_{i}$ given in table 1 and the index $i$ runs over the
used data points; $\delta^{i}_{\rm CS}$ is the relative uncertainty of the
cross section derived from eq. 10, which depends on the index $i$ whether a Cr
or Ar source has been used. In order to take into account the asymmetric cross
section errors we use for $\delta^{i}_{\rm CS}$ the upper (lower) error if the
value of the pull parameter $\xi_{\rm CS}$ at the minimum is larger (smaller)
than zero. The results of this test are summarized in table 2, where we give
the $\chi^{2}$ of the null-hypothesis for using only the two BEST data points
or for combining all 6 gallium data points. We see that for both cross
sections, very low $p$-values are obtained, corresponding roughly to $5\sigma$
significance, with CS2 leading to slightly higher significances.

|  $\chi^{2}_{\text{null}}/$dof |  $p$-value
---|---|---
CS1, BEST | 32.1/2 |  $1.1\times 10^{-7}$ (5.3$\sigma$)
CS1, all | 36.3/6 |  $2.4\times 10^{-6}$ (4.7$\sigma$)
CS2, BEST | 34.7/2 |  $2.9\times 10^{-8}$ (5.5$\sigma$)
CS2, all | 38.4/6 |  $9.4\times 10^{-7}$ (4.9$\sigma$)
Table 2: Evaluating the null-hypothesis $R=1$ for the BEST experiments (inner
and outer volumes combined) and for all gallium experiments, for the two
recommended cross sections CS1 and CS2 from Haxton et al. [16]. We give the
$\chi^{2}$/dof for the null-hypothesis and the corresponding $p$-values. In
the bracket the $p$-values are converted into two-sided Gaussian standard
deviations. The analysis includes experimental uncertainties as well as the
cross section uncertainties as provided in [16].

###  3.2 Fitting gallium data with the decoherence model

To test the decoherence model introduced in section 2, we modify the
$\chi^{2}$ definition from eq. 11 in the following way:

$\displaystyle\chi^{2}$ $\displaystyle={\rm
min}_{\xi_{\alpha}}\chi^{2}(\xi_{\alpha})\,,\qquad\alpha={\rm
CS},\theta_{12},\theta_{13}\,,$ (12) $\displaystyle\chi^{2}(\xi_{\alpha})$
$\displaystyle=\sum_{i}\frac{1}{\sigma_{i}^{2}}\left[\left(1+\delta^{i}_{\rm
CS}\xi_{\rm CS}\right)\langle
P_{ee}\rangle_{i}+\pi^{i}_{\theta_{12}}\xi_{\theta_{12}}+\pi^{i}_{\theta_{13}}\xi_{\theta_{13}}-R_{i}\right]^{2}+\sum_{\alpha={\rm
CS},\theta_{12},\theta_{13}}\xi_{\alpha}^{2}\,,$ (13)
$\displaystyle\pi^{i}_{\theta_{jk}}$
$\displaystyle=\delta_{s^{2}_{jk}}\frac{\partial\langle
P_{ee}\rangle_{i}}{\partial s^{2}_{jk}}\,,\qquad
s^{2}_{jk}\equiv\sin^{2}\theta_{jk}\,,\quad jk=(12,13)\,,$ (14)

where $\langle P_{ee}\rangle_{i}$ is the $\nu_{e}$ survival probability
averaged over the detector volume as well as the neutrino energy lines
corresponding to each data point $i$, for details see [6, 10]. As before, we
take into account the asymmetric cross section uncertainties by chosing
$\delta^{i}_{\rm CS}$ depending on the sign of $\xi_{\rm CS}$ at the minimum,
and we include the uncertainties on the leptonic mixing angles
$\theta_{12},\theta_{13}$ by introducing the pull parameters
$\xi_{\theta_{12}},\xi_{\theta_{13}}$, and
$\delta_{s^{2}_{12}},\delta_{s^{2}_{13}}$ are the $1\sigma$ errors on
$\sin^{2}\theta_{12},\sin^{2}\theta_{13}$ from NuFit-5.2 [44, 45].

| $r=2$ | $r=12$
---|---|---
|  $\chi^{2}_{\text{min}}/$dof |  $p$-val. | $\Delta\chi^{2}$ | #$\sigma$ |  $\lambda_{12}$ [m] |  $\chi^{2}_{\text{min}}/$dof |  $p$-val. | $\Delta\chi^{2}$ | #$\sigma$ |  $\lambda_{12}$ [m]
CS1, BEST | 2.0/1 | 0.16 | 30.1 | 5.1 | 1.44 | 1.7/1 | 0.19 | 30.4 | 5.2 | 1.44
CS1, all | 7.7/5 | 0.17 | 28.6 | 5.0 | 1.74 | 8.3/5 | 0.14 | 28.0 | 4.9 | 2.10
CS2, BEST | 2.6/1 | 0.11 | 32.1 | 5.3 | 1.19 | 2.2/1 | 0.14 | 32.5 | 5.4 | 1.44
CS2, all | 8.4/5 | 0.14 | 30.0 | 5.1 | 1.44 | 9.2/5 | 0.10 | 29.2 | 5.0 | 1.74
Table 3: Best fit results for the decoherence model with $r=2$ (left) and
$r=12$ (right) for the BEST experiment (inner and outer volumes combined) and
for all gallium experiments, for the two recommended cross sections CS1 and
CS2 from Haxton et al. [16]. We give the $\chi^{2}$/dof at the best fit point
where we assume one effective fit parameter (namely $\lambda_{12}$, see text
for explanations), the corresponding $p$-values of the best fit points, the
$\Delta\chi^{2}$ to the null hypothesis, the number of two-sided Gaussian
standard deviations when converting the $\Delta\chi^{2}$ into a confidence
level for 2 dof, and the value of $\lambda_{12}$ at the best fit point. The
best fit for $\lambda_{13}$ is in all cases at 0.04 m, which corresponds to
the lower boundary of the considered range.

Figure 1: Allowed regions for the decoherence lengths $\lambda_{12}$ and
$\lambda_{13}$ at $1,2,3\sigma$ for 2 dof obtained by fitting combined gallium
data. The left (right) panel corresponds to an energy dependence of the
decoherence parameter with the power $r=2$ (12). We use the CS2 cross section.
The black-solid contours/blue regions assume
$\gamma_{23}=\gamma_{12}+\gamma_{13}-2\sqrt{\gamma_{12}\gamma_{13}}$ whereas
the dashed contours use
$\gamma_{23}=\gamma_{12}+\gamma_{13}+2\sqrt{\gamma_{12}\gamma_{13}}$.

The results of the fit are provided in table 3 for the two recommended cross
section from eq. 10 and fitting either only the two BEST data points or all
gallium data combined. Figure 1 shows the allowed parameter range for the
decoherence lengths $\lambda_{12}$ and $\lambda_{13}$ using all gallium data
and the CS2 cross section (other combinations give similar allowed regions).
We consider two representative examples for the power law, namely $r=2$ and
$r=12$. As we will see below, consistency with neutrino oscillation data
requires that decoherence effects become weak very quickly as the neutrino
energy increases, requiring values of $r\gtrsim 10$.

We find that the best fit point for $\lambda_{13}$ is driven towards the
boundary of our considered region, at $\lambda_{13}=0.04$ m, which effectively
means full decoherence at the distances relevant for gallium experiments. In
this limit the survival probability becomes

$P_{ee}^{\rm gal}\approx
1-\frac{1}{2}\sin^{2}2\theta_{13}-\frac{1}{2}\cos^{4}\theta_{13}\sin^{2}2\theta_{12}\left(1-e^{-\gamma_{12}L}\right)\qquad(\lambda_{13}\to
0)\,,$ (15)

where we have used $\gamma_{23}\approx\gamma_{13}\gg\gamma_{12}$. Since
$0.5\sin^{2}2\theta_{13}\approx 0.043$, the suppression due to decoherence of
the 3rd mass state is not enough to account for the $\simeq 20\%$ suppression
in gallium experiments, and therefore we need to invoke decoherence in the 12
sector corresponding to the last term in eq. 15. Numerically we have
$0.5\cos^{4}\theta_{13}\sin^{2}2\theta_{12}\approx 0.404$. Hence, we need
partial decoherence in the 12 sector to obtain $P_{ee}\simeq 0.8$. This is
reflected in the allowed region for $\lambda_{12}$ visible in fig. 1,
indicating values $\lambda_{12}\simeq 1-2$ m, comparable to the typical sizes
of gallium experiments. From fig. 1 we also see, that the results are very
similar for both sign options to determine $\lambda_{23}$ according to eq. 8,
and they become identical in the limits $\lambda_{12}\gg\lambda_{13}$ and
$\lambda_{12}\ll\lambda_{13}$. For definiteness we will adopt the negative
sign for the following discussion.

Figure 2: Predicted event ratios at the best fit point for the combined
gallium data for $r=2$ and the CS2 cross section (red lines). The red shaded
boxes indicate the $1\sigma$ correlated cross section uncertainty on the
predictions. Black data points show the observed ratios with error bars at
$1\sigma$ including statistical and experimental systematic errors.

In table 3 we provide the $\chi^{2}$ values at the best fit points. To
calculate the corresponding $p$-value to evaluate the goodness-of-fit we
assume one effective free parameter. The justification for this is that
$\lambda_{13}$ is driven to small values, where predictions become independent
of it, see eq. 15. In all cases shown in the table we find $p$-values in the
range between 10% and 20%. While this is a huge improvement compared to the
$p$-values of the null hypothesis (see table 2) the fit is not perfect. This
is related to the partial decoherence in the 12 sector, which is required for
the reasons discussed above. It leads to a distance dependence on the scale of
gallium experiments which in particular predicts different event ratios in the
inner and outer detector volumes of the BEST experiment. We illustrate this on
one example fit in fig. 2 which compares the predicted ratios at the best fit
point to the observed values. While currently this is acceptable within
uncertainties, the distance dependence of $P_{ee}$ at the scale of 1 m and few
100 keV neutrino energies is a specific prediction of this scenario.

In table 3 we also provide the $\Delta\chi^{2}$ of the best fit points with
respect to the null hypothesis. Here we use 2 dof to evaluate these values as
both parameters, $\lambda_{12}$ and $\lambda_{13}$, have to be changed to move
from the best fit point to the null hypothesis which corresponds to
$\lambda_{12,13}\to\infty$. We obtain that the decoherence model is preferred
over the null hypothesis at the level of around $5\sigma$ in all cases
considered in the table.

| $r=2$ | $r=12$
---|---|---
|  $\chi^{2}_{\text{min}}/$dof |  $p$-val. | $\Delta\chi^{2}$ | #$\sigma$ |  $\lambda_{12}$ [m] |  $\chi^{2}_{\text{min}}/$dof |  $p$-val. | $\Delta\chi^{2}$ | #$\sigma$ |  $\lambda_{12}$ [m]
CS1, BEST | 3.0/1 | 0.08 | 29.1 | 5.4 | 0.99 | 2.6/1 | 0.11 | 29.5 | 5.4 | 1.12
CS1, all | 9.1/5 | 0.10 | 27.2 | 5.2 | 1.27 | 10.3/5 | 0.07 | 26.0 | 5.1 | 1.44
CS2, BEST | 3.5/1 | 0.06 | 31.2 | 5.6 | 0.87 | 3.1/1 | 0.08 | 31.6 | 5.6 | 0.93
CS2, all | 9.8/5 | 0.08 | 28.6 | 5.4 | 1.05 | 10.3/5 | 0.07 | 28.1 | 5.3 | 1.44
Table 4: Same as table 3 but setting $\lambda_{13}\to\infty$. The number of
standard deviations relative to the null hypothesis are obtained by evaluating
$\Delta\chi^{2}$ for 1 dof.

We note that decoherence in the 13 sector is actually not required by the fit;
the allowed regions at $1\sigma$ extend up to $\lambda_{13}\to\infty$, c.f.
fig. 1. In table 4 we give the properties of the best fit results when fixing
$\lambda_{13}\to\infty$, i.e., $\gamma_{13}=0$. In this case, we have
$\lambda_{12}=\lambda_{23}$ and the survival probability relevant for gallium
data becomes111An equivalent solution is obtained for
$\lambda_{12}=\lambda_{13}$ with $\lambda_{23}\to\infty$. This case
corresponds to replacing $|U_{e2}|^{2}\to|U_{e1}|^{2}$ in eq. 16.

$P_{ee}^{\rm gal}\approx
1-2|U_{e2}|^{2}(1-|U_{e2}|^{2})\left(1-e^{-\gamma_{12}L}\right)\qquad(\lambda_{13}\to\infty)\,.$
(16)

Comparing the $\chi^{2}_{\rm min}$ values from tables 3 and 4, we see that the
$\chi^{2}$ is increased only by about 1 unit, the goodness-of-fit is around or
slightly below 10% for all cases, whereas the preference compared to the null
hypothesis is above $5\sigma$ in all cases.

##  4 Consistency with oscillation data

In this section we show that our proposed explanation of the gallium anomaly
does not impact the various observations of neutrino oscillations. We focus
first on solar neutrinos in section 4.1, which require special care due to the
matter effect in the sun. This will lead us to a very steep energy dependence
of the decoherence coefficients with $r\gtrsim 10$. The remaining oscillation
data is discussed in section 4.2 where we also consider short-baseline
anomalies and argue that in certain decoherence scenarios the LSND anomaly
could be explained as well. The relevant length scales are illustrated in fig.
3.

Figure 3: Comparison of relevant length scales at different neutrino energies.
Lines in black show the decoherence length $1/\gamma$ for $\lambda=2$ m and
$r=10,11,12$; the region around and above these lines is affected by the
decoherence terms. Blue lines show the vacuum oscillation lengths due to
$\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$. Furthermore, we show
approximately the regions probed by gallium experiments (red star), short-,
medium-, and long-baseline reactor experiments (green regions), atmospheric
neutrinos (cyan region), as well as accelerator experiments including the
long-baseline experiments T2K, NOvA, DUNE (purple) and short-baseline
experiments LSND and MiniBooNE (magenta). The red curve shows the distance of
the MSW resonance inside the sun from the solar center. We also indicate the
energy of the 7Be solar neutrino line and the size of the matter potential at
the center of the sun converted into a distance (grey), as well as the energy
range relevant for 8B solar neutrinos (red region).

###  4.1 Solar neutrinos

As discussed in section 2, if matter effects can be neglected, the decoherence
terms lead to a damping of the off-diagonal elements of the density matrix in
the mass basis, which results in an incoherent mixture of neutrino mass
eigenstates. In other words, they suppress the interference terms of the
oscillation probability, see eq. 5. For distances larger than the oscillation
length, these terms average to zero even for standard evolution with
$\gamma_{ij}=0$. This means that decoherence does not change the oscillation
probability in such a case. In other words, for the range well above the blue
solid line in fig. 3, the oscillatory terms in the oscillation probability
average to zero and decoherence effects cannot in practice be resolved,
regardless of whether we are above the black lines or not. As a result, the
effect of decoherence on the oscillation of solar neutrinos from the Sun
surface to the Earth surface (as well as for the supernova or cosmic neutrinos
outside the source) will not be observable.

However, inside the Sun and the Earth, the matter effects [46, 47] change the
picture [23, 25, 28, 29]. While $D$ commutes with the Hamiltonian in vacuum,
it will not commute with the effective Hamiltonian inside matter. In such a
situation the decoherence terms will push $\rho$ towards a matrix proportional
to the identity matrix. This can be understood because in this limit both
$\mathcal{D}$ and the commutator of $\rho$ and the Hamiltonian vanish, making
$\rho\propto I$ an asymptotic solution of eq. 1 when $[D_{n},H]\neq 0$. Let us
discuss the various parts of the solar neutrino spectrum in turn.

Low energy: Solar $pp$ neutrinos with energies $\lesssim 0.4$ MeV will be
strongly affected by decoherence. However, for neutrinos with these energies,
matter effects are small and their survival probability is determined by
vacuum oscillations. As mentioned above, in this case, the decoherence effects
are indistinguishable from standard averaging and hence we expect no
modification of low energy solar neutrinos compared to the standard
oscillation picture.

High energy: Let us now focus on 8B neutrinos with energies above the SK
detection threshold of 4.5 MeV [48]. The relevant region is indicated by the
red-shaded box in fig. 3.222The $L$ range for this box is only for
illustration purposes and has been chosen as $[0.02R_{\odot},R_{\odot}]$, with
$R_{\odot}$ denoting the solar radius and $0.02R_{\odot}$ is approximately the
production region for 8B neutrinos inside the Sun. When the high-energy solar
neutrinos propagate out from the center of the sun to the surface, the
evolution follows adiabatically the effective mass eigenstates in matter until
they cross the MSW resonance. After the resonance we have basically
propagation of the vacuum mass states. The red curve in fig. 3 shows the
location of the MSW resonance in the Sun as a function of neutrino energy.
Below 2 MeV, the density even in the Sun center will be too low for a
resonance. In order to be consistent with the success of the MSW mechanism we
need to make sure, that the decoherence effects do not affect the evolution as
long as matter dominates. Hence, we require that the decoherence length
$1/\gamma$ must be larger than the path-length during which matter dominates.
We can see from the figure, for a decoherence length as required to explain
gallium data at $E_{\nu}\simeq 0.75$ MeV, we need $r\gtrsim 10$ to have
$1/\gamma$ larger than the resonance location for $E_{\nu}>4.5$ MeV. We have
verified also by numerical calculations, that for these values, the
decoherence effects do not modify significantly the $\nu_{e}$ survival
probability in this energy range. Note that this requirement ensures also that
the day-night effect for the 8B solar neutrinos will not be modified compared
to the standard theory, as the decoherence length is many orders of magnitudes
longer than the Earth diameter.

Intermediate energy: As indicated by the grey line in fig. 3, at
$E_{\nu}=0.862$ MeV, which is the energy of 7Be line measured by Borexino [49,
50], $\gamma$ can be sizable. Since this energy is close to the one in gallium
experiments, $\gamma$ for the 7Be line is not significantly suppressed
relative to that at the gallium experiments. However, in this case, the matter
effects are subdominant: $[\Delta
m_{21}^{2}/(2E_{\nu})]/(\sqrt{2}G_{F}n_{e}|_{\rm Sun~{}center})\sim 0.1$, and
we expect the decoherence effects to be approximately similar to the pure
vacuum case. Numerical computation shows that the deviation of $P_{ee}$ from
the standard prediction is at the level of $10\%$ which is of the same size as
the current experimental uncertainty at $1\sigma$ and therefore compatible
with observations: $P_{ee}(0.862\,{\rm MeV})=0.53\pm 0.05$ [50].

In summary, we conclude that by choosing $r\gtrsim 10$ we can make our gallium
explanation consistent with solar neutrino data. Notice that we cannot avoid
this solar neutrino bound on $r$ by taking different decoherence parameters
for neutrinos and antineutrinos. In future, with precise measurements of the
vacuum-to-matter transition region of the solar $\nu_{e}$ survival probability
we may be able to detect deviations from the standard MSW prediction. We leave
a dedicated investigation of this possibility for future work.

###  4.2 Other oscillation data

We now comment on the impact of decoherence on other neutrino oscillation
experiments. We note that values of $r\neq 1$ imply violation of Lorentz
symmetry. Therefore, it may be expected that decoherence effects are also CPT
violating and $\gamma_{ij}$ could be different for neutrinos and antineutrinos
[21]. In such a case, we make no prediction for parameters for antineutrinos
(see also the discussion of LSND below on this point).

If we assume that decoherence is the same for neutrinos and antineutrinos,
reactor experiments put also a sever constraint on the energy dependence of
$\gamma_{ij}$. In fig. 3 the green boxes indicate the ranges probed by various
classes of reactor experiments: short-baseline experiments at $L\lesssim 100$
m, medium-baseline experiments, such as DayaBay, RENO, DoubleChooz at $L\sim
1$ km, and the long-baseline experiment KamLAND with $L\sim 180$ km. The
spectral distortion observed in the latter [51] poses a sever constraint on
decoherence effects (see e.g., [23, 52] for related analyses). Figure 3
suggests, that values $r\gtrsim 12$ are required for not affecting KamLAND.
Figure 4 shows the survival probability relevant for reactor experiments as a
function of distance, assuming that neutrinos and antineutrinos are subject to
the same decoherence effects. We see that in order to be consistent with
KamLAND we need a very steep energy dependence, $r\gtrsim 10$, in order to
compensate the factor $L_{\rm KamL}/L_{\rm Gal}\sim 200\,{\rm km}/(2\,{\rm
m})=10^{5}$ by the factor $(0.75\,{\rm MeV}/E_{\nu})^{r}$. The future JUNO
reactor experiment at $L\simeq 60$ km may be able to further strengthen the
requirement on $r$.

Figure 4: Survival probability $P_{ee}$ as a function of the baseline $L$ for
$E_{\nu}=3$ MeV with the decoherence lengths $\lambda_{12}=1.44$ m (both
panels) and $\lambda_{13}=0.04$ m ($\infty$) for the left (right) panel and
for several values of $r$. The black dashed curve corresponds to the standard
three flavour oscillation probability, which overlaps with the $r=12$ curve.
Oscillation parameters are taken at the NuFit-5.2 best fit point [45].
Probabilities are averaged over a Gaussian energy resolution of
$0.03\%\sqrt{{\rm MeV}/E_{\nu}}$.

From fig. 3 it is clear that for all the other oscillation experiments,
including atmospheric and accelerator neutrino experiments, decoherence
effects on our model will be negligible, if the power law extends to
$E_{\nu}\gtrsim 0.1$ GeV.

LSND, MiniBooNE and short-baseline reactors. From figs. 3 and 4 it becomes
clear, that in our scenario short-baseline reactor experiments are not
affected: decoherence effects at short baselines would spoil the oscillation
signatures observed at medium and long-baseline reactor experiments.
Similarly, we cannot explain the MiniBooNE anomaly [33], see magenta bar
around $10^{3}$ MeV in fig. 3: decoherence at such small baselines would
distort the oscillation signatures observed in atmospheric and long-baseline
accelerator experiments. In both cases (short-baseline reactor experiments and
MiniBooNE), decoherence effects are completely negligible under the power law
assumption with $r\gtrsim 10$.

The LSND experiment, reporting evidence for $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$
transitions [32], corresponds to the magenta bar around 30 MeV in fig. 3. If
we assume the same decoherence parameters for neutrinos and antineutrinos and
the power law with $r\gtrsim 10$, it is clear that no effect is predicted for
LSND. However, as there are no other observations in this energy range333Note
that within the standard model, coherent neutrino–nucleus scattering as
observed by COHERENT [53] is a flavour-universal neutral-current process and
is therefore not expected to be affected by flavour transitions due to
decoherence. However, in the presence of decoherence, the bounds on new
physics such as non-standard neutrino interactions with non-universal
couplings should be reconsidered., we can introduce decoherence effects there
to explain LSND as well, for instance adopting a scenario as in ref. [26].
This could be achieved in the following two ways:

  * •

We could assume that the decoherence effects violate the CPT symmetry and
postulate different decoherence parameters for neutrinos and antineutrinos. To
explain the gallium anomaly we need decoherence in neutrinos at $E_{\nu}\simeq
0.75$ MeV getting quickly suppressed for higher energies. For LSND we need
decoherence around $E_{\nu}\simeq 30$ MeV in antineutrinos, being suppressed
both at lower and at higher energies [26].

  * •

To explain both gallium and LSND anomalies with the same parameters for
neutrinos and antineutrinos, we may consider a double-peak structure of the
decoherence effects, occurring both at $E_{\nu}\simeq 0.75$ MeV and around
$30$ MeV, but being strongly suppressed in between and above these energies.
For example, this can be achieved by setting $d_{1}=0$, $d_{2}$ a Gaussian
peaked around 0.75 MeV and $d_{3}$ another Gaussian peaked around 30 MeV (see
eq. 4 for the relation between $d_{i}$ and the decoherence prameters
$\gamma_{ij}$).

To identify possible UV completions for such scenarios is beyond the scope of
the present article.

##  5 Summary

We have proposed an explanation of the gallium anomaly based on quantum
decoherence. Our scenario does not require sterile neutrinos, but we postulate
that at the relevant neutrino energies of $E_{\nu}^{\rm gallium}\simeq 0.75$
MeV the neutrino mass states $\nu_{1}$ and $\nu_{2}$ decohere at length scales
comparable to the size of the gallium detectors, of order 2 m. In order to be
consistent with other oscillation data, in particular with solar neutrinos and
(if equal decoherence parameters for neutrinos and antineutrinos are assumed)
the KamLAND reactor experiment, decoherence effects have to decrease quickly
for energies larger than $E_{\nu}^{\rm gallium}$. If we assume a power law
behaviour with neutrino energy, the decoherence parameters should scale as
$E_{\nu}^{-r}$ with $r\gtrsim 10-12$. While explanations of the gallium
anomaly in terms of sterile neutrinos with eV-scale mass-squared differences
suffer from severe tension with solar neutrinos and reactor data, the
explanation proposed here is consistent with these data. Furthermore, we
expect that cosmology is not changed compared to the standard three-flavour
neutrino case.

With the power law energy dependence mentioned above, all other data on
neutrino oscillations will be unaffected and proceed as in the standard three-
flavour scenario, including experiments at short baselines. However, it may be
possible to reconcile our proposal for gallium also with an explanation of
LSND in terms of decoherence, if we allow for different decoherence parameters
for neutrinos and antineutrinos or by adopting a peaked energy dependence for
the decoherence parameters.

A testable prediction of our scenario is a distance dependent deficit at the
radioactive source experiments. We predict a $\nu_{e}$ survival probability of
about 0.86 in the inner detector volume of the BEST experiment and 0.75 in the
outer volume. Although the current BEST results do not show evidence for such
a behaviour, our prediction is consistent with the BEST data within the
errors. If more precise measurements in the future confirm a distance-
independent suppression at the scale of 1–2 meters, our proposal can be ruled
out. Another signature of this model is the modification of the solar neutrino
survival probability in the transition region between the vacuum and matter
dominated energy regimes. The possibility to test this model by future high-
precision solar neutrino observations requires further study.

### Acknowledgement

This work has been supported by the European Union’s Framework Programme for
Research and Innovation Horizon 2020 under grant H2020-MSCA-
ITN-2019/860881-HIDDeN. YF has received financial support from Saramadan under
contract No. ISEF/M/401439. She would like to acknowledge support from ICTP
through the Associates Programme and from the Simons Foundation through grant
number 284558FY19 as well as under the Marie Skłodowska-Curie Staff Exchange
grant agreement No 101086085 – ASYMMETRY.

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#  Linear dynamics of an operator associated to the Collatz map

Vincent Béhani  Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the dynamics of an operator $\mathcal{T}$ naturally
associated to the so-called Collatz map, which maps an integer $n\geq 0$ to
$n/2$ if $n$ is even and $3n+1$ if $n$ is odd. This operator $\mathcal{T}$ is
defined on certain weighted Bergman spaces $\mathcal{B}^{2}_{\omega}$ of
analytic functions on the unit disk. Building on previous work of Neklyudov,
we show that $\mathcal{T}$ is hypercyclic on $\mathcal{B}^{2}_{\omega}$,
independently of whether the Collatz Conjecture holds true or not. Under some
assumptions on the weight $\omega$, we show that $\mathcal{T}$ is actually
ergodic with respect to a Gaussian measure with full support, and thus
frequently hypercyclic and chaotic.

###### Key words and phrases:

Collatz conjecture, Hypercyclicity, Chaotic operators, Frequent
Hypercyclicity, Ergodicity

###### 1991 Mathematics Subject Classification:

47A16, 47A35, 30H20, 30B10, 15A18

This work was supported in part 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).

I am grateful to Sophie Grivaux for our discussions and her helpful
proofreadings.

##  1\. Introduction

Our aim in this paper is to investigate from the point of view of linear
dynamics the properties of an operator naturally associated to the so-called
Collatz map. This map $T_{0}\colon\mathbf{Z}_{+}\to\mathbf{Z}_{+}$ is defined
as

$T_{0}(n)=\left\\{\begin{array}[]{l l}n/2&\text{if $n$ is even;}\\\
3n+1&\text{if $n$ is odd.}\end{array}\right.$

The dynamics of this simple-looking map are still very mysterious, and the
famous Collatz Conjecture (called also the $3n+1$-Conjecture, or the Syracuse
Conjecture) states that the orbit of any integer $k\geq 1$ under the action of
$T_{0}$ eventually reaches the point 1.

We will use in this paper the following modified Collatz map
$T\colon\mathbf{Z}_{+}\to\mathbf{Z}_{+}$, defined as

$T(n)=\left\\{\begin{array}[]{l l}n/2&\text{if $n$ is even;}\\\
(3n+1)/2&\text{if $n$ is odd}\end{array}\right.$

whose dynamics are the same as those of $T_{0}$ as far as the Collatz
Conjecture is concerned.

We will here give some of the etablished results about the Collatz Conjecture.
We refer the reader to the survey [lag10] written by Lagarias for an overview
of this problem.

The conjecture had been empirically verified for every $k\leq 20\times 10^{9}$
and is now known to be true for every $k\leq 20\times 2^{58}\approx
5.764\times 10^{18}$ ([oli10]).

Since $T(1)=2$ and $T(2)=1$, $(1,2)$ is called the trivial cycle of $T$ and
Eliahou showed in [eli93] that a non-trivial cycle of $T$ must have a length
greater than 10 439 860 591.

A possible approach to the Collatz Conjecture is to study the proportion of
positive integers $k$ having a $T$-orbit that reaches 1. Krasikov and Lagarias
proved for instance in [kra02] that if $X$ is sufficiently large, the number
of integers $1\leq k\leq X$ whose $T$-orbit reaches 1 is at least $X^{0.84}$.

It is also of interest to investigate the time that it takes to a $T$-orbit to
reach 1. Applegate and Lagarias proved in [app02] that infinitely many
positive integers $k$ have a $T$-orbit that reaches 1 in at least
$6.143\log(k)$ steps.

Moreover, Tao showed in [tao22] that ”almost all” $T$-orbits attain almost
bounded values. That is to say, if we fix a function
$f\colon\mathbf{N}\to\mathbf{R}$ such that $f(k)\to+\infty$ as $k\to+\infty$,
the infinimum of the $T$-orbit of $k$ is less than $f(k)$ for almost every
integer $k\geq 1$, in the sense of logarithmic density.

It is possible to associate to $T$ in a natural way bounded operators on some
Hilbert spaces of analytic functions on the unit disk $\mathbf{D}$, and to
link the dynamics of $T$ to those of these operators. An approach of this kind
was first proposed by Berg and Meinardus in [ber94]. Consider the operator
$\mathcal{F}$ defined on the space $Hol(\mathbf{D})$ of holomorphic functions
on the unit disk $\mathbf{D}$ in the following way: for every $f\in
Hol(\mathbf{D})$ with $f(z)=\sum_{n=0}^{\infty}c_{n}z^{n},$

$\mathcal{F}f(z)=\sum_{n=0}^{\infty}c_{T(n)}z^{n},\quad z\in\mathbf{D}.$

Berg and Meinardus expressed in [ber94] the Collatz Conjecture in terms of
functional equations and obtained that it is equivalent to the fact that 1 is
an eigenvalue of $\mathcal{F}$ of multiplicity 2.

Then Neklyudov took in [nek21] a different perpective, and considered the
adjoint $\mathcal{T}$, for the Hardy space $\mathcal{H}^{2}(\mathbf{D})$, of
the operator $\mathcal{F}$: for every $f\in\mathcal{H}^{2}(\mathbf{D})$ with
$f(z)=\sum_{n=0}^{\infty}c_{n}z^{n},$

$\mathcal{T}f(z)=\sum_{n=0}^{\infty}c_{n}z^{T(n)},\quad z\in\mathbf{D}.$

This operator $\mathcal{T}$ is considered in [nek21] as acting on the Bergman
space

$\mathcal{B}^{2}=\left\\{f\colon z\mapsto\sum_{n=0}^{\infty}c_{n}z^{n}\in
Hol(\mathbf{D});\|f\|^{2}=\sum_{n=0}^{\infty}\frac{\pi\left|c_{n}\right|^{2}}{n+1}<+\infty\right\\}$

on $\mathbf{D}$. Since $T(0)=0$, $T(1)=2$ and $T(2)=1$, the vector space
$\mathcal{H}_{0}=\text{span}[1,z,z^{2}]$ is invariant under the action of
$\mathcal{T}$, and $\mathcal{T}$ induces a bounded operator on the quotient
space $\mathcal{X}=\mathcal{B}^{2}/\mathcal{H}_{0}$, which we still denote by
$\mathcal{T}$. Considering this quotient space allows to avoid the trivial
cycle $(1,2)$ and the fixed point 0 of the Collatz map, and simplifies the
study of the dynamics of $\mathcal{T}$. Neklyudov proved in [nek21] several
results pertaining to the behaviour of the iterates of $\mathcal{T}$, in
connection with the Collatz Conjecture. For instance, he showed that if $T$
has no non-trivial cycle, then $\mathcal{T}$ is hypercyclic, i.e. admits a
vector with dense orbit [nek21, Theorem 2.2]. He also undertook a study of the
periodic points of $\mathcal{T}$ and showed that each hypothetical cycle of
$T$ and each hypothetical diverging $T$-orbit can be associated to a fixed
point of $\mathcal{T}$. He also constructed fixed points of $\mathcal{T}$
independently of any cycle or diverging orbit of $T$.

Neklyudov asked in [nek21] about the existence of a criterion allowing to
distinguish these fixed points and also whether $\mathcal{T}$ is chaotic (i.e.
hypercyclic and admitting a dense set of periodic points).

In this work, we continue this study of operators on Hilbert spaces of
analytic functions associated to the Collatz map and give an affirmative
answer to this question of Neklyudov. We consider the operator $\mathcal{T}$
above as acting on weighted Bergman spaces of holomorphic functions on
$\mathbf{D}$ of the form

$\mathcal{B}^{2}_{\omega}=\left\\{f\colon
z\mapsto\sum_{n=0}^{\infty}c_{n}z^{n}\in
Hol(\mathbf{D});\|f\|^{2}_{\omega}=\sum_{n=0}^{\infty}\frac{\left|c_{n}\right|^{2}}{\omega(n)}<+\infty\right\\}$

where $\omega\colon\mathbf{Z}_{+}\to(0,+\infty)$ is a positive weight. Just as
above, it will be more natural to study the action of $\mathcal{T}$ on the
quotient space
$\mathcal{X}_{\omega}=\mathcal{B}^{2}_{\omega}/\mathcal{H}_{0}$. Let
$\omega_{0}(n)=(n+1)/\pi$ for every $n\geq 0$. Then
$\mathcal{B}^{2}_{\omega_{0}}$ is the classical Bergman space
$\mathcal{B}^{2}$. We write $\mathcal{X}_{\omega_{0}}=\mathcal{X}$.

Whenever the weight $\omega$ is such that $\mathcal{T}$ defines a bounded
operator on $\mathcal{X}_{\omega}$, it is of interest to study the dynamics of
$\mathcal{T}$. The main properties that we will consider are those of
hypercyclicity (existence of a dense orbit, or, equivalently, topological
transitivity), chaos, frequent hypercyclicity, ergodicity with respect to an
invariant measure with full support… We give a brief overview of these notions
in Section 3 of the present paper, and refer the reader to one of the works
[bay09] and [gro11] for more on linear dynamics.

Our first main result states that under some mild assumptions on the weight
$\omega$, the operator $\mathcal{T}$ is hypercyclic.

######  Theorem 1.1.

If $\omega$ is bounded from below and if $\omega(k2^{n})\to+\infty$ as
$n\to+\infty$ for every $k\geq 3$, then $\mathcal{T}$ is hypercyclic.

Theorem 1.1 applies in particular to the weight $\omega_{0}$, i.e. to the case
where $\mathcal{T}$ acts on the Bergman space, and shows that the result
[nek21, Theorem 2.2] of Neklyudov is in fact not conditional to the Collatz
map having no non-trivial cycle.

Our second main result shows, under slightly stronger assumptions on $\omega$,
that $\mathcal{T}$ acting on $\mathcal{X}_{\omega}$ enjoys some strong
dynamical properties:

######  Theorem 1.2.

If $\omega$ is bounded from below and if
$\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, then
$\mathcal{T}$ is chaotic, fequently hypercyclic and ergodic with respect to a
Gaussian invariant measure with full support.

In particular, $\mathcal{T}$ acting on the Bergman space $\mathcal{X}$ is
chaotic. This provides an affirmative answer to Neklyudov’s question.

The paper is organized as follows. We first provide in Section 2 conditions on
the weight $\omega$ ensuring that $\mathcal{T}$ acts as a bounded operator on
$\mathcal{X}_{\omega}$ (Proposition 2.1); building on a description given in
[nek21] of some eigenvectors of $\mathcal{T}$, we show (still under some mild
conditions on $\omega$) that these eigenvectors span a dense subspace of
$\mathcal{X}_{\omega}$ (Theorem 2.4) and that the adjoint $\mathcal{T}^{*}$ of
$\mathcal{T}$ on $\mathcal{X}_{\omega}$ has empty point spectrum (Theorem
2.7). After recalling in Section 3 some background in linear dynamics, we
prove Theorems 1.1 and 1.2 above. The proofs of these two results rely heavily
on the properties of the eigenvectorfields of $\mathcal{T}$ presented in
Section 2, and in particular on Theorem 2.4. Section 4 collects some
additional results as well as some open questions.

##  2\. Boundedness of $\mathcal{T}$ and properties of its eigenvectors

Let $\omega$ be a positive weight on $\mathbf{Z}_{+}$. We first consider the
operator $\mathcal{T}$ acting on the space
$\mathbf{C}[\xi]/\text{span}[1,\xi,\xi^{2}]$, defined by

$\mathcal{T}\sum_{n=3}^{d}c_{n}\xi^{n}=\sum_{3\leq n\leq d,T(n)\geq
3}c_{n}\xi^{T(n)}\quad\text{for every
}\sum_{n=3}^{d}c_{n}\xi^{n}\in\mathbf{C}[\xi]/\text{span}[1,\xi,\xi^{2}],$

and give conditions on $\omega$ implying that $\mathcal{T}$ can be extended to
a bounded operator on $\mathcal{X}_{\omega}$.

######  Proposition 2.1.

The operator $\mathcal{T}$, defined for every $f\in\mathcal{X}_{\omega}$ with
$f(z)=\sum_{n=3}^{\infty}c_{n}z^{n}$ by

$\mathcal{T}\sum_{n=3}c_{n}z^{n}=\sum_{T(j)\geq 3}c_{j}z^{T(j)},\quad
z\in\mathbf{D},$

is a bounded operator acting on $\mathcal{X}_{\omega}$ if and only if the
three sequences $(\omega(6m)/\omega(3m))_{m\geq 1}$,
$(\omega(6m+2)/\omega(3m+1))_{m\geq 1}$ and
$((\omega(6m+4)+\omega(2m+1))/\omega(3m+2))_{m\geq 1}$ are bounded. In this
case

$\|\mathcal{T}\|_{\omega}^{2}=\max\left\\{\sup_{m\geq
1}\frac{\omega(6m)}{\omega(3m)},\sup_{m\geq
1}\frac{\omega(6m+2)}{\omega(3m+1)},\sup_{m\geq
1}\frac{\omega(6m+4)+\omega(2m+1)}{\omega(3m+2)}\right\\}.$

Moreover for every $n\geq 0$

$\|\mathcal{T}^{n}\|_{\omega}^{2}=\sup_{k\geq
3}\sum_{T^{n}(j)=k}\frac{\omega(j)}{\omega(k)}.$

Taking $\omega=\omega_{0}$ we obtain in particular that $\mathcal{T}$ is
bounded on $\mathcal{X}$ and that $\|\mathcal{T}\|^{2}=8/3$.

###### Proof.

Suppose that the three sequences above are bounded, which means that the
maximum

$\max\left\\{\sup_{m\geq 1}\frac{\omega(6m)}{\omega(3m)},\sup_{m\geq
1}\frac{\omega(6m+2)}{\omega(3m+1)},\sup_{m\geq
1}\frac{\omega(6m+4)+\omega(2m+1)}{\omega(3m+2)}\right\\}=\sup_{k\geq
3}\sum_{T(j)=k}\frac{\omega(j)}{\omega(k)}$

is finite. Let $n\geq 0$ and $f\in\mathcal{X}_{\omega}$ with
$f(z)=\sum_{k=3}^{\infty}c_{k}z^{k}$. We have

$\displaystyle\mathcal{T}^{n}f(z)=\sum_{T^{n}(j)\geq
3}c_{j}z^{T^{n}(j)}=\sum_{k=3}^{\infty}\sum_{T^{n}(j)=k}c_{j}z^{k},\quad
z\in\mathbf{D},$

and

$\|\mathcal{T}^{n}f\|_{\omega}^{2}=\sum_{k=3}^{\infty}\frac{1}{\omega(k)}\left|\sum_{T^{n}(j)=k}c_{j}\right|^{2}=\sum_{k=3}^{\infty}\frac{1}{\omega(k)}\left|\sum_{T^{n}(j)=k}\sqrt{\omega(j)}\frac{c_{j}}{\sqrt{\omega(j)}}\right|^{2}.$

By the Cauchy-Schwarz’s inequality

$\|\mathcal{T}^{n}f\|^{2}_{\omega}\leq\sum_{k=3}^{\infty}\sum_{T^{n}(j)=k}\omega(j)\sum_{T^{n}(j)=k}\frac{\left|c_{j}\right|^{2}}{\omega(j)}\leq\left(\sup_{k\geq
3}\sum_{T^{n}(j)=k}\frac{\omega(j)}{\omega(k)}\right)\|f\|^{2}_{\omega},$

which gives that $\mathcal{T}$ is bounded if the sequence
$(\sum_{T(j)=k}\omega(j)/\omega(k))_{k\geq 3}$ is bounded and that
$\|\mathcal{T}^{n}\|^{2}_{\omega}\leq\sup_{k\geq
3}\sum_{T^{n}(j)=k}\omega(j)/\omega(k).$

Conversely, let $n\geq 0$ and suppose that $\mathcal{T}$ is a bounded operator
on $\mathcal{X}_{\omega}$. Let $(k_{p})_{p\geq 1}$ such that
$\sum_{T^{n}(j)=k_{p}}\omega(j)/\omega(k_{p})\to\sup_{k\geq
3}\sum_{T^{n}(j)=k}\omega(j)/\omega(k)$ as $p\to+\infty$. Consider the
function $f_{p}\in\mathcal{X}_{\omega}$ with
$f(z)=\sum_{T^{n}(j)=k_{p}}\omega(j)z^{j}$ satisfying
$\|f_{p}\|^{2}_{\omega}=\sum_{T^{n}(j)=k_{p}}\left|\omega(j)\right|^{2}/\omega(j)=\sum_{T^{n}(j)=k_{p}}\omega(j)$
and

$\|\mathcal{T}^{n}f_{p}\|_{\omega}^{2}=\frac{1}{\omega(k_{p})}\left|\sum_{T^{n}(j)=k_{p}}\omega(j)\right|^{2}=\left(\sum_{T^{n}(j)=k_{p}}\frac{\omega(j)}{\omega(k_{p})}\right)\|f_{p}\|_{\omega}^{2}.$

So we have
$\|\mathcal{T}^{n}f_{p}\|_{\omega}^{2}/\|f_{p}\|_{\omega}^{2}\to\sup_{k\geq
3}\sum_{T^{n}(j)=k}\omega(j)/\omega(k)$ as $p\to+\infty$ and it follows that
$\sup_{k\geq
3}\sum_{T^{n}(j)=k}\omega(j)/\omega(k)\leq\|\mathcal{T}^{n}\|^{2}_{\omega}<+\infty$
for every $n\geq 0$, which concludes the proof. ∎

For instance, it follows from Proposition 2.1 that if $\omega$ has polynomial
growth, then $\mathcal{T}$ is a bounded operator on $\mathcal{X}_{\omega}$.

Eigenvectors play a very important role in linear dynamics. We will thus be
interested in the eigenvalues and the eigenvectors of $\mathcal{T}$, and will
provide some condition on the weight $\omega$ ensuring that the eigenvectors
of $\mathcal{T}$ span a dense subspace of $\mathcal{X}_{\omega}$.

######  Definition 2.2.

For every $\mu\in\mathbf{C}$ and every $m\geq 1$, consider the analytic
functions on $\mathbf{D}$

$h_{m}(\mu,\cdot)\colon
z\mapsto\sum_{n=0}^{\infty}\mu^{n}\left(z^{(6m+4)2^{n}}-z^{(2m+1)2^{n}}\right)\quad\text{and}\quad
h_{0}(\mu,\cdot)\colon z\mapsto\sum_{n=0}^{\infty}\mu^{n}z^{2^{n+2}}.$

These functions has been introduced by Neklyudov in [nek21, Theorem 2.3] and
are the only eigenvectors that will be needed in order to obtain results
concerning the spanning of dense subspaces.

######  Proposition 2.3.

For every $m\geq 0$ and every $\mu\in\mathbf{C}$, the vector
$h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ if and only if the series
$\sum_{n\geq 0}\left|\mu\right|^{2n}/\omega((2m+1)2^{n})$ and $\sum_{n\geq
0}\left|\mu\right|^{2n}/\omega((6m+4)2^{n})$ converge. In this case
$\mathcal{T}h_{m}(\mu,\cdot)=\mu h_{m}(\mu,\cdot)$.

In particular if $\omega$ is bounded from below then $h_{m}(\mu,\cdot)$
belongs to $\mathcal{X}_{\omega}$ for every $m\geq 0$ and every
$\mu\in\mathbf{D}$. Moreover if $\omega=\omega_{0}$ then $h_{m}(\mu,\cdot)$
belongs to $\mathcal{X}$ for every $m\geq 0$ and every $\mu\in\mathbf{C}$ such
that $\left|\mu\right|<\sqrt{2}$.

###### Proof.

Let $\mu\in\mathbf{C}$ and $m\geq 1$. The function $h_{m}(\mu,\cdot)$ belongs
to $\mathcal{X}_{\omega}$ if and only if
$\sum_{n=0}^{\infty}\left|\mu\right|^{2n}/\omega((6m+4)2^{n})+\sum_{n=0}^{\infty}\left|\mu\right|^{2n}/\omega((2m+1)2^{n})<+\infty$.
And $h_{0}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ if and only if the
series $\sum_{n=0}^{\infty}\left|\mu\right|^{2n}/\omega(2^{n+2})<+\infty$.

Moreover for every $m\geq 1$ and every $z\in\mathbf{D}$ we have

$\displaystyle\mathcal{T}h_{m}(\mu,\cdot)(z)$
$\displaystyle=\sum_{n=0}^{\infty}\mu^{n}\left(z^{T((6m+4)2^{n})}-z^{T((2m+1)2^{n})}\right)$
$\displaystyle=\sum_{n=1}^{\infty}\mu^{n}\left(z^{(6m+4)2^{n-1}}-z^{(2m+1)2^{n-1}}\right)+z^{3m+2}-z^{3m+2}$
$\displaystyle=\mu h_{m}(\mu,z)$

and

$\displaystyle\mathcal{T}h_{0}(\mu,\cdot)(z)=\sum_{n=0}^{\infty}\mu^{n}z^{T(2^{n+2})}=\sum_{n=1}^{\infty}\mu^{n}z^{2^{n+1}}+z^{2}=\mu
h_{0}(\mu,z),$

which shows that the functions $h_{m}(\mu,\cdot)$, $m\geq 0$, are eigenvectors
of $\mathcal{T}$ associated to the eigenvalue $\mu$ as soon as they belong to
$\mathcal{X}_{\omega}$. ∎

Our aim is now to show that if the weight $\omega$ is bounded from below, the
eigenvectors $h_{m}(\mu,\cdot)$, $m\geq 0$, $\mu\in\mathbf{D}$, span a dense
subspace of $\mathcal{X}_{\omega}$.

######  Theorem 2.4.

If $\omega$ is bounded from below, then $\text{span}[h_{m}(\mu,\cdot);m\geq
0,\mu\in\mathbf{D}]$ is dense in $\mathcal{X}_{\omega}$.

In particular if $\omega=\omega_{0}$ then $\\{h_{m}(\mu,\cdot);m\geq
0,\mu\in\mathbf{D}\\}$ spans a dense subspace of $\mathcal{X}$.

The proof of Theorem 2.4 relies on the following lemma:

######  Lemma 2.5.

Let $f\in\mathcal{X}_{\omega},$ with $f(z)=\sum_{k=3}^{\infty}c_{k}z^{k},$
such that for every $n\geq 0$ and every $m\geq 1$

$\frac{c_{(6m+4)2^{n}}}{\omega((6m+4)2^{n})}=\frac{c_{(2m+1)2^{n}}}{\omega((2m+1)2^{n})}\quad\text{
and }\quad c_{2^{n+2}}=0.$

Then for every $k\geq 3$ such that $k\notin\\{2^{i};i\geq 2\\}$, there exist
integer sequences $(m_{n})_{n\geq 1}$, $(p_{n})_{n\geq 1}$ and $(j_{n})_{n\geq
1}$ depending on $k$ such that:

  1. (i)

$c_{k}=(\omega((2m_{1}+1)2^{p_{1}})/\omega((6m_{n}+4)2^{p_{n}}))c_{(6m_{n}+4)2^{p_{n}}}$
for every $n\geq 1$;

  2. (ii)

$m_{n}\geq 1$ for every $n\geq 1$;

  3. (iii)

$3m_{n}+2=T^{j_{n}}(k)$ for every $n\geq 1$;

  4. (iv)

$j_{n+1}>j_{n}$ if and only if $3m_{n}+2\notin\\{2^{i};i\geq 2\\}$ for every
$n\geq 1$;

  5. (v)

$(j_{n})_{n\geq 1}$ and $((6m_{n}+4)2^{p_{n}})_{n\geq 1}$ are either both
strictly increasing or both stationary.

###### Proof.

Since $k\geq 3$ and $k\notin\\{2^{i};i\geq 2\\}$, there exist $m_{1}\geq 1$
and $p_{1}\geq 0$ such that $k=(2m_{1}+1)2^{p_{1}}$. Then

$c_{k}=c_{(2m_{1}+1)2^{p_{1}}}=\frac{\omega((2m_{1}+1)2^{p_{1}})}{\omega((6m_{1}+4)2^{p_{1}})}c_{(6m_{1}+4)2^{p_{1}}}$

and

$3m_{1}+2=T(2m_{1}+1)=T^{p_{1}+1}((2m_{1}+1)2^{p_{1}})=T^{p_{1}+1}(k).$

We set then $j_{1}=p_{1}+1$.

Suppose now that $(m_{i})_{1\leq i\leq n}$, $(p_{i})_{1\leq i\leq n}$ and
$(j_{i})_{1\leq i\leq n}$ have been defined so as to satisfy the following
properties:

  1. (i)

$c_{k}=(\omega((2m_{1}+1)2^{p_{1}})/\omega((6m_{i}+4)2^{p_{i}}))c_{(6m_{i}+4)2^{p_{i}}}$
for every $1\leq i\leq n$;

  2. (ii)

$m_{i}\geq 1$ for every $1\leq i\leq n$;

  3. (iii)

$3m_{i}+2=T^{j_{i}}(k)$ for every $1\leq i\leq n$;

  4. (iv)

$j_{i+1}>j_{i}$ if and only if $3m_{i}+2\notin\\{2^{p};p\geq 2\\}$ for every
$1\leq i\leq n-1$.

Let us construct $m_{n+1}$, $p_{n+1}$ and $j_{n+1}$.

If $3m_{n}+2\in\\{2^{i};i\geq 2\\}$, then we set $m_{n+1}=m_{n}$,
$p_{n+1}=p_{n}$ and $j_{n+1}=j_{n}$. Otherwise there exist $m_{n+1}\geq 1$ and
$q_{n+1}\geq 0$ such that $3m_{n}+2=(2m_{n+1}+1)2^{q_{n+1}}.$ So

$(6m_{n}+4)2^{p_{n}}=(3m_{n}+2)2^{p_{n}+1}=(2m_{n+1}+1)2^{q_{n+1}+p_{n}+1}.$

We set then $p_{n+1}=q_{n+1}+p_{n}+1>p_{n}$. Moreover

$\displaystyle c_{k}$
$\displaystyle=\frac{\omega((2m_{1}+1)2^{p_{1}})}{\omega((2m_{n+1}+1)2^{p_{n+1}})}c_{(2m_{n+1}+1)2^{p_{n+1}}}$
$\displaystyle=\frac{\omega((2m_{1}+1)2^{p_{1}})}{\omega((2m_{n+1}+1)2^{p_{n+1}})}\frac{\omega((2m_{n+1}+1)2^{p_{n+1}})}{\omega((6m_{n+1}+4)2^{p_{n+1}})}c_{(6m_{n+1}+4)2^{p_{n+1}}}$
$\displaystyle=\frac{\omega((2m_{1}+1)2^{p_{1}})}{\omega((6m_{n+1}+4)2^{p_{n+1}})}c_{(6m_{n+1}+4)2^{p_{n+1}}}.$

Remark that

$3m_{n+1}+2=T^{p_{n+1}+1}((2m_{n+1}+1)2^{p_{n+1}})=T^{p_{n+1}+1}((6m_{n}+4)2^{p_{n}})=T^{p_{n+1}}((3m_{n}+2)2^{p_{n}}).$

So $3m_{n+1}+2=T^{p_{n+1}-p_{n}}(3m_{n}+2)=T^{p_{n+1}-p_{n}+j_{n}}(k)$ and we
set $j_{n+1}=p_{n+1}-p_{n}+j_{n}>j_{n}$. We also have
$(6m_{n+1}+4)2^{p_{n+1}}>(2m_{n+1}+1)2^{p_{n+1}}=(6m_{n}+4)2^{p_{n}}.$

Defined this way, the sequences $(m_{n})_{n\geq 1}$, $(p_{n})_{n\geq 1}$ and
$(j_{n})_{n\geq 1}$ satisfy the first four properties and it remains to prove
the last one. On the one hand, if there exists $n_{0}\geq 1$ such that
$3m_{n_{0}}+2\in\\{2^{i};i\geq 2\\}$, then the sequences $(j_{n})_{n\geq 1}$
and $((6m_{n}+4)2^{p_{n}})_{n\geq 1}$ are stationary by construction. On the
other hand if $3m_{n}+2\notin\\{2^{i};i\geq 2\\}$ for every $n\geq 1$, then
the sequences $(j_{n})_{n\geq 1}$ and $((6m_{n}+4)2^{p_{n}})_{n\geq 1}$ are
stricly increasing by construction, which proves that the fifth property is
satisfied. ∎

###### Proof of Theorem 2.4.

By Proposition 2.3, $h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ for
every $m\geq 0$ and every $\mu\in\mathbf{D}$ since $\omega$ is bounded from
below. Let $f\in\text{span}[h_{m}(\mu,\cdot);m\geq
0,\mu\in\mathbf{D}]^{\perp}$ in $\mathcal{X}_{\omega}$ with
$f(z)=\sum_{k=3}^{\infty}c_{k}z^{k}$. Our aim is to show that $f=0$. For every
$\mu\in\mathbf{D}$ and every $m\geq 1$

$\displaystyle\varphi_{m}(\mu)\colon\\!=\langle
h_{m}(\mu,\cdot),f\rangle=\sum_{n=0}^{\infty}\left(\frac{\overline{c_{(6m+4)2^{n}}}}{\omega((6m+4)2^{n})}-\frac{\overline{c_{(2m+1)2^{n}}}}{\omega((2m+1)2^{n})}\right)\mu^{n}=0,$
$\displaystyle\varphi_{0}(\mu)\colon\\!=\langle
h_{0}(\mu,\cdot),f\rangle=\sum_{n=0}^{\infty}\frac{\overline{c_{2^{n+2}}}}{\omega(2^{n+2})}\mu^{n}=0.$

For every $k\geq 3$ the radius of convergence of the power series $\sum_{n\geq
0}(\overline{c_{k2^{n}}}/\omega(k2^{n}))z^{n}$ is greater than 1. Indeed by
the Cauchy-Schwarz’s inequality

$\left(\sum_{n=0}^{\infty}\left|\frac{\overline{c_{k2^{n}}}}{\omega(k2^{n})}z^{n}\right|\right)^{2}\leq\sum_{n=0}^{\infty}\frac{\left|c_{k2^{n}}\right|^{2}}{\omega(k2^{n})}\sum_{n=0}^{\infty}\frac{\left|z\right|^{2n}}{\omega(k2^{n})}<+\infty$

for every $z\in\mathbf{D}$ because $f$ belongs to $\mathcal{X}_{\omega}$ and
$\omega$ is supposed to be bounded from below. Thus the functions
$\varphi_{m}$ and $\varphi_{0}$ are holomorphic and they identically vanish on
$\mathbf{D}$ for every $m\geq 1$. So for every $m\geq 1$

$\frac{c_{(6m+4)2^{n}}}{\omega((6m+4)2^{n})}=\frac{c_{(2m+1)2^{n}}}{\omega((2m+1)2^{n})}\quad\text{
and }\quad c_{2^{n+2}}=0\quad\text{for every $n\geq 0$.}$

######  Claim 2.6.

Let $\omega$ be a weight on $\mathbf{Z}_{+}$ which is bounded from below, and
let $f\in\mathcal{X}_{\omega}$ with $f(z)=\sum_{k=3}^{\infty}c_{k}z^{k}$.
Suppose that for every $m\geq 1$

$\frac{c_{(6m+4)2^{n}}}{\omega((6m+4)2^{n})}=\frac{c_{(2m+1)2^{n}}}{\omega((2m+1)2^{n})}\quad\text{
and }\quad c_{2^{n+2}}=0\quad\text{for every $n\geq 0$,}$

Then $c_{k}=0$ for every $k\geq 3$.

###### Proof.

Fix $k\geq 3$. If $k\in\\{2^{i};i\geq 2\\}$ then $c_{k}=0$. Otherwise consider
the sequences $(m_{n})_{n\geq 1}$, $(p_{n})_{n\geq 1}$ and $(j_{n})_{n\geq 1}$
associated to $k$ given by Lemma 2.5.

On the one hand suppose that $\\{T^{n}(k);n\geq 0\\}\cap\\{2^{i};i\geq
0\\}=\emptyset$, then in particular $3m_{n}+2\notin\\{2^{i};i\geq 2\\}$ for
every $n\geq 1$ by (iii). So by (iv), $(j_{n})_{n\geq 1}$ is strictly
increasing and then so is $((6m_{n}+4)2^{p_{n}})_{n\geq 1}$ by (v). Therefore
by (i)

$\displaystyle\|f\|_{\omega}^{2}=\sum_{j=3}^{\infty}\frac{\left|c_{j}\right|^{2}}{\omega(j)}\geq\sum_{n=1}^{\infty}\frac{\left|c_{(6m_{n}+4)2^{p_{n}}}\right|^{2}}{\omega((6m_{n}+4)2^{p_{n}})}=\sum_{n=1}^{\infty}\frac{\omega((6m_{n}+4)2^{p_{n}})}{\omega((2m_{1}+1)2^{p_{1}})^{2}}\left|c_{k}\right|^{2}.$

Since $\omega$ is bounded from below, the previous series can converge only in
the case where $c_{k}=0$.

On the other hand suppose that $\\{T^{n}(k);n\geq 0\\}\cap\\{2^{i};i\geq
0\\}\neq\emptyset$, then $(j_{n})_{n\geq 1}$ is stationary. Indeed suppose
that $(j_{n})_{n\geq 1}$ is not stationary, that is to say strictly increasing
by (v). There exists $N\geq 1$ such that $T^{N}(k)\in\\{2^{i};i\geq 0\\}$, so
$T^{n}(k)\in\\{2^{i};i\geq 0\\}$ for every $n\geq N$ and
$T^{j_{N}}(k)\in\\{2^{i};i\geq 0\\}$ since $j_{N}\geq N$. Then
$T^{j_{N}}(k)=3m_{N}+2\in\\{2^{i};i\geq 2\\}$ by (ii) and (iii), which implies
that $j_{N+1}=j_{N}$. This is impossible, so $(j_{n})_{n\geq 1}$ is indeed
stationary. Thus there exists $N\geq 1$ such that $3m_{N}+2\in\\{2^{i};i\geq
2\\}$ by (iv), which implies by (i) that

$c_{k}=\frac{\omega((2m_{1}+1)2^{p_{1}})}{\omega((6m_{N}+4)2^{p_{N}})}c_{(6m_{N}+4)2^{p_{N}}}=0$

because $(6m_{N}+4)2^{p_{N}}=(3m_{N}+2)2^{p_{N}+1}\in\\{2^{i};i\geq 2\\}$.

We have shown that $c_{k}=0$ for every $k\geq 3$. ∎

Thus Claim 2.6 gives that $f=0$, which concludes the proof of Theorem 2.4. ∎

We conclude this section by proving that the adjoint $\mathcal{T}^{*}$ of
$\mathcal{T}$ acting on $\mathcal{X}_{\omega}$ does not admit any eigenvalue.
We will denote by $\sigma_{p}(\mathcal{T}^{*})$ the set of its eigenvalues.

######  Theorem 2.7.

If $\omega$ is bounded from below then
$\sigma_{p}(\mathcal{T}^{*})=\emptyset$.

In particular if $\omega=\omega_{0}$ then
$\sigma_{p}(\mathcal{T}^{*})=\emptyset$.

The proof proceeds along the same lines as the proof of Theorem 2.4. We first
need the following Lemma, whose proof is similar to the proof of Lemma 2.5 and
will be omitted here.

######  Lemma 2.8.

Let $\mu\in\mathbf{C}\backslash\\{0\\}$ and let $f\in\mathcal{X}_{\omega}$
with $f(z)=\sum_{k=3}^{\infty}c_{k}z^{k}\in\mathcal{X}_{\omega}$. Suppose that
for every $m\geq 1$ and every $n\geq 0$

$\frac{\mu^{-1}c_{(3m+2)2^{n}}}{\omega((3m+2)2^{n})}=\frac{c_{(2m+1)2^{n}}}{\omega((2m+1)2^{n})}\quad\text{and}\quad
c_{2^{n+2}}=0.$

Then for every $k\geq 3$ such that $k\notin\\{2^{i};i\geq 2\\}$, there exist
integer sequences $(m_{n})_{n\geq 1}$, $(p_{n})_{n\geq 1}$ and $(j_{n})_{n\geq
1}$ such that:

  1. (i)

$c_{k}=(\mu^{-n}\omega((2m_{1}+1)2^{p_{1}})/\omega((3m_{n}+2)2^{p_{n}}))c_{(3m_{n}+2)2^{p_{n}}}$
for every $n\geq 1$;

  2. (ii)

$m_{n}\geq 1$ for every $n\geq 1$;

  3. (iii)

$3m_{n}+2=T^{j_{n}}(k)$ for every $n\geq 1$;

  4. (iv)

$j_{n+1}>j_{n}$ if and only if $3m_{n}+2\notin\\{2^{i};i\geq 2\\}$ for every
$n\geq 1$;

  5. (v)

$(j_{n})_{n\geq 1}$ and $((3m_{n}+2)2^{p_{n}})_{n\geq 1}$ are either both
strictly increasing or both stationary.

###### Proof of Theorem 2.7.

For every $f,g\in\mathcal{X}_{\omega}$ with
$f(z)=\sum_{n=3}^{\infty}c_{n}(f)z^{n}$ and with
$g(z)=\sum_{n=3}^{\infty}c_{n}(g)z^{n}$, we have

$\displaystyle\langle\mathcal{T}f\mid
g\rangle=\sum_{n=3}^{\infty}\frac{c_{n}(\mathcal{T}f)\overline{c_{n}(g)}}{\omega(n)}=\sum_{n=3}^{\infty}\frac{\sum_{T(j)=n}c_{j}(f)\overline{c_{n}(g)}}{\omega(n)}=\sum_{j\in\\{k\geq
3;T(k)\geq 3\\}}\frac{c_{j}(f)\overline{c_{T(j)}(g)}}{\omega(T(j))},$

so

$\displaystyle\langle\mathcal{T}f\mid
g\rangle=\frac{c_{3}(f)\overline{c_{5}(g)}}{\omega(5)}+\sum_{k=5}^{\infty}\frac{c_{k}(f)\overline{c_{T(k)}(g)}}{\omega(T(k))}.$

Then $\mathcal{T}^{*}$ is defined in the following way:

$\mathcal{T}^{*}\sum_{n=3}^{\infty}c_{n}z^{n}=\frac{\omega(3)}{\omega(5)}c_{5}z^{3}+\sum_{k=5}^{\infty}\frac{\omega(k)}{\omega(T(k))}c_{T(k)}z^{k}\quad\text{for
every }f\colon z\mapsto\sum_{n=3}^{\infty}c_{n}z^{n}\in\mathcal{X}_{\omega}.$

Let $\mu\in\mathbf{C}$, let $f\in\mathcal{X}_{\omega}$ with
$f(z)=\sum_{n=3}^{\infty}c_{n}z^{n}$ and suppose that $\mathcal{T}^{*}f=\mu
f$. We have

$\displaystyle\left\\{\begin{array}[]{l l}(\omega(3)/\omega(5))c_{5}=\mu
c_{3}\\\ (\omega(k)/\omega(T(k)))c_{T(k)}=\mu c_{k},&k\geq 5\\\ 0=\mu
c_{4},&\end{array}\right.$

which is equivalent to

$\displaystyle\left\\{\begin{array}[]{l l}(\omega(2m)/\omega(m))c_{m}=\mu
c_{2m},&m\geq 3\\\ (\omega(2m+1)/\omega(3m+2))c_{3m+2}=\mu c_{2m+1},&m\geq
1\\\ 0=\mu c_{4}.\end{array}\right.$

Firstly if $\mu=0$ then $c_{m}=0$ for every $m\geq 3$. So $0$ does not belong
to $\sigma_{p}(\mathcal{T}^{*})$.

We suppose now that $\mu\neq 0$. Then we have $\mathcal{T}^{*}f=\mu f$ if and
only if

$\displaystyle\left\\{\begin{array}[]{l
l}c_{m2^{n}}=\mu^{-n}(\omega(m2^{n})/\omega(m))c_{m},&m\geq 3,n\geq 0\\\
\mu^{-1}c_{(3m+2)2^{n}}/\omega((3m+2)2^{n})=c_{(2m+1)2^{n}}/\omega((2m+1)2^{n}),&m\geq
1,n\geq 0\\\ c_{2^{n+2}}=0,&n\geq 0\end{array}\right.$

because for every $m\geq 1$ and every $n\geq 0$

$\displaystyle
c_{(2m+1)2^{n}}=\frac{\mu^{-1}\omega((2m+1)2^{n})}{\omega((2m+1)2^{n-1})}c_{(2m+1)2^{n-1}}=\ldots=\frac{\mu^{-n}\omega((2m+1)2^{n})}{\omega(2m+1)}c_{2m+1},$

so

$\displaystyle
c_{(2m+1)2^{n}}=\frac{\mu^{-n-1}\omega((2m+1)2^{n})}{\omega(3m+2)}c_{3m+2}=\ldots=\frac{\mu^{-n-1+n}\omega((2m+1)2^{n})}{\omega((3m+2)2^{n})}c_{(3m+2)2^{n}}.$

Fix $k\geq 3$. If $k\in\\{2^{i};i\geq 2\\}$ then $c_{k}=0$. Suppose now that
$k\notin\\{2^{i};i\geq 2\\}$. If $\left|\mu\right|\leq 1$, we can remark that

$\|f\|_{\omega}^{2}=\sum_{j=3}^{\infty}\frac{\left|c_{j}\right|^{2}}{\omega(j)}\geq\sum_{n=0}^{\infty}\frac{\left|c_{k2^{n}}\right|^{2}}{\omega(k2^{n})}=\sum_{n=0}^{\infty}\frac{\left|\mu\right|^{-2n}\omega(k2^{n})}{\omega(k)^{2}}\left|c_{k}\right|^{2}.$

Since $\omega$ is bounded from below, the previous series can converge only in
the case where $c_{k}=0$ and it follows that $f=0$. Thus $\mu$ does not belong
to $\sigma_{p}(\mathcal{T}^{*})$. If $\left|\mu\right|>1$, consider the
sequences $(m_{n})_{n\geq 1}$, $(p_{n})_{n\geq 1}$ and $(j_{n})_{n\geq 1}$
associated to $k$ given by Lemma 2.8.

On the one hand suppose that $\\{T^{n}(k);n\geq 0\\}\cap\\{2^{i};i\geq
0\\}=\emptyset$, then in particular $3m_{n}+2\notin\\{2^{i};i\geq 2\\}$ for
every $n\geq 1$ by (iii). So by (iv), $(j_{n})_{n\geq 1}$ is strictly
increasing and then so is $((3m_{n}+2)2^{p_{n}})_{n\geq 1}$ by (v). Therefore
by (i)

$\|f\|_{\omega}^{2}=\sum_{j=3}^{\infty}\frac{\left|c_{j}\right|^{2}}{\omega(j)}\geq\sum_{n=1}^{\infty}\frac{\left|c_{(3m_{n}+2)2^{p_{n}}}\right|^{2}}{\omega((3m_{n}+2)2^{p_{n}})}=\sum_{n=1}^{\infty}\frac{\left|\mu\right|^{2n}\omega((3m_{n}+2)2^{p_{n}})}{\omega((2m_{1}+1)2^{p_{1}})^{2}}\left|c_{k}\right|^{2}.$

Since $\omega$ is bounded from below, the previous series converges only in
the case where $c_{k}=0$.

On the other hand suppose that $\\{T^{n}(k);n\geq 0\\}\cap\\{2^{i};i\geq
0\\}\neq\emptyset$, we will follow the lines of the proof of Theorem 2.4 and
claim that $(j_{n})_{n\geq 1}$ is stationary. Indeed suppose that
$(j_{n})_{n\geq 1}$ is not stationary, that is to say is strictly increasing
by (v). There exists $N\geq 1$ such that $T^{N}(k)\in\\{2^{i};i\geq 0\\}$, so
$T^{n}(k)\in\\{2^{i};i\geq 0\\}$ for every $n\geq N$ and
$T^{j_{N}}(k)\in\\{2^{i};i\geq 0\\}$ since $j_{N}\geq N$. Then
$T^{j_{N}}(k)=3m_{N}+2\in\\{2^{i};i\geq 2\\}$ by (ii) and (iii), which implies
that $j_{N+1}=j_{N}$. This is impossible. So $(j_{n})_{n\geq 1}$ is stationary
and there exists $N\geq 1$ such that $3m_{N}+2\in\\{2^{i};i\geq 2\\}$ by (iv).
This implies by (i) that

$c_{k}=\frac{\mu^{-N}\omega((2m_{1}+1)2^{p_{1}})}{\omega((3m_{N}+2)2^{p_{N}})}c_{(3m_{N}+2)2^{p_{N}}}=0$

because $(3m_{N}+2)2^{p_{N}}\in\\{2^{i};i\geq 2\\}$. We have shown that
$c_{k}=0$ for every $k\geq 3$. Thus $f=0$ and so $\mu$ does not belong to
$\sigma_{p}(\mathcal{T}^{*})$ either. ∎

##  3\. Linear dynamics of the operator $\mathcal{T}$

Our aim is now to understand the links between the dynamics of the Collatz map
and those of the operator $\mathcal{T}$. We begin by recalling briefly the
definition of some important properties in linear dynamics (hypercyclicity,
chaos, frequent hypercyclicity and ergodicity), before investigating whether
the operator $\mathcal{T}$ acting on $\mathcal{X}_{\omega}$ satisfies each of
these properties.

###  3.1. Hypercyclicity

A much studied dynamical property is called hypercyclicity. We refer the
reader to the books [bay09] and [gro11] for an in-depth study of this notion.
Let $X$ be a separable infinite-dimensional complex Banach (or Fréchet) space.
We will denote by $\mathcal{B}(X)$ the set of bounded (or continuous) linear
operators on $X$.

######  Definition 3.1 ([gro11, Definition 2.15]).

An operator $\mathcal{A}\in\mathcal{B}(X)$ is said to be hypercyclic if there
exists $x\in X$ such that its orbit $\\{\mathcal{A}^{n}x;n\geq 0\\}$ under
$\mathcal{A}$ is dense in $X.$ In this case $x$ is called a hypercyclic vector
for $\mathcal{A}$.

Recall that a $G_{\delta}$-set in $X$ is a countable intersection of open sets
of $X$. The following theorem shows that either the hypercyclic vectors of an
operator $\mathcal{A}\in\mathcal{B}(X)$ forms a dense $G_{\delta}$-set in $X$
or $\mathcal{A}$ does not admit any hypercyclic vectors.

######  Theorem 3.2 (Birkhoff’s transitivity theorem, [gro11, Theorems 1.16
and 2.19]).

An operator $\mathcal{A}\in\mathcal{B}(X)$ is hypercyclic if and only if for
every non-empty open sets $U$ and $V$ in $X$, there exists $n\geq 0$ such that
$\mathcal{A}^{n}(U)\cap V\neq\emptyset.$ In this case, the hypercyclic vectors
for $\mathcal{A}$ form a dense $G_{\delta}$-set in $X$.

Important examples of hypercyclic operators can be found in the literature. On
the Fréchet space $Hol(\mathbf{C})$ of entire functions endowed with the
seminorms defined by
$\|f\|_{n,\infty}=\sup\\{\left|f(z)\right|;\left|z\right|\leq n\\}$ for $n\geq
1$, the Birkhoff’s operator $T^{(a)}\colon f\mapsto f(\cdot+a)$ is hypercyclic
if and only if $a\neq 0$. The derivation operator $D$ acting on
$Hol(\mathbf{C})$ is also hypercyclic. In the Banach space setting, the
simplest examples of hypercyclic operators are given by the Rolewicz’s
operators $\lambda B$ for $\left|\lambda\right|>1$, where $B$ is the backward
shift on $\ell^{2}(\mathbf{N})$ defined by $B(x_{n})_{n\geq
1}=(x_{n+1})_{n\geq 1}$.

The following criterion provides a pratical mean to prove the hypercyclicity
of an operator.

######  Theorem 3.3 (Hypercyclicity Criterion, [gro11, Theorem 3.12]).

Let $\mathcal{A}\in\mathcal{B}(X)$. Suppose that there exist dense subsets
$X_{0}$ and $Y_{0}$ of $X$, an increasing sequence $(n_{k})_{k\geq 1}$ of
integers and a sequence $(S_{n_{k}}\colon Y_{0}\to X)_{k\geq 1}$ of maps such
that:

  1. (i)

$\mathcal{A}^{n_{k}}x\to 0$ as $k\to+\infty$ for every $x\in X_{0}$;

  2. (ii)

$S_{n_{k}}y\to 0$ as $k\to+ \infty$ for every $y\in Y_{0}$;

  3. (iii)

$\mathcal{A}^{n_{k}}S_{n_{k}}y\to y$ as $k\to+\infty$ for every $y\in Y_{0}.$

Then the operator $\mathcal{A}$ is hypercyclic.

In the case where $\omega=\omega_{0}$, and provided that the map $T$ does not
admit any non-trivial cycle, it is shown is [nek21, Theorem 2.2] that
$\mathcal{T}$ satisfies the Hypercyclicity Criterion, and is thus hypercyclic.
Our first theorem generalizes this result by showing that $\mathcal{T}$ is
hypercyclic on $\mathcal{X}_{\omega}$ under a rather mild hypothesis on the
weight $\omega$, independently of any assumption on the existence of cycles
for the Collatz map.

######  Theorem 3.4.

If $\omega$ is bounded from below and if $\omega(k2^{n})\to+\infty$ as
$n\to+\infty$ for every $k\geq 3$, then $\mathcal{T}$ is hypercyclic.

In particular if $\omega=\omega_{0}$ then $\mathcal{T}$ is hypercyclic on
$\mathcal{X}$.

###### Proof.

It suffices to prove that $\mathcal{T}$ satisfies the Hypercyclicity Criterion
(Theorem 3.3). Consider the subspace
$X_{0}=\text{span}[h_{m}(\mu,\cdot);\mu\in\mathbf{D},m\geq 0]$, which is dense
in $\mathcal{X}_{\omega}$ by Theorem 2.4. We have
$\mathcal{T}^{n}h_{m}(\mu,\cdot)=\mu^{n}h_{m}(\mu,\cdot)\to 0$ as
$n\to+\infty$ for every $\mu\in\mathbf{D}$ and every $m\geq 0$. Thus this is
true for any linear combination in $X_{0}$. Consider now the dense subspace
$Y_{0}=\text{span}[z^{k};k\geq 3]$ in $\mathcal{X}_{\omega}$ and the map
$S\colon Y_{0}\to Y_{0}$ defined by $Sz^{k}=z^{2k}$ for every $k\geq 3$. Then
$\|S^{n}z^{k}\|_{\omega}^{2}=\|z^{k2^{n}}\|_{\omega}^{2}=1/\omega(k2^{n})\to
0$ as $n\to+\infty$ and $\mathcal{T}Sz^{k}=\mathcal{T}z^{2k}=z^{T(2k)}=z^{k}$
for every $k\geq 3$. Thus this is still true for any linear combination in
$Y_{0}$. Then $\mathcal{T}$ satisfies the Hypercyclicity Criterion, so
$\mathcal{T}$ is hypercyclic. ∎

Actually, the role played by eigenvectors in the hypercyclicity of an operator
appears explicitly in the following criterion.

######  Theorem 3.5 (Godefroy-Shapiro Criterion, [gro11, Theorem 3.1]).

Let $\mathcal{A}\in\mathcal{B}(X)$. Suppose that the subspaces

$X_{0}=\text{span}[\ker(\mathcal{A}-\mu);\left|\mu\right|<1]\quad\text{and}\quad
Y_{0}=\text{span}[\ker(\mathcal{A}-\mu);\left|\mu\right|>1]$

are dense in $X$. Then $\mathcal{A}$ is hypercyclic.

This criteria is satisfied as soon as there are enough complex numbers $\mu$
such that $h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ for every
$m\geq 0$.

######  Theorem 3.6.

If $\omega$ is bounded from below and if there exists $\rho>1$ such that for
every $k\geq 3$ the sequence $(\rho^{n}/\omega(k2^{n}))_{n\geq 0}$ is bounded,
then $\mathcal{T}$ satisfies the Godefroy-Shapiro Criterion.

In particular if $\omega=\omega_{0}$, then taking $\rho=2$ gives us that
$\mathcal{T}$ satisfies the Godefroy-Shapiro Criterion.

###### Proof.

By Proposition 2.3, $h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ for
every $m\geq 0$ and every $\mu\in\mathbf{C}$ such that
$\left|\mu\right|<\sqrt{\rho}$. By Theorem 2.4, it suffices to show that the
subspace $Y_{0}=\text{span}[h_{m}(\mu,\cdot);m\geq
0,1<\left|\mu\right|<\sqrt{\rho}]$ is dense in $\mathcal{X}_{\omega}$. This
can be proved by following the steps of the proof of Theorem 2.4 and
considering the holomorphic functions $\varphi_{m}$ and $\varphi_{0}$ on
$\\{\mu\in\mathbf{C};\left|\mu\right|<\sqrt{\rho}\\}$ instead of $\mathbf{D}$.
∎

###  3.2. Frequent hypercyclicity and ergodicity

The notion of frequent hypercyclicity is a reinforcement of that of
hypercyclicity. It quantifies the frequency with which the orbit of a vector
visits a non-empty open set. We refer the reader to [bay06] for more on this
notion.

######  Definition 3.7 ([gro11, Definition 9.2]).

An operator $\mathcal{A}\in\mathcal{B}(X)$ is said to be frequently
hypercyclic if there exists a vector $x\in X$ such that for every non-empty
open set $U$ in $X$

$\liminf_{N\to+\infty}\frac{\text{card}\\{0\leq n\leq N;\mathcal{A}^{n}x\in
U\\}}{N+1}>0.$

In this case $x$ is called a frequently hypercyclic vector for $\mathcal{A}$.

The Hypercyclicity Criterion admits a frequently hypercyclic version, which is
the following theorem.

######  Theorem 3.8 (Frequent Hypercyclicity Criterion, [gro11, Theorem 9.9]).

Let $\mathcal{A}\in\mathcal{B}(X)$. Suppose that there exist a dense set
$X_{0}$ and a map $S\colon X_{0}\to X_{0}$ such that for every $x\in X_{0}$:

  1. (1)

$\sum_{n\geq 0}\mathcal{A}^{n}x$ converges unconditionally;

  2. (2)

$\sum_{n\geq 0}S^{n}x$ converges unconditionally;

  3. (3)

$\mathcal{A}Sx=x$.

Then $\mathcal{A}$ is frequently hypercyclic.

The three hypercyclic operators presented before are actually frequently
hypercyclic. The operators $\lambda B$, $D$ and $T^{(a)}$ are frequently
hypercyclic respectively on $\ell^{2}(\mathbf{N})$ and $H(\mathbf{C})$ if
$\left|\lambda\right|>1$ and if $a\neq 0$.

In order to prove that $\mathcal{T}$ is frequently hypercyclic, we will
actually show that $\mathcal{T}$ is ergodic with respect to a Gaussian measure
of full support.

Let $H$ be a complex Hilbert space, let $\mathcal{B}$ be the $\sigma$-algebra
of Borel subsets of $H$ and let $m$ be a probability measure on
$(H,\mathcal{B})$.

######  Definition 3.9 ([bay06, Definition 3.9]).

A transformation $\mathcal{A}\colon(H,\mathcal{B},m)\to(H,\mathcal{B},m)$ in
$\mathcal{B}(H)$ is said to preserve the measure $m$ if
$m(\mathcal{A}^{-1}(B))=m(B)$ for every $B\in\mathcal{B}$. A measure-
preserving transformation
$\mathcal{A}\colon(H,\mathcal{B},m)\to(H,\mathcal{B},m)$ is said to be ergodic
if for every $B\in\mathcal{B}$, $\mathcal{A}^{-1}(B)=B$ implies that
$m(B)\in\\{0,1\\}$.

The eigenvectors of an operator play an important role in the study of its
dynamics. The fact that the eigenvectors of $\mathcal{T}$ span a dense
subspace of $\mathcal{X}_{\omega}$ allowed us to show that $\mathcal{T}$ is
hypercyclic under some assumptions on the weight $\omega$. In order to show
that $\mathcal{T}$ is ergodic with respect to a Gaussian measure with full
support, we will rely on the properties of the eigenvectors of $\mathcal{T}$
associated to unimodular eigenvalues. We first recall a few revelant
definitions.

######  Definition 3.10 ([bay06, Definition 3.1]).

An operator $\mathcal{A}\in\mathcal{B}(H)$ is said to have a perfectly
spanning set of eigenvectors associated to unimodular eigenvalues if there
exists a continuous probability measure $\sigma$ on the unit circle
$\mathbf{T}$ such that the eigenvectors of $\mathcal{A}$ associated to
eigenvalues $\mu\in A$ span a dense subspace in $H$ for every
$\sigma$-measurable subset $A$ of $\mathbf{T}$ satistying $\sigma(A)=1$.

######  Definition 3.11 ([bay06, Definitions 3.13 and 3.14]).

Let $(\Omega,\mathcal{F},P)$ be a probability space. A measurable function
$f\colon(\Omega,\mathcal{F},P)\to\mathbf{C}$ is said to have complex symmetric
Gaussian distribution if $\Re(f)$ and $\Im(f)$ have independent centered
Gaussian distribution and the same variance.

A probability measure $m$ on $(H,\mathcal{B})$ is said to be a Gaussian
measure if the function $y\mapsto\langle y,x\rangle$ has complex symmetric
Gaussian distribution for every $x\in H$.

A measure $m$ on $(H,\mathcal{B})$ is said to have a full support if $m(U)>0$
for every open set $U\in\mathcal{B}$.

######  Theorem 3.12 ([bay06, Theorem 3.22]).

If an operator $\mathcal{A}\in\mathcal{B}(H)$ admits a perfectly spanning set
of eigenvectors associated to unimodular eigenvalues, then there exists a
Gaussian invariant measure with full support $m$ on $H$ such that
$\mathcal{A}\colon(H,\mathcal{B},m)\to(H,\mathcal{B},m)$ is ergodic.

We are now ready to prove the ergodicity of $\mathcal{T}$.

######  Theorem 3.13.

If $\omega$ is bounded from below and if
$\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, then
$\mathcal{T}$ is ergodic with respect to a Gaussian invariant measure with
full support on $\mathcal{X}_{\omega}$.

This is in particular the case when $\omega=\omega_{0}$.

###### Proof.

By Theorem 3.12, it suffices to prove that $\mathcal{T}$ admits a perfectly
spanning set of unimodular eigenvectors. Since
$\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, the
function $h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ for every $m\geq
0$ and every $\mu\in\mathbf{C}$ such that $\left|\mu\right|\leq 1$ by
Proposition 2.3. We will prove that for every Borel set $A\subset\mathbf{T}$
such that $\sigma(A)=1$, where $\sigma$ is the Lebesgue measure on the unit
circle, the subspace $\text{span}[h_{m}(\mu,\cdot);m\geq 0,\mu\in A]$ is dense
in $\mathcal{X}_{\omega}$.

######  Claim 3.14.

Let $\omega$ be a weight on $\mathbf{Z}_{+}$ which is bounded from below and
such that $\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$.
Then the subspace $\text{span}[h_{m}(\mu,\cdot);m\geq 0,\mu\in D]$ is dense in
$\mathcal{X}_{\omega}$ for every dense subset $D$ of $\mathbf{T}$.

###### Proof.

Let $f\in\text{span}[h_{m}(\mu,\cdot);m\geq 0,\mu\in D]^{\perp}$ in
$\mathcal{X}_{\omega}$ with $f(z)=\sum_{k=3}^{\infty}c_{k}z^{k}$. Our aim is
to prove that $f=0$. For every $\mu\in D$ and every $m\geq 1$

$\varphi_{m}(\mu)\colon\\!=\langle
h_{m}(\mu,\cdot),f\rangle=\sum_{n=0}^{\infty}\left(\frac{\overline{c_{(6m+4)2^{n}}}}{\omega((6m+4)2^{n})}-\frac{\overline{c_{(2m+1)2^{n}}}}{\omega((2m+1)2^{n})}\right)\mu^{n}=0$

and

$\varphi_{0}(\mu)\colon\\!=\langle
h_{0}(\mu,\cdot),f\rangle=\sum_{n=0}^{\infty}\frac{\overline{c_{2^{n+2}}}}{\omega(2^{n+2})}\mu^{n}=0.$

For every $k\geq 3$ the power series
$\sum_{n=0}^{\infty}(\overline{c_{k2^{n}}}/\omega(k2^{n}))z^{n}$ is uniformly
convergent on the closed disk $\overline{\mathbf{D}}$. Indeed for every $n\geq
0$ and every $z\in\overline{\mathbf{D}}$

$\left|\frac{\overline{c_{k2^{n}}}}{\omega(k2^{n})}z^{n}\right|\leq\frac{\left|c_{k2^{n}}\right|}{\sqrt{\omega(k2^{n})}}\frac{1}{\sqrt{\omega(k2^{n})}}$

and by the Cauchy-Schwarz’s inequality

$\left(\sum_{n=0}^{\infty}\frac{\left|c_{k2^{n}}\right|}{\sqrt{\omega(k2^{n})}}\frac{1}{\sqrt{\omega(k2^{n})}}\right)^{2}\leq\sum_{n=0}^{\infty}\frac{\left|c_{k2^{n}}\right|^{2}}{\omega(k2^{n})}\sum_{n=0}^{\infty}\frac{1}{\omega(k2^{n})}<+\infty$

since $f$ belongs to $\mathcal{X}_{\omega}$ and by assumption on $\omega$.
Then $\varphi_{m}$ and $\varphi_{0}$ are holomorphic on $\mathbf{D}$,
continuous on $\overline{\mathbf{D}}$ and they vanish on $D$ for every $m\geq
1$. Since $D$ is dense in the unit circle $\mathbf{T}$, the functions
$\varphi_{m}$ and $\varphi_{0}$ vanish on $\mathbf{T}$ by continuity. By the
maximum modulus principle, $\varphi_{m}$ and $\varphi_{0}$ identically vanish
on $\mathbf{D}$ for every $m\geq 1$ and

$\frac{c_{(6m+4)2^{n}}}{\omega((6m+4)2^{n})}=\frac{c_{(2m+1)2^{n}}}{\omega((2m+1)2^{n})}\quad\text{and}\quad
c_{2^{n+2}}=0\quad\text{for every $n\geq 0$.}$

It follows then from Claim 2.6 that $f=0$, which gives that
$\text{span}[h_{m}(\mu,\cdot);m\geq 0,\mu\in D]$ is dense in
$\mathcal{X}_{\omega}$. ∎

Thus since a Borel set $A\subset\mathbf{T}$ satisfying $\sigma(A)=1$ is dense
in $\mathbf{T}$, Claim 3.14 gives that $\mathcal{T}$ admits a perfectly
spanning set of unimodular eigenvectors, which concludes the proof. ∎

We now will link the ergodicity of an operator to its frequent hypercyclicity.
Let $\mathcal{A}\in\mathcal{B}(H)$ be ergodic with respect to an invariant
measure with full support. First it follows from the Birkhoff’s transitivity
theorem that $\mathcal{A}$ is hypercyclic. Then it follows from Birkhoff’s
pointwise ergodic theorem that:

######  Theorem 3.15 ([bay09, Proposition 6.23]).

If an operator $\mathcal{A}\in\mathbf{B}(H)$ is ergodic with respect to a
probability measure $m$ on $H$ with full support, then $\mathcal{A}$ is
frequently hypercyclic. Moreover the frequently hypercyclic vectors of
$\mathcal{A}$ form a set of full measure for $m$.

We deduce from Theorems 3.13 and 3.15 the following result:

######  Theorem 3.16.

If $\omega$ is bounded from below and if
$\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, then
$\mathcal{T}$ is frequently hypercyclic on $\mathcal{X}_{\omega}$.

In particular if $\omega=\omega_{0}$, $\mathcal{T}$ is frequently hypercyclic
on $\mathcal{X}$.

We finish this section by proving that, under the assumptions of Theorem 3.13,
$\mathcal{T}$ is chaotic.

######  Definition 3.17 ([gro11, Definition 2.29]).

An operator $\mathcal{A}\in\mathcal{B}(X)$ is said to be chaotic if
$\mathcal{A}$ is hypercyclic and has a dense set of periodic points.

######  Theorem 3.18.

If $\omega$ if bounded from below and if
$\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, then
$\mathcal{T}$ is chaotic on $\mathcal{X}_{\omega}$.

In particular if $\omega=\omega_{0}$, then $\mathcal{T}$ is chaotic on
$\mathcal{X}$.

###### Proof.

Since $\sum_{n=0}^{\infty}1/\omega(k2^{n})<+\infty$ for every $k\geq 3$, the
function $h_{m}(\mu,\cdot)$ belongs to $\mathcal{X}_{\omega}$ for every $m\geq
0$ and every $\mu\in\mathbf{C}$ such that $\left|\mu\right|\leq 1$ by
Proposition 2.3. By Theorem 3.4, $\mathcal{T}$ is hypercyclic and it suffices
to show that set of its perdiodic points Per($\mathcal{T}$) is dense in
$\mathcal{X}_{\omega}$. It follows from [gro11, Proposition 2.33] that
$\text{Per}(\mathcal{T})=\text{span}[h_{m}(\mu,\cdot);m\geq
0,\mu\in\\{e^{\alpha i\pi};\alpha\in\mathbf{Q}\\}],$ and its density is given
by Claim 3.14. ∎

##  4\. Open questions

We finish this paper by presenting a few open questions connected to the
results we have presented.

######  Question 4.1.

In the case where $\omega=\omega_{0}$, it follows from Propositions 2.1 and
2.3 that

$\\{z\in\mathbf{C};\left|z\right|<\sqrt{2}\\}\subset\sigma_{p}(\mathcal{T})\subset\sigma(\mathcal{T})\subset\\{z\in\mathbf{C};\left|z\right|\leq
8/3\\}.$

What is the exact value of the spectral radius $\rho(\mathcal{T})$ of
$\mathcal{T}$ ? Is it true that $\rho(\mathcal{T})=\sqrt{2}$ ?

In order to answer this question, one could perhaps build on the following
proposition, which provides an explicit expression for the norm of the
iterates $\mathcal{T}^{n}$ of $\mathcal{T}$:

######  Proposition 4.2.

If $\omega=\omega_{0}$, then for every $n\geq 1$

$\|\mathcal{T}^{n}\|^{2}=\max_{0\leq r\leq 3^{n}-1}\sum_{a\xi+b\in
P^{(n)}_{n,r}\subset\mathbf{Q}[\xi]}\frac{a}{3^{n}}$

where $P^{(n)}_{0,r}=\\{3^{n}\xi+r\\}$ and $P^{(n)}_{k+1,r}=\\{2P;P\in
P^{(n)}_{k,r}\\}\cup\\{(2P-1)/3;P\in P^{(n)}_{k,r},P(0)\equiv 2\mod 3\\}$ for
every $k\geq 0$ and $0\leq r\leq 3^{n}-1$.

###### Proof.

According to Proposition 2.1

$\|\mathcal{T}^{n}\|^{2}=\max_{0\leq r\leq
3^{n}-1}\left\\{\sup_{m\geq\delta_{r}}\sum_{T^{n}(j)=3^{n}m+r}\frac{j+1}{3^{n}m+r+1}\right\\}$

where $\delta_{r}=1$ if $r\leq 2$ and $\delta_{r}=0$ otherwise. Since $T(j)=k$
if and only if $j=2k$ or $j=(2k-1)/3$ if $k\equiv 2\mod 3$, the sets
$P^{(n)}_{n,r}$ are such that $T^{n}(j)=3^{n}m+r$ if and only if
$j\in\\{P(m);P\in P^{(n)}_{n,r}\\}$ for every $0\leq r\leq 3^{n}-1$ and every
$m\geq\delta_{r}$. Then

$\displaystyle\|\mathcal{T}^{n}\|^{2}$ $\displaystyle=\max_{0\leq r\leq
3^{n}-1}\left\\{\sup_{m\geq\delta_{r}}\sum_{a\xi+b\in
P^{(n)}_{n,r}}\frac{am+b+1}{3^{n}m+r+1}\right\\}.$

Fix $0\leq r\leq 3^{n}-1$. If $\alpha,\beta,\gamma,\delta>0$, one can remark
that a sequence $((\alpha m+\beta)/(\gamma m+\delta))_{m\geq 0}$ is strictly
increasing or constant if and only if $\alpha\delta\geq\beta\gamma$. We will
prove by recursion that $a(r+1)\geq 3^{n}(b+1)$ for every $a\xi+b \in
P^{(n)}_{n,r}$. Indeed this is firstly true for
$P^{(n)}_{0,r}=\\{3^{n}\xi+r\\}$. Besides if $P^{(n)}_{k,r}$ satisfies it, let
$P\in P^{(n)}_{k+1,r}$. Either $P=2a\xi+2b$ with $a\xi+b\in P^{(n)}_{k,r}$, or
$P=2a\xi/3+(2b-1)/3$ with $a\xi+b\in P^{(n)}_{k,r}$ and $b\equiv 2\mod 3$. On
the one hand we would have $2a(r+1)\geq 3^{n}(2b+2)>3^{n}(2b+1)$, and on the
other hand we would have $2a(r+1)/3\geq 3^{n}(2b+2)/3\geq 3^{n}((2b-1)/3+1)$.
Then this is true for $P^{(n)}_{k+1,r}$, which proves that $P^{(n)}_{n,r}$
satisfies it. So for every $a\xi+b\in P^{(n)}_{n,r}$

$\sup_{m\geq\delta_{r}}\frac{am+b+1}{3^{n}m+r+1}=\lim_{m\to+\infty}\frac{am+b+1}{3^{n}m+r+1}=\frac{a}{3^{n}},$

which concludes the proof. ∎

######  Question 4.3.

If $\liminf_{n\to+\infty}\omega(n)=0,$ could $\mathcal{T}$ be bounded,
hypercyclic or ergodic ? Could $\mathcal{T}^{*}$ have any eigenvalue ?

The operator $\mathcal{T}$ can actually be seen as acting on a space of
sequences $(c_{n})_{n\geq 3}$ instead of on a space of holomorphic functions
$f\colon z\mapsto\sum_{n=3}^{\infty}c_{n}z^{n}$.

######  Question 4.4.

What would remain from these results if we considered $\mathcal{T}$ as acting
on the space of complex sequences $\mathbf{C}^{\mathbf{N}}$ ?

## References