# Ion Dynamics Across a Low Mach Number Bow Shock

###### Abstract

A thorough understanding of collisionless shocks requires knowledge of how
different ion species are accelerated across the shock. We investigate a bow
shock crossing using the Magnetospheric Multiscale spacecraft after a coronal
mass ejection crossed Earth, which led to solar wind consisting of protons,
alpha particles, and singly charge helium ions. The low Mach number of the bow
shock enabled the ions to be distinguished upstream and sometimes downstream
of the shock. Some of the protons are specularly reflected and produce quasi-
periodic fine structures in the velocity distribution functions downstream of
the shock. Heavier ions are shown to transit the shock without reflection.
However, the gyromotion of the heavier ions partially obscures the fine
structure of proton distributions. Additionally, the calculated proton moments
are unreliable when the different ion species are not distinguished by the
particle detector. The need to high time-resolution mass-resolving ion
detectors when investigating collisionless shocks is discussed.

JGR-Space Physics

Swedish Institute of Space Physics, Uppsala, Sweden. Space and Plasma Physics,
Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden.
Southwest Research Institute, San Antonio, TX, USA. University of Texas at San
Antonio, San Antonio, TX, USA.

D. B<EMAIL_ADDRESS>
Protons, singly charged helium ions, and alpha particles are observed upstream
and downstream of a bow shock crossing.

The alpha particles and helium ions partly obscure the fine structure of the
downstream proton distributions.

High time-resolution mass-resolving ion detectors are needed to study the ion
dynamics across collisionless shocks.

## Plain Language Summary

One of the ongoing challenges when investigating collisionless shocks is
determining the energy partition between electromagnetic fields and different
particle species. Resolving this question requires detailed observations of
the electromagnetic fields and particle distributions, and is challenging when
multiple ion species are present. We investigate a crossing of Earth’s bow
shock for unusual solar wind conditions; three ion species are observed in the
solar wind and behind the bow shock, namely protons, alpha particles, and
singly charged helium ions. We investigate the ion dynamics and show that a
small fraction of protons are reflected by the electric field associated with
the shock, which results in complex ion distributions. However, since the
highest time-resolution ion detectors cannot distinguish between different ion
species, the heavier ions partly obscure the fine structure of the protons.
The heavier ions lead to errors when calculating the bulk properties (e.g.,
moments) of protons. These observations illustrate the need for high time-
resolution ion detectors, which can distinguish different ion species when
studying shocks.

##  1 Introduction

Shocks are ubiquitous throughout the universe and are regions of intense
particle acceleration. Earth’s bow shock is the most studied collisionless
shock using spacecraft observations. At the bow shock, the supersonic solar
wind is heated and compressed to form the magnetosheath plasma, which convects
around Earth’s magnetic field. Near Earth, the solar wind is primarily
composed of electrons, protons, and alpha particles. Protons and alpha
particles crossing the bow shock typically undergo different dynamics
[Fuselier Schmidt (1994)]. In rare cases, significant contributions of singly
charged helium ions have been observed following interplanetary shocks
[Gosling . (1980), Schwenn . (1980), Zwickl . (1982)]. Understanding the
dynamics of the different ion species is crucial to understanding how energy
is distributed among the different particle species.

At Earth’s bow shock, the plasma is essentially collisionless so the
dissipation required to maintain the shock must come from other mechanisms,
such as wave-particle interactions [Wu . (1984), Wilson III . (2014)]. For
supercritical shocks, additional dissipation from ion reflection becomes
crucial [Paschmann . (1980)]. Previous observations have shown that ion
reflection is extremely common at the bow shock [Paschmann . (1982)] and can
occur for subcritical shocks. For quasi-perpendicular shocks, ions reflected
by the shock are turned by the magnetic field and accelerated by the
convection electric field, so they gain energy and are transmitted through the
shock [Paschmann . (1982)]. Generally, protons are reflected, while the
heavier alpha particles cross the shock without reflection, although there are
cases where reflected alpha particles are observed [Broll . (2018)].

Investigating the fine structure of ion distributions across and downstream of
the bow shock has proved difficult from spacecraft observations. Most ion
detectors have relied on the spacecraft spin to construct three-dimensional
(3D) ion distributions, limiting the cadence to the spin period. Reflected
ions have typically been observed in the shock foot [Sckopke . (1983)], while
downstream of the shock the reflected ions are observed as an energetic
component of the distribution [Sckopke . (1990)]. However, with the
Magnetospheric Multiscale (MMS) mission, ion distributions are resolved at
substantially higher cadence, enabling distributions to be resolved over
kinetic scales by the Fast Plasma Investigation (FPI) instrument [Pollock .
(2016)]. However, FPI does not distinguish between different ion species,
making it challenging to investigate the behavior of different ion species. At
present, ion detectors that distinguish different ion species operate at low
cadences and cannot resolve ion kinetic scales at the bow shock.

Here we investigate a low Mach number bow shock observed by MMS. Protons,
alpha particles, and singly charged Helium ions are observed upstream and
downstream of the bow shock. We investigate the dynamics of the three ion
species across the bow shock using observations and numerical modelling. We
show that the ion dynamics can be investigated due to the low degree of
heating across the shock. We discuss the need for high-resolution mass-
resolving particle instruments when investigating collisionless shocks.

##  2 Observations

We use data from the MMS spacecraft; magnetic field ${\bf B}$ data from
Fluxgate Magnetometer (FGM) [Russell . (2016)], electric field ${\bf E}$ data
from Electric field Double Probes (EDP) [Lindqvist . (2016), Ergun . (2016)],
particle distributions and moments from Fast Plasma Investigation FPI [Pollock
. (2016)], and ion data from Hot Plasma Composition Analyzer (HPCA) [Young .
(2016)]. We use both fast and burst mode data. In burst mode FPI measures ion
distributions and moments every $150$ ms. Particles are sorted by energy per
charge. In contrast, HPCA measures distributions and moments of different ion
species (protons, Helium ions, alpha particles, and Oxygen) but the temporal
resolution is limited to $10$ s in both fast and burst mode. Both ion
detectors cover all directions when constructing 3D distributions, and as a
result lack the angular and energy resolutions to accurately resolve the fast
and cold ion distributions of the solar wind. For FPI the angular channels are
separated by $11.25^{\circ}$, while the energy resolution is $\Delta
E/E\approx 0.13$ with logarithmically spaced energy channels. We primarily use
data from MMS1.

We investigate a bow shock crossing observed on 24 April 2023 at 03:50:12 UT.
The spacecraft were located at $(13.7,-15.9,-9.3)$ $R_{E}$ in Geocentric Solar
Ecliptic (GSE) coordinates, where $R_{E}$ is Earth’s radius. Prior to the bow
shock crossings encountered on this day, MMS observed a Coronal Mass Ejection
(CME) shock in the solar wind at 17:34:32 UT on 23 April 2023. At
$\sim$01:46:20 UT on 24 April 2023 MMS transitioned from the CME sheath to the
magnetic cloud, shown in Figures 1a–1c, which provide an overview of the bow
shock crossings. The solar wind plasma becomes substantially colder and ${\bf
B}$ becomes steady and large amplitude. In the CME sheath and magnetic cloud
${\bf B}$ is unusually large compared with the typical solar wind at 1 AU.
After the transition to the magnetic cloud, we observe five bow shock
crossings, characterized by sharp changes in ${\bf B}$, broadening of the ion
energy fluxes, and increased energy of the electron energy fluxes. During the
solar wind magnetic cloud intervals we often see three distinct lines in the
ion energy flux (Figure 1b), which are most clearly seen just after 03:50 UT.
These three peaks correspond to solar wind protons (H+) (lowest energy line),
alpha particles (He2+), and singly charge Helium (He+) ions (highest energy
line). Although He+ are generally not observed in the solar wind, He+ can
occur in magnetic clouds behind CMEs [Schwenn . (1980), Zwickl . (1982)].

Figure 1: Overview of the solar wind and bow shock crossings observed on 24
April 2023 by MMS1. (a)–(c) Solar wind and bow shock crossings. (a) ${\bf B}$
in GSE coordinates. (b) Ion omni-directional energy flux. (c) Electron omni-
directional energy flux. (d)–(h) Bow shock crossing observed at the time
indicated by the yellow-shaded region in panel (a). (d) ${\bf B}$ in GSE
coordinates. (e) Ion omni-directional energy flux. (f) Electron omni-
directional energy flux. (g) Ion number densities $n_{i}$ and (h)
$x$-component of bulk ion velocities ${\bf V}_{i}$. The black lines are FPI
moments and the blue, red, and green lines are HPCA moments for H+, He2+, and
He+, respectively.

The shock crossing is characterized by a sharp decrease in $|{\bf B}|$,
corresponding to the shock ramp. The ion differential energy flux shows three
bands upstream of the shock corresponding to H+, He2+, and He+ (Figure 1e).
Downstream of shock the fluxes of He2+ and He+ exhibit quasi-periodic
fluctuations in energy due to their gyromotion. The helium ions have
approximately half the frequency of the alpha particle fluctuations due to the
lower gyroperiod. The He2+ fluxes often overlap in energy with the H+ fluxes.
The three flux components remain narrow in energy across the shock, meaning
the ion components remain relatively cold downstream of the shock. The
electron fluxes show that electrons are heated across the shock (Figure 1f).

Figures 1g and 1h show the ion density $n_{i}$ and sunward component of the
ion bulk velocity ($V_{i,x}$) measured by FPI and HPCA. The $n_{i}$ calculated
from FPI is significantly larger than the H+ density $n_{H+}$ measured by
HPCA. There are significant number densities of He2+ and He+ upstream and
downstream of the shock. Both FPI and HPCA cannot fully resolve solar wind
distributions, so there is significant uncertainty in the density moments.
Figure 1h shows that the ion velocities of H+, He2+, and He+ agree with each
other across the shock, confirming that there are He+ ions in the solar wind.
We note that $V_{i,x}$ calculated from FPI is significantly larger than that
calculated from HPCA, primarily due to the He2+ and He+ fluxes at higher
energies being included in the calculation of the moments, which results in an
overestimate of ${\bf V}_{i}$.

To estimate the H+ velocity we compute the one-dimensional (1D) reduced
distributions from FPI and take the velocity where the phase-space density
$f_{i}$ peaks. In the spacecraft frame we obtain ${\bf V}_{u}=(-510,40,-100)$
km s-1 upstream of the shock and ${\bf V}_{d}=(-370,60,-150)$ km s-1
downstream of the shock in GSE coordinates. These velocities are smaller than
the FPI moments, but slightly larger than those from HPCA. Using these
velocities and the average upstream and downstream ${\bf B}$ we can define a
shock-aligned coordinate system. Using the mixed-mode methods based on changes
in ${\bf B}$ and ${\bf V}$ we calculate the shock normal direction to be
$\hat{\bf n}=(0.93,0.10,-0.36)$ in GSE coordinates [Abraham-Shrauner (1972)].
From $\hat{\bf n}$ we obtain a shock normal angle of $\theta_{Bn}=87^{\circ}$,
corresponding to a nearly perpendicular shock. Based on four-spacecraft timing
of ${\bf B}$ we estimate the shock speed to be $V_{sh}\approx-130$ km s-1
along $\hat{\bf n}$. In the shock frame $V_{u,n}=-320$ km s-1 and
$V_{d,n}=-170$ km s-1.

Figure 2 provides an overview of the bow shock crossing observed by MMS1 in
$(\hat{\bf n},\hat{\bf t}_{1},\hat{\bf t}_{2})$ coordinates, where $\hat{\bf
t}_{1}$ is along the upstream ${\bf B}$, and $\hat{\bf t}_{2}$ completes the
right-hand coordinate system. Figure 2b shows that there is a sharp increase
in the electron number density $n_{e}$ across the shock coinciding with the
increase in $|{\bf B}|$. Figures 2c–2e show the 1D reduced ion distributions
$f_{i}$ along $v_{n}$, $v_{t1}$, and $v_{t2}$, respectively. To calculate the
reduced distributions we convert energy $E$ to speed using
$v=\sqrt{2eE/m_{p}}$, where $e$ is the unit charge and $m_{p}$ is the proton
mass. This will result in the speeds of He2+ and He+ being overestimated by a
factor of $\sqrt{2}$ and $2$, respectively. Across the shock we observe that
$f_{i}$ sharply drops to lower $v_{n}$. In the downstream region we observe
the quasi-periodic fluctuations in He2+ and He+ in the $v_{n}$ and $v_{t2}$
components, although both He2+ and He+ are observed for $v_{n}<0$ and
$v_{t2}>0$. There is significant overlap between the H+ and He2+ species.
Likewise, the H+ and He+ components overlap, although there are intervals
where the ions are well separated in velocity space. Figure 2d shows that
$f_{i}(v_{t1})$ does not change significantly across the shock. Thus ion
heating is largely perpendicular to ${\bf B}$. Additionally, quasi-periodic
fine structures in $f_{i}$ are observed for $v_{n}>0$ and $v_{t2}<0$
downstream of the shock. In Figures 2c–2e we overplot ${\bf V}_{i}$ computed
from FPI. Due to the presence of He2+ and He+, ${\bf V}_{i}$ overestimates the
bulk H+ speed. This is most evident in Figure 2c, where $V_{i,n}$ is offset
from the peak in $f_{i}$.

Figure 2: Overview of the bow shock crossing observed by MMS1. (a) ${\bf B}$
in ($\hat{\bf n}$,$\hat{\bf t}_{1}$,$\hat{\bf t}_{2}$) coordinates. (b)
$n_{e}$. (c)–(e) 1D reduced ion distributions in the $\hat{\bf n}$, $\hat{\bf
t}_{1}$, and $\hat{\bf t}_{2}$ directions, respectively. The black lines show
the ion bulk velocity calculated from FPI. Panels (f)–(i) show a zoomed in
interval across the ramp indicated by the yellow shading in panels (a) and
(b). (f) ${\bf B}$ in ($\hat{\bf n}$,$\hat{\bf t}_{1}$,$\hat{\bf t}_{2}$)
coordinates. (g) Reduced ion distributions along $\hat{\bf n}$. (h) 1D reduced
electron distributions in the direction parallel to ${\bf B}$. (i) Frequency-
time spectrogram of ${\bf E}$. The white line is $f_{pe}$ calculated from
electron moments.

Figures 2f–2i show the properties of the shock ramp. There is an enhancement
in $f_{i}$ for $v_{n}>0$ ahead of the ramp observed over two ion distributions
$\sim 0.3$ s (Figure 2g) due to ion reflection at the ramp. Figure 2h shows
the 1D reduced electron distribution as a function of $v_{e,||}$, where
$v_{e,||}$ is the electron speed parallel to ${\bf B}$. Across the shock
$f_{e}$ broadens, corresponding to electron heating. Ahead of the ramp we
observe weak electron beams at $v_{e,||}\sim 10^{4}$ km s-1. These are
reflected and accelerated electrons, which are typical of quasi-perpendicular
shocks. Figure 2i shows the frequency-time spectrogram of the high-frequency
${\bf E}$ across the shock. The downstream spectrum is characterized by
broadband fluctuations with frequency $f$ below $10$ kHz. Just upstream of the
shock we observe Langmuir waves at the local electron plasma frequency
$f_{pe}$, likely generated by the reflected electrons. We estimate the
upstream density from the frequency of the Langmuir waves to be $n_{e}=9$
cm-3, corresponding to $f_{pe}=27$ kHz. Thus, $n_{e}$ is underestimated by FPI
upstream of the shock. Assuming flux conservation across the shock, we
estimate $n_{e}=17$ cm-3 downstream of the shock, in agreement with the
observed $n_{e}$. Using these $n_{e}$ and the scalar electron pressure $P_{e}$
calculated from FPI, we calculate the upstream and downstream electron
temperatures to be $T_{e}=10$ eV and $T_{e}=37$ eV, respectively. We estimate
the Alfvén and fast magnetosonic Mach numbers to be $M_{A}\approx 1.4$ and
$M_{f}\approx 1.4$, which are very low values for Earth’s bow shock [Lalti .
(2022)]. Due to the presence of multiple ion species and the unreliability of
the temperature measurements for very cold ion populations, it is difficult to
calculate meaningful ion temperatures $T_{i}$ for this event, although $T_{i}$
does not significantly affect the calculation of $M_{f}$ in this case. The
shock properties correspond to a subcritical shock [Kennel . (1985)], so ion
reflection is expected to be minimal.

##  3 Ion motion

We investigate the motion of the different ion species across the bow shock
using a test-particle model. We perform the calculations in the normal
incidence frame (NIF), and transform the particle motion to the spacecraft
frame for direct comparison with observations. To model the bow shock we
assume a perpendicular shock with ${\bf B}$, $n_{e}$, and $P_{e}$ profiles
given by

$B_{t1}(n)=-B_{0}\tanh{\left(\frac{n}{l}\right)}+B_{1},$ (1)
$n_{e}(n)=-n_{0}\tanh{\left(\frac{n}{l}\right)}+n_{1},$ (2)
$P_{e}(n)=-P_{0}\tanh{\left(\frac{n}{l}\right)}+P_{1},$ (3)

where $l$ is the characteristic width of the ramp, $B_{0}$, $B_{1}$, $n_{0}$,
$n_{1}$, $P_{0}$, and $P_{1}$ determine the jump conditions, and $n$ is the
distance along $\hat{\bf n}$. To model the observed shock we use $B_{0}=13.5$
nT, $B_{1}=45.5$ nT, $n_{0}=4$ cm-3, $n_{1}=13$ cm-3, $P_{0}=4.25\times
10^{-2}$ nPa, $P_{1}=5.75\times 10^{-2}$ nPa, and $l=10$ km. The current
density ${\bf J}$ is then given by

$J_{t2}(n)=-\frac{B_{0}\mathrm{sech}^{2}\left(\frac{n}{l}\right)}{\mu_{0}l}.$
(4)

The electric field along $\hat{\bf n}$ and $\hat{\bf t}_{2}$ is given by

$E_{n}=-\frac{1}{en_{e}(n)}\left(J_{t2}(n)B_{t1}(n)+\frac{\partial
P_{e}(n)}{\partial n}\right),$ (5)
$E_{t2}=-\frac{V_{u,n}(n_{1}-n_{0})B_{t1}(n)}{n_{e}(n)},$ (6)

where $V_{u,n}$ is the upstream speed in the NIF. The gradient in $P_{e}$
contributes to only $\approx 10$ % of the total $E_{n}$. Equation (6)
determines the average ${\bf V}_{i}$ upstream and downstream of the shock via
${\bf E}\times{\bf B}$ drift. The particle velocities and trajectories are
calculated by integrating the usual equations of motion

$\frac{d{\bf x}}{dt}={\bf v},\frac{d{\bf v}}{dt}=\frac{Ze}{m_{i}}\left({\bf
E}+{\bf v}\times{\bf B}\right),$ (7)

where $Z$ is the charge number of the ion and $m_{i}$ is the ion mass.

Figure 3: Ion motion across the model bow shock. (a) Profiles of $B_{t1}$,
$E_{n}$, and $E_{t2}$. (b) $v_{n}$ versus $n$ and (c) $v_{t2}$ versus $n$. (d)
$v_{n}$ versus $n$ and (e) $v_{t2}$ versus $n$ in the MMS-FPI frame. The blue,
red, and yellow lines indicate H+, He2+, and He+, respectively. (f) ${\bf B}$
observed by MMS1. (g) and (h) 1D reduced ion distributions along $v_{n}$ and
$v_{t2}$.

Figure 3a shows the profiles of the model ${\bf B}$ and ${\bf E}$. For these
shock conditions $E_{t2}$ is uniform across the shock. In Figures 3b and 3c we
plot the velocities $v_{n}$ and $v_{t2}$ of H+, He2+, and He+ versus $n$ in
the NIF. For each species the upstream initial condition is
$v_{n}=V_{u,n}=-320$ km s-1, so there is no force on the particles upstream of
the shock and ${\bf v}$ is constant. At the ramp there is sudden drop in
$v_{n}$ due to the cross-shock potential $\phi=-\int E_{n}dn=512$ V. After
this decrease in $v_{n}$ due to $\phi$ the ions begin cyclotron turning
because they are no longer moving at the ${\bf E}\times{\bf B}$ velocity. This
initially results in a further decrease in $v_{n}$ and $v_{t2}<0$. After this
point the ions undergo periodic gyromotion, with $v_{n}$ remaining negative
while $v_{t2}$ fluctuates around $0$.

We now consider how the ion motion will be observed in the spacecraft frame
using FPI. This conversion is given by

${\bf v}_{i,sc}=\sqrt{\frac{Zm_{i}}{m_{p}}}\left({\bf v}_{i}+{\bf
V}_{sh}\right),$ (8)

where ${\bf V}_{sh}$ = (-130, 120, 240) km s-1 is the velocity of the NIF in
($\hat{\bf n}$,$\hat{\bf t}_{1}$,$\hat{\bf t}_{2}$) coordinates relative to
the spacecraft. The dependence on $m_{i}$ and $Z$ result from assuming the H+
mass and charge when converting particle energy to speed. The results of
equation (8) are shown in Figures 3d and 3e. The frame transformation shifts
the ions apart in velocity compared with the NIF where all the ions overlap.
Upstream of the shock, the ions are well separated in velocity allowing them
to be distinguished by FPI. Downstream of the shock there are intervals where
the different ions are well separated from each other in velocity space. In
particular, He+ is largely separated from the other ions. Figures 3g and 3h
show the reduced distributions along $v_{n}$ and $v_{t2}$ for comparison.
Qualitatively, these periodic motions in He2+ and He+ are seen in the reduced
distributions, along with the separation between the ions upstream of the
shock.

##  4 Ion distributions

We now extend the test-particle model to investigate the predicted
distributions downstream of the shock. To model the ion distributions
downstream of the shock we use Liouville’s theorem, which states that the
phase-space density is constant along particle trajectories. This is expressed
as

$f({\bf x}_{1},{\bf v}_{1})=f({\bf x}_{0},{\bf v}_{0}),$ (9)

where ${\bf x}_{1}$ and ${\bf v}_{1}$ are solutions to equation (7), and ${\bf
x}_{0}$ and ${\bf v}_{0}$ refer the initial conditions, the solar wind in this
case. We assume that the fields governing the equations of motion and upstream
solar wind distributions do not change in time, so we can neglect the time
dependence. We assume the solar wind ion distributions are Maxwellian. In this
model, there are no forces acting on the ions along $\hat{\bf t}_{1}$ so we
use a two-dimensional (2D) reduced distribution for the solar wind given by

$f_{i}(v_{n},v_{t2})=\frac{n_{i}}{\pi
v_{i}^{2}}\exp{\left(-\frac{(v_{n}-V_{sw,n})^{2}+v_{t2}^{2}}{v_{i}^{2}}\right)},$
(10)

where $v_{i}=\sqrt{2k_{B}T_{i}/m_{i}}$ is the ion thermal speed and $k_{B}$ is
Boltzmann’s constant. Figure 2d shows that there is little change in the
distribution along $v_{t1}$ across the bow shock, supporting the use of the
reduced 2D distribution for the model. For the solar wind we assume a H+
temperature of $T_{p}=3$ eV and that He2+ and He+ have the same thermal speed
as H+ ($T_{i}=12$ eV). For the H+ number density we use $n_{H+}=8$ cm-3 and
for He2+ and He+ we assume densities of $n_{He2+}=0.1n_{H+}$ and
$n_{He+}=0.01n_{H+}$. We compute $f({\bf x}_{1},{\bf v}_{1})$ by integrating
backwards in time using ${\bf x}_{1}$ and ${\bf v}_{1}$ as initial conditions.
For trajectories reaching the upstream, equations (9) and (10) are used to
compute $f({\bf x}_{1},{\bf v}_{1})$, otherwise $f({\bf x}_{1},{\bf v}_{1})=0$
is used.

Figure 4: Ion distributions across the shock predicted from Liouville mapping
and comparison with distributions observed by MMS1. (a) Profile of $B_{t1}$
and $\phi$ of the model bow shock. (b)–(d) 1D reduced distributions along
$v_{n}$ for H+, He2+, and He+, respectively. (e)–(g) 1D reduced distributions
along $v_{t2}$ for H+, He2+, and He+, respectively. These distributions are
plotted in the NIF. (h)–(j) ${\bf B}$, 1D reduced distributions along $v_{n}$
and $v_{t2}$ observed by MMS1. (k) and (l) Modelled 1D reduced distributions
along $v_{n}$ and $v_{t2}$ in the MMS-FPI frame. The H+, He2+, and He+
distributions are converted to energies measured by FPI.

The resulting distributions of H+, He2+, and He+ across the shock are shown in
Figure 4. Figures 4b–4d show the distributions along $v_{n}$ in the NIF for
H+, He2+, and He+, and Figures 4e–4g show the distributions along $v_{t2}$ in
the same format. The H+ distributions downstream of the shock consist of
directly transmitted H+ and a small fraction of reflected H+ ($\sim 10$ % in
the model). The H+ transmitted without reflection undergo periodic
fluctuations along $v_{n}$ and $v_{t2}$. The reflected H+ are specularly
reflected by $\phi$ into the upstream region where they are accelerated by the
convection electric field $E_{t2}$, and have significantly higher velocities
when they return to the shock and are transmitted across the ramp. These
reflected H+ are responsible for the fine structures seen in Figures 4b and
4e, and are consistent with those observed by FPI (Figures 4i and 4j). We note
that there are some differences between the observed reflected H+ and in the
model. In observations for $v_{n}>0$ we see hook-shaped structures rather than
the full loop associated with reflected H+. For $v_{t2}<0$ we observe two
narrow bands associated with reflected H+ that meet at the largest values of
$|v_{t2}|$ in the model, while in observations we only observe a single line.
These differences may result from the angular and energy resolution of FPI,
which may not fully resolve these fine structures.

For He2+ and He+ the distributions are decelerated by $\phi$ and compressed
across the ramp. The shock potential is too small to reflect these ions. The
distributions gyrate downstream of the shock, which is consistent with
observations. Additionally, the densities and temperatures fluctuate and peak
where $v_{n}$ is closest to $0$, primarily due to broadening in $f_{i}$ along
$v_{t2}$. This results in non-gyrotropic ion distributions. The He2+ and He+
distributions averaged over multiple gyrations result in ring-like
distributions in the $v_{n}$–$v_{t2}$ plane.

In Figures 4k and 4l we convert the ion distributions to the MMS-FPI frame and
overlay the three species together to model the observed distributions. When
compared with the observed distributions by FPI over a comparable spatial
scale (Figures 4i and 4j), we find very good agreement between observations
and the model. For He2+ and He+ the transformation broadens the ion
distributions, and is most evident for He+. Figure 4k shows that along $v_{n}$
the He+ distribution overlaps with the reflected H+ for $v_{n}<0$, obscuring
the observation of the fine structures associated with reflected H+.
Similarly, along $v_{t2}$ the He+ distribution overlaps with the reflected H+
for $v_{t2}>0$. The He2+ distributions obscure the periodic fluctuations in
the distribution of transmitted H+.

Figure 5: 2D ion distributions in the $v_{n}$–$v_{t2}$ plane in the MMS
spacecraft reference frame. (a)–(e) Ion distributions observed by FPI at the
times indicated by the magenta lines in Figure 4i. (f)–(j) Model ion
distributions in the MMS-FPI frame at the locations indicated by the magenta
lines in Figure 4k. Model distributions present all ion species. (e) and (j)
correspond to the shock foot, while the remaining distributions are behind the
ramp with increasing distances from right to left. The blue dots indicate the
solar wind speed and the blue circle indicates the velocities where specularly
reflected H+ should be observed.

To further investigate the ion distributions downstream of the shock we
compare observed 2D reduced distributions in the $v_{n}$–$v_{t2}$ plane with
those predicted from the model, shown in Figure 5. Figures 5a–5e show ion
distributions observed by FPI at the times indicated by the magenta lines in
Figure 4i, while Figures 5f–5j show model distributions at the positions
indicated by the magenta lines in Figure 4k. For direct comparison, the model
distributions are presented in the MMS-FPI frame and taken from positions
corresponding to the observed distributions. Figures 5e and 5j show the ion
distributions in the foot, where reflected H+ are observed. In observations
and the model the reflected H+ occur in the same region of velocity space and
have velocities in good agreement with the theoretical prediction of speeds of
$|{\bf v}-{\bf V}_{u}|=2V_{u,n}$ for specularly reflected H+ in the NIF
(indicated by the blue circles in Figure 5). The incoming ion distributions
approximately correspond to the solar wind ion distributions, and are
relatively well separated from each other.

Figures 5a–5d and 5f–5i show distributions just downstream of the shock. In
each case the relative positions of the H+, He2+, and He+ distributions agree
between observations and the model. Figures 5d and 5i show distributions
around where the He2+ temperature is expected to peak. The H+ and He2+
distributions overlap significantly, while the He+ distribution is relatively
well separated from the other distributions. Figures 5c and 5h show
distributions where the He+ temperature is expected to peak. The model shows a
crescent-shaped distribution of He+, which overlaps with the H+ and He2+
distributions. The observed distribution shows the same overlaps with the H+
and He2+ distributions, although the crescent shape of He+ is not resolved.
The remaining distributions show significant overlap between H+ and He2+,
while the He+ distribution is more separated from these distributions.

In each of the observed distributions we see the reflected H+ in different
regions of velocity space. Comparing with the model we see that these
reflected H+ occur around the same velocities as predicted by the model. In
the model some of the reflected H+ occur in or near the same region of
velocity space as the He+ distributions, specifically in Figures 5f, 5h, and
5i. Thus, the He+ ions partly obscure some of the reflected H+ in the observed
distributions.

In summary, we find excellent agreement between the observed and modeled ion
distributions. A small fraction of H+ are specularly reflected at the shock
due to $\phi$, resulting in quasi-periodic fine structures in the observed ion
distributions. In contrast, He2+ and He+ transit the shock without any
reflection. Quasi-periodic fluctuations in He2+ and He+ are observed, which
are due to the gyromotion of these ions downstream of the shock. These
gyrating ions are observed by MMS and partly obscure the reflected H+.

##  5 Discussion

We now discuss the effects of He2+ and He+ on the particle moments downstream
of the shock. When computing ion moments with FPI it is generally assumed that
H+ are the dominant or only ion species and thus heavier ions do not
significantly affect the calculation. The effect of the heavier ions on ${\bf
V}_{i}$ was evident in Figure 1, where FPI overestimated the bulk velocity. In
particular, the sunward ion speed estimated from the ion moments in the solar
wind is $V_{sw,x}=-580$ km s-1, while we estimate ${\bf V}_{sw,x}=-510$ km s-1
from the peak in $f_{i}$. This overestimate of ${\bf V}_{sw}$ calculated from
the particle moments will result in overestimated Mach numbers and could
influence estimates of $\theta_{Bn}$ using mixed-mode methods [Abraham-
Shrauner (1972)]. However, reasonable estimates of ${\bf V}_{sw}$ can be
obtained by isolating the energy ranges associated with H+, when the solar
wind is relatively cold. Similarly, ${\bf V}_{i}$ is overestimated by the
moments downstream of the shock.

Additionally, ion temperatures downstream of the shock become extremely
difficult to estimate due the significant overlap of the different ions.
Figure 5 shows that the H+, He2+, He+ distributions typically overlap, and
this overlap is most significant at the positions where the He2+ and He+
temperatures peak according to the model. Since the contributions from He2+
and He+ are significant for this shock the temperature computed from FPI will
likely not match the actual H+. We can consider this effect by comparing with
the model predictions. From FPI we obtain a median scalar ion temperature of
$T_{i}\approx 50$ eV (for simplicity we discuss the scalar temperature rather
than the full temperature tensor). From the model the median $T_{i}$ of all H+
is $\approx 120$ eV. This high $T_{i}$ is due to the reflected H+ and thus
$T_{i}$ depends strongly on the fraction of reflected H+. From the model,
reflected H+ constitute $\approx 16$ % of the total average downstream
$n_{H+}$. The fraction of reflected H+ is very sensitive to $\phi$ and the
properties of the upstream H+ distribution, such as bulk velocity,
temperature, and the shape of the distribution function. If we only consider
the H+ transmitted through the shock without reflection we obtain
$T_{i}\approx 8$ eV, compared with upstream $T_{i}=3$ eV. Thus, there is only
a small amount of heating associated with transmitted H+.

For He2+, $T_{i}$ ranges from $16$ eV to $29$ eV, with a median value of $18$
eV, while for He+, $T_{i}$ ranges from $13$ eV to $59$ eV, with a median value
of $17$ eV. For both He2+ and He+, $T_{i}=12$ eV is assumed for upstream
distributions. The peaks in $T_{i}$ are due to gyrophase bunching and occur
where the ion distributions overlap with each other in observations (Figure
5), so it is difficult to verify these $T_{i}$ changes in observations.
Finally, we can calculate the model $T_{i}$ for the combined ion distributions
(Figures 4k and 4l) in the spacecraft frame. For these distributions the
median $T_{i}$ is $120$ eV, while if we neglect reflected H+ we obtain
$T_{i}\approx 35$ eV. Thus, the case without reflected H+ is a closer, albeit
lower, approximation to the observed $T_{i}$. This might suggest that in the
model the fraction of reflected H+ could be overestimated compared with
observations. However, it is also possible that FPI does not fully resolve the
reflected H+, which occur in narrow regions of velocity space, because of the
finite angular and energy resolution of the instrument. From the above
discussion we conclude that He2+ and He+ contribute to the measured $T_{i}$
(typically assumed to be H+) and illustrates the need for high-resolution mass
resolving particle instruments, as was proposed for ESA’s THOR mission
[Vaivads . (2016)] and for Plasma Observatory [Retinò . (2022)].

This bow shock crossing is an unusual case because three ion species are
resolved and $M_{A}$ is very low. Thus $T_{i}$ remains relatively small
allowing the different ion species to be partly resolved downstream of the
shock by FPI. For Earth’s bow shock typical Mach numbers are $5\lesssim
M_{A}\lesssim 10$ [Lalti . (2022)], so ion heating will be substantially
stronger. Moreover, these shocks are typically highly dynamic, so the quasi-
periodic fluctuations in the ion distributions are unlikely to occur. Thus,
different ion species would not be resolved at ion kinetic scales.

Finally, around the ramp we observe intense electrostatic waves reaching $\sim
300$ mV m-1. These waves are consistent with ion-acoustic-like waves, which
can potentially interact with ions. The fact that the fine structures in the
ion distributions associated with the reflected H+ remain well downstream of
the ramp suggests that the electrostatic waves do not play a substantial role
in ion scattering. The ion dynamics are well explained by the laminar ${\bf
B}$ and ${\bf E}$ of the shock.

##  6 Conclusions

In this paper we have investigated the ion dynamics of a low Mach number bow
shock crossing, where protons, alpha particles, and singly charged helium ions
are observed upstream and downstream of the shock. The key results are:

  * •

Highly structured proton distributions are observed downstream of the bow
shock and are caused by a small fraction of the incoming protons being
reflected by the cross-shock potential. The gyromotion of the reflected
protons results in quasi-periodic striations in the 1D reduced ion
distributions.

  * •

Alpha particles and singly charged helium ions do not undergo reflection but
are compressed and heated across the bow shock.

  * •

The gyromotion of the alpha particles and singly charged helium ions
downstream of the shock partly obscures the fine structures associated with
the proton distributions.

These results demonstrate the need for high time-resolution mass resolving
particle instruments when investigating collisionless shocks in space. At
present, mass resolving particle detectors onboard spacecraft cannot resolve
ion distributions associated with shocks at or below ion kinetic scales. Thus,
the complicated ion distributions associated with reflected protons cannot be
resolved. At present the highest time-resolution ion detector in space, namely
FPI onboard MMS, does not distinguish between different ion species. Thus,
when significant concentrations of heavier ions are present, such as He2+ and
He+, these ions prevent accurate proton moments from being calculated.

## Data Availability Statement

MMS data are available at https://lasp.colorado.edu/mms/sdc/public.

###### Acknowledgements.

We thank the entire MMS team and instrument PIs for data access and support.
This work is supported by the European Union’s Horizon 2020 research and
innovation program under grant agreement number 101004131 (SHARP).

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convergence to zero follows from our assumption that the sequences
$a_{1},\ldots,a_{\ell}$ are good for irrational equidistribution.

So it remains to show that if ${\mathbb{E}}(f_{i}|\mathcal{K}(T))=0$, for some
$i\in\\{1,\ldots,\ell\\}$, then the averages (52) converge to $0$ in
$L^{2}(\mu)$. Without loss of generality we can assume that $i=1$, hence
${\mathbb{E}}(f_{1}|\mathcal{K}(T))=0$. Since the sequences
$a_{1},\ldots,a_{\ell}$ are very good for seminorm estimates and
${\mathbb{E}}(f|\mathcal{Z}_{s}(T))=0$ implies
$\lvert\\!|\\!|f|\\!|\\!\rvert_{s+1,T}=0$, we can assume that all functions
$f_{1},\ldots,f_{\ell}$ are $\mathcal{Z}_{s}(T)$-measurable for some
$s\in{\mathbb{N}}$. In this case we can assume that
$\mathcal{X}=\mathcal{Z}_{s}(T)$,131313The assumption that the sequences
$a_{1},\ldots,a_{\ell}$ are very good for seminorm estimates (versus simply
“good”) is needed in order to get this reduction. hence using Theorem 5.1 and
an approximation argument we can assume that the system $(X,\mu,T)$ has finite
rational spectrum, in which case there exists $k\in{\mathbb{N}}$ such the
system $(X,\mu,T^{k})$ has trivial rational spectrum.

For $i=1,\ldots,\ell$ we let

$b_{i}(n):=\sum_{r=0}^{k-1}{\bf
1}_{k{\mathbb{Z}}+r}(a_{i}(n))\,\frac{a_{i}(n)-r}{k},\quad n\in{\mathbb{Z}}.$

By Lemma 5.2, the sequences $b_{1},\ldots,b_{\ell}$ are good for irrational
equidistribution and seminorm estimates. Then for $i=1,\ldots,\ell$ and
$S:=T^{k}$ we have that

$T^{a_{i}(n)}f_{i}=\sum_{r=0}^{k-1}{\bf
1}_{k{\mathbb{Z}}+r}(a_{i}(n))\,(T^{k})^{\frac{a_{i}(n)-r}{k}}(T^{r}f_{i})=\sum_{r=0}^{k-1}{\bf
1}_{k{\mathbb{Z}}+r}(a_{i}(n))\,S^{b_{i}(n)}(T^{r}f_{i}),\quad
n\in{\mathbb{N}}.$

We insert this identity in (52) and expand the product. We deduce that it
suffices to show the following: If the (not necessarily ergodic) system
$(X,\mu,S)$ has trivial rational spectrum, the sequences
$b_{1},\ldots,b_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ are good for
irrational equidistibution and seminorm estimates, and
${\mathbb{E}}(f_{1}|\mathcal{K}(T))=0$, then for every bounded sequence
$(c_{n})$ we have

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,c_{n}\cdot
S^{b_{1}(n)}f_{1}\cdot\ldots\cdot S^{b_{\ell}(n)}f_{\ell}=0$

in $L^{2}(\mu)$.

To prove this, we first remark that since the system $(X,\mu,S)$ has trivial
rational spectrum, the same holds for the system $(X\times
X,\mu\times\mu,S\times S)$. Furthermore, it is known that
${\mathbb{E}}(f_{1}|\mathcal{K}(S))=0$ implies that
${\mathbb{E}}(f_{1}\otimes\overline{f}_{1}|\mathcal{I}(S\times S))=0$.
Combining these facts (whose proof follows for example from [28, Lemma 4.18])
we deduce that if

$\mu\times\mu=\int(\mu\times\mu)_{(x,y)}\,d(\mu\times\mu)$

is the ergodic decomposition of the measure $\mu\times\mu$ with respect to the
transformation $S\times S$, then for $(\mu\times\mu)$-almost every $(x,y)\in
X\times X$ the system $(X\times X,(\mu\times\mu)_{(x,y)},S\times S)$ is
ergodic, has trivial rational spectrum, and $\int
f_{1}\otimes\overline{f}_{1}\,d(\mu\times\mu)_{(x,y)}=0$. By Theorem 1.1 we
deduce that for $(\mu\times\mu)$-almost every $(x,y)\in X\times X$ we have

${\mathbb{E}}_{n\in[N]}\,(S\times
S)^{b_{1}(n)}(f_{1}\otimes\overline{f}_{1})\cdot\ldots\cdot(S\times
S)^{b_{\ell}(n)}(f_{\ell}\otimes\overline{f}_{\ell})\to^{L^{2}((\mu\times\mu)_{(x,y)})}0$

as $N\to\infty$. This implies that

${\mathbb{E}}_{n\in[N]}\,(S\times
S)^{b_{1}(n)}(f_{1}\otimes\overline{f}_{1})\cdot\ldots\cdot(S\times
S)^{b_{\ell}(n)}(f_{\ell}\otimes\overline{f}_{\ell})\to^{L^{2}(\mu\times\mu)}0$

as $N\to\infty$, and using the Cauchy-Schwarz inequality we deduce that

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,\int(f_{0,N}\otimes\overline{f}_{0,N})\cdot(S\times
S)^{b_{1}(n)}(f_{1}\otimes\overline{f}_{1})\cdot\ldots\cdot(S\times
S)^{b_{\ell}(n)}(f_{\ell}\otimes\overline{f}_{\ell})\,d(\mu\times\mu)=0$

for all functions $f_{0,N}\in L^{\infty}(\mu)$, $N\in{\mathbb{N}}$, that are
uniformly bounded. Hence,

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,\Big{|}\int f_{0,N}\cdot
S^{b_{1}(n)}f_{1}\cdot\ldots\cdot S^{b_{\ell}(n)}f_{\ell}\,d\mu\Big{|}^{2}=0,$

which implies using the Cauchy-Schwarz inequality that

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,c_{n}\,\int f_{0,N}\cdot
S^{b_{1}(n)}f_{1}\cdot\ldots\cdot S^{b_{\ell}(n)}f_{\ell}\,d\mu=0$

for every bounded sequence $(c_{n})$. If we let

$f_{0,N}:=\overline{{\mathbb{E}}_{n\in[N]}\,c_{n}\cdot
S^{b_{1}(n)}f_{1}\cdot\ldots\cdot S^{b_{\ell}(n)}f_{\ell}},\quad
N\in{\mathbb{N}},$

we deduce that

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,c_{n}\cdot
S^{b_{1}(n)}f_{1}\cdot\ldots\cdot S^{b_{\ell}(n)}f_{\ell}=0$

in $L^{2}(\mu)$. This completes the proof. ∎

###  5.2. Proof of Corollary 1.7

Let $A\in\mathcal{X}$ and $\varepsilon>0$. Let $\mathcal{K}_{r}$ denote the
closed subspace of $L^{2}(\mu)$ consisting of all $T^{r}$-invariant functions.
Then there exists $r\in{\mathbb{N}}$ such that

(53) $\left\|{\mathbb{E}}({\bf 1}_{A}|\mathcal{K}_{rat})-{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r})\right\|_{L^{2}(\mu)}\leq\frac{\varepsilon}{\ell}.$

Let

$S_{r}=\\{n\in{\mathbb{N}}\colon a_{1}(n)\equiv 0,\ldots,a_{\ell}(n)\equiv
0\\!\\!\\!\pmod{r}\\}.$

By assumption, we have $\bar{d}(S_{r})>0$, hence there exist $N_{k}\to\infty$
such that

(54) $\lim_{k\to\infty}\frac{|S_{r}\cap[N_{k}]|}{N_{k}}>0.$

First, we claim that $\mathcal{K}_{rat}(T)$ is a characteristic factor for the
averages

(55) ${\mathbb{E}}_{n\in S_{r}\cap[N_{k}]}\,T^{a_{1}(n)}f_{1}\cdot\ldots\cdot
T^{a_{\ell}(n)}f_{\ell},$

meaning, if ${\mathbb{E}}(f_{j_{0}}|\mathcal{K}_{rat}(T))=0$ for some
$j_{0}\in\\{1,\ldots,\ell\\}$, then the averages converge to zero in
$L^{2}(\mu)$ as $k\to\infty$. To see this, note that for every
$k\in{\mathbb{N}}$ the averages (55) are equal to

(56)
$\frac{N_{k}}{|S_{r}\cap[N_{k}]|}\cdot{\mathbb{E}}_{n\in[N_{k}]}\,(T\times
R)^{a_{1}(n)}(f_{1}\otimes g)\cdot\ldots\cdot T^{a_{\ell}(n)}(f_{\ell}\otimes
g),$

where $R$ is the shift transformation on a cyclic group of order $r$ and $g$
is the indicator function of the identity element in this cyclic group. Then
${\mathbb{E}}(f_{j_{0}}\otimes g|\mathcal{K}_{rat}(T\times R))=0$, and since
by assumption the sequences $a_{1},\ldots,a_{\ell}$ are very good for seminorm
estimates and good for irrational equidistribution, we get by Theorem 1.6 that
the averages in (55) converge to $0$ in $L^{2}(\mu)$ as $k\to\infty$. We
deduce from this and (54) that the averages (56) converge to $0$ in
$L^{2}(\mu)$ as $k\to\infty$, completing the proof of our claim.

It follows from what we just proved that the limit

$\liminf_{k\to\infty}{\mathbb{E}}_{n\in S_{r}\cap[N_{k}]}\,\mu(A\cap
T^{-a_{1}(n)}A\cap\cdots\cap T^{-a_{\ell}(n)}A)$

is equal to the limit

$\liminf_{k\to\infty}{\mathbb{E}}_{n\in S_{r}\cap[N_{k}]}\int{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{rat}(T))\cdot T^{a_{1}(n)}{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{rat}(T))\cdot\ldots\cdot T^{a_{\ell}(n)}{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{rat}(T))\,d\mu.$

Using (53) and telescoping, we get that the last limit is greater or equal
than

$\liminf_{k\to\infty}{\mathbb{E}}_{n\in S_{r}\cap[N_{k}]}\int{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T))\cdot T^{a_{1}(n)}{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T))\cdot\ldots\cdot T^{a_{\ell}(n)}{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T))\,d\mu-\varepsilon.$

Note that for $n\in S_{r}$ we have $T^{a_{j}(n)}{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T))={\mathbb{E}}({\bf 1}_{A}|\mathcal{K}_{r}(T))$,
$j=1,\ldots,\ell$, hence the last limit is equal to

$\int({\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T)))^{\ell+1}\,d\mu\geq\Big{(}\int{\mathbb{E}}({\bf
1}_{A}|\mathcal{K}_{r}(T))\,d\mu\Big{)}^{\ell+1}=(\mu(A))^{\ell+1}.$

Combining the above estimates we get that

$\liminf_{k\to\infty}{\mathbb{E}}_{n\in S_{r}\cap[N_{k}]}\,\mu(A\cap
T^{-a_{1}(n)}A\cap\cdots\cap
T^{-a_{\ell}(n)}A)\geq(\mu(A))^{\ell+1}-\varepsilon,$

completing the proof of Corollary 1.7.

##  6\. Joint ergodicity of special sequences

In this section we prove the results of Sections 1.5 and 1.6.

###  6.1. Definition of Hardy fields

Let $B$ be the collection of equivalence classes of real valued functions
defined on some half line $[c,+\infty)$, where we identify two functions if
they agree eventually.141414The equivalence classes just defined are often
called “germs of functions”. We choose to use the word function when we refer
to elements of $B$ instead, with the understanding that all the operations
defined and statements made for elements of $B$ are considered only for
sufficiently large values of $t\in{\mathbb{R}}$. A _Hardy field_ $\mathcal{H}$
is a subfield of the ring $(B,+,\cdot)$ that is closed under differentiation.
For the purposes of this section we assume that all Hardy fields $\mathcal{H}$
considered are contained in some other Hardy field $\mathcal{H}^{\prime}$ that
satisfies property (8). A particular example of such a Hardy field
$\mathcal{H}$ is the collection of _logarithmic-exponential functions_ ,
meaning all functions defined on some half line $[c,+\infty)$ by a finite
combination of the symbols $+,-,\times,:,\log,\exp$, operating on the real
variable $t$ and on real constants; linear combinations of functions of the
form $t^{a}(\log{t})^{b}c^{t}$, $a,b\in{\mathbb{R}}$, $c>0$, are examples of
such functions. The reader can find more information about Hardy fields in
[12] and the references therein.

###  6.2. Good equidistribution properties for Hardy field sequences

We will use the following equidistribution result:

######  Theorem 6.1 (Boshernitzan [12]).

Let $a\colon[c,+\infty)\to{\mathbb{R}}$ be a Hardy field function with at most
polynomial growth. Then the sequence $(a(n))$ is equidistributed on
${\mathbb{T}}$ if and only if it stays logarithmically away from rational
polynomials (see definition in Section 1.5).

We will also use the following reduction, variants of which have been
frequently used in the literature. We give its proof for completeness.

######  Lemma 6.2.

Let $a_{1},\ldots,a_{\ell}\colon[c,+\infty)\to{\mathbb{R}}$ be such that for
every $t_{1},\ldots,t_{\ell}\in{\mathbb{R}}$, not all of them zero, we have

(57)
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,e(a_{1}(n)t_{1}+\cdots+a_{\ell}(n)t_{\ell})=0.$

Then the sequences $[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are good for
equidistribution.

######  Remark.

If we assume that (57) holds for all $t_{1},\ldots,t_{\ell}\in{\mathbb{R}}$,
not all of them rational, then a similar argument gives that the sequences
$[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are good for irrational equidistribution.

###### Proof.

We first remark that it suffices to show the following: If
$F\colon{\mathbb{T}}\to{\mathbb{C}}$ and
$G\colon{\mathbb{T}}^{\ell}\to{\mathbb{R}}$ are Riemann-integrable, then for
all $t_{1},\ldots,t_{\ell}\in[0,1)$, not all of them $0$, we have

(58)
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,F(a_{1}(n)t_{1}+\cdots+a_{\ell}(n)t_{\ell})\,G(a_{1}(n),\ldots,a_{\ell}(n))=\int
F\,dm_{\mathbb{T}}\cdot\int G\,dm_{{\mathbb{T}}^{\ell}}.$

Indeed, if this is the case, then using (58) for the continuous function
$F(x):=e(x)$, $x\in{\mathbb{T}}$, and the Riemann integrable function
$G(x_{1},\ldots,x_{\ell}):=e(-\\{x_{1}\\}t_{1}-\cdots-\\{x_{\ell}\\}t_{\ell})$,
$x_{1},\ldots,x_{\ell}\in{\mathbb{T}}$, we get that

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,e([a_{1}(n)]t_{1}+\cdots+[a_{\ell}(n)]t_{\ell})=0$

for all $t_{1},\ldots,t_{\ell}\in[0,1)$, not all of them $0$.

We move now to the proof of (58). After approximating from above and below by
continuous functions, we can assume that $F\in C({\mathbb{T}})$ and $G\in
C({\mathbb{T}}^{\ell})$. After a further approximation by trigonometric
polynomials we can assume that $F(x)=e(kx)$, $x\in{\mathbb{T}}$, and
$G(x_{1},\ldots,x_{\ell})=e(k_{1}x_{1}+\cdots+k_{\ell}x_{\ell})$,
$x_{1},\ldots,x_{\ell}\in{\mathbb{T}}$, for some
$k,k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$. If $k=k_{1}=\cdots=k_{\ell}=0$, then
the identity is obvious. Hence, it suffices to show that for all
$k,k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, not all of them zero, and
$t_{1},\ldots,t_{\ell}\in[0,1)$, not all of them zero, we have

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,e((k_{1}+kt_{1})a_{1}(n)+\cdots+(k_{\ell}+kt_{\ell})a_{\ell}(n))=0.$

Since $k_{1}+kt_{1},\ldots,k_{\ell}+kt_{\ell}$ are not all of them zero, this
follows from our assumptions, completing the proof. ∎

Combining the previous two results we get the following:

######  Proposition 6.3.

Let $a_{1},\ldots,a_{\ell}\colon[c,+\infty)\to{\mathbb{R}}$ be functions of at
most polynomial growth from a Hardy field such that every non-trivial linear
combination of these functions stays logarithmically away from rational
polynomials. Then the sequences $[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are good for
equidistribution.

In a similar fashion, using the variant recorded on the remark following Lemma
6.2, we get the following:

######  Proposition 6.4.

Let $a_{1},\ldots,a_{\ell}\colon[c,+\infty)\to{\mathbb{R}}$ be functions of at
most polynomial growth from a Hardy field such that every non-trivial linear
combination of these functions, with at least one irrational coefficient,
stays logarithmically away from rational polynomials. Then the sequences
$[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are good for irrational equidistribution.

###  6.3. Good seminorm estimates for Hardy field sequences

We will use the following known result.151515As far as we know this result is
proved in detail only under the assumption (8), which is the reason why we
impose this assumption on all Hardy fields considered in this section.

######  Proposition 6.5.

Let $a_{1},\ldots,a_{\ell}\colon[c,+\infty)\to{\mathbb{R}}$ be functions from
a Hardy field such that the functions and their pairwise differences are non-
constant functions in $\mathcal{T}+\mathcal{P}$. Then the sequences
$[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are very good for seminorm estimates.

If $\mathcal{T}$ is replaced with the class of functions
$a\colon{\mathbb{R}}_{+}\to{\mathbb{R}}$ that satisfy the slightly more
restrictive growth condition $t^{k+\varepsilon}\prec a(t)\prec t^{k+1}$ for
some $k\in{\mathbb{Z}}_{+}$, then the argument used to prove [21, Theorem 2.9]
can be applied without any change to prove Proposition 6.5. For the more
extended class of functions used above one can employ the argument used to
prove [10, Theorem 4.2] without essential changes. We omit the details.

###  6.4. Proof of Theorem 1.8

By Proposition 6.5 the sequences $[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are good
for seminorm estimates, and by Proposition 6.3 they are also good for
equidistribution. Hence by Theorem 1.1 they are jointly ergodic.

###  6.5. Proof of Theorem 1.9

By Proposition 6.5 the sequences $[a_{1}(n)],\ldots,[a_{\ell}(n)]$ are very
good for seminorm estimates and by Proposition 6.4 they are good for
irrational equidistribution. Hence, by Theorem 1.1 (or Corollary 1.2) they are
jointly ergodic for totally ergodic systems and by Theorem 1.6 the rational
Kronecker factor is characteristic for these sequences.

###  6.6. Proof of Theorem 1.10

By Corollary 1.4 it suffices to show that these collections of sequences are
good for equidistribution. By Lemma 6.2 it suffices to show that, after
removing the integer parts, every non-trivial linear combination of the given
collections of sequences is equidistributed on the circle.

For the sequences in Part $(i)$ this follows by combining [14, Lemma 9] (which
is the main result in [35]) with [14, Lemma 16].

For the sequences in Parts $(ii)$ and $(iii)$ this follows from [3, Theorem
2.2] and [4, Theorem 4.1].

##  7\. Joint ergodicity for flows

In this section we prove Theorem 1.11, which we repeat for convenience.

######  Theorem 7.1.

Let $a_{1},\ldots,a_{\ell}\colon[c,+\infty)\to{\mathbb{R}}_{+}$ be functions
from a Hardy field. Suppose that there exists $\delta>0$ such that
$t^{\delta}\prec a_{1}(t)$ and $(a_{j+1}(t))^{\delta}\prec
a_{j}(t)\prec(a_{j+1}(t))^{1-\delta}$ for $j=1,\ldots,\ell-1$. Then for all
measure preserving actions $T_{1}^{t},\ldots,T_{\ell}^{t}$,
$t\in{\mathbb{R}}$, on a probability space $(X,\mathcal{X},\mu)$ and
$f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$, we have

$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}f_{1}(T_{1}^{a_{1}(t)}x)\cdot\ldots\cdot
f_{\ell}(T_{\ell}^{a_{\ell}(t)}x)\,dt=\tilde{f_{1}}\cdots\tilde{f_{\ell}}$

pointwise for $\mu$-almost every $x\in X$, where for $j=1,\ldots,\ell$ we
denote by $\tilde{f_{j}}$ the orthogonal projection of $f_{j}$ on the space of
functions that are $T_{j}^{t}$-invariant for every $t\in{\mathbb{R}}$.

We start with the following crucial change of variables property (a variant of
this property also appears in [1]):

######  Lemma 7.2.

Let $a\colon{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ be a function from a Hardy
field such that $t^{\delta}\prec a(t)$ for some $\delta>0$. Let $f\in
L^{\infty}(m_{\mathbb{R}})$ and suppose that the following limit exists

$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}f(t)\,dt.$

Then also the following limit exists

$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}f(a(t))\,dt$

and the two limits are equal.

######  Remark.

More generally, our argument works if there exist $\delta,M>0$ such that the
functions $a,a^{-1}$ are three times differentiable, their derivatives are
non-zero on $[M,+\infty)$, and also $a(t)\geq t^{\delta}$ and
$\frac{a(t)}{ta^{\prime}(t)}$ is bounded for $t\geq M$.

###### Proof.

We can assume that the first limit is $0$. Let

$F(t)=\frac{1}{t}\int_{0}^{t}f(s)\,ds,\quad t>0.$

Let $\varepsilon>0$. Using our assumption we have that there exists $M>0$ such
that $|F(t)|\leq\varepsilon$ for $t>M$ and also $a,a^{-1}\in
C^{3}([M,+\infty))$. Since $f$ is bounded, it suffices to show that
$\lim_{y\to+\infty}\frac{1}{y}\int_{a^{-1}(M)}^{y}f(a(t))\,dt=0$. We assume
that $y$ is large enough so that $a(y)\geq M$. Using the change of variables
$s=a(t)$ we get

$\frac{1}{y}\int_{a^{-1}(M)}^{y}f(a(t))\,dt=\frac{1}{y}\int_{M}^{a(y)}f(t)\cdot(a^{-1})^{\prime}(t)\,dt.$

Since $f(t)=(tF(t))^{\prime}$ for Lebesgue almost every $t\in{\mathbb{R}}$, we
have

$\frac{1}{y}\int_{M}^{a(y)}f(t)\cdot(a^{-1})^{\prime}(t)\,dt=\frac{1}{y}\int_{M}^{a(y)}(tF(t))^{\prime}\cdot(a^{-1})^{\prime}(t)\,dt.$

Integration by parts ($tF(t)$ and $(a^{-1})^{\prime}(t)$ are absolutely
continuous on $[M,a(y)]$) gives that the last integral is equal to

(59)
$\frac{a(y)F(a(y))(a^{-1})^{\prime}(a(y))}{y}-\frac{C}{y}-\frac{1}{y}\,\int_{M}^{a(y)}tF(t)\cdot(a^{-1})^{\prime\prime}(t)\,dt$

for some $C\in{\mathbb{R}}_{+}$.

The first term in (59) is equal to

$\frac{a(y)F(a(y))(a^{-1})^{\prime}(a(y))}{y}=\frac{\int_{0}^{a(y)}f(t)\,dt}{ya^{\prime}(y)}=\frac{\int_{0}^{a(y)}f(t)\,dt}{a(y)}\cdot\frac{a(y)}{ya^{\prime}(y)}.$

Notice that since $a(t)$ is a Hardy field function with $a(t)\succ
t^{\delta}$, we have that

$\lim_{y\to+\infty}\frac{a(y)}{ya^{\prime}(y)}=\lim_{y\to+\infty}\frac{\log{y}}{\log{a(y)}}<\frac{1}{\delta}.$

Moreover, by assumption we have
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}f(t)\,dt=0$ and
$\lim_{y\to+\infty}a(y)=+\infty$. We deduce that the first term in (59)
converges to $0$ as $y\to+\infty$.

It remains to show that the limsup as $y\to+\infty$ of last term in (59) is
bounded by a constant multiple of $\varepsilon$. Since $|F(t)|\leq\varepsilon$
for $t>M$, it suffices to show that the limsup as $y\to+\infty$ of the
expression

$\frac{1}{y}\int_{M}^{a(y)}t\cdot|(a^{-1})^{\prime\prime}(t)|\,dt$

is bounded by a quantity that is independent of $\varepsilon$ . Since
$(a^{-1})^{\prime\prime}$ has eventually constant sign, say in
$[M_{1},+\infty)$, we can replace $M$ with $M_{1}$ and remove the absolute
value. After doing so, integration by parts leads to the expression

$\frac{1}{y}\Big{(}a(y)\cdot(a^{-1})^{\prime}(a(y))-C_{1}-\int_{M_{1}}^{a(y)}(a^{-1})^{\prime}(t)\,dt\Big{)}$

for some $C_{1}\in{\mathbb{R}}_{+}$. The last expression is equal to

$\frac{a(y)}{ya^{\prime}(y)}-\frac{C_{1}}{y}-\frac{y-a^{-1}(M_{1})}{y}.$

As we showed before, the limit of the first term is bounded and the limit of
the other two terms as $y\to+\infty$ is $1$. This completes the proof. ∎

We review some basic facts from the spectral theory of unitary
${\mathbb{R}}$-actions that we will use. Proofs of the stated facts can be
found for example in [19] (we use Theorem 9.58 and the variant of Theorems
9.17 that applies to flows). If $(X,\mu,T^{t})$, $t\in{\mathbb{R}}_{+}$, is a
measure preserving flow, then for every $f\in L^{2}(\mu)$ there exists a
positive and bounded measure $\sigma_{f}$ on ${\mathbb{R}}$, that is called
the spectral measure of $f$, such that

(60) $\int T^{t}f\cdot\overline{f}\,d\mu=\int e(ts)\,d\sigma_{f}(s),\quad
t\in{\mathbb{R}}.$

More generally, for every $f,g\in L^{2}(\mu)$ there exists a complex measure
$\sigma_{f,g}$, with bounded variation, such that

$\int T^{t}f\cdot\overline{g}\,d\mu=\int e(ts)\,d\sigma_{f,g}(s),\quad
t\in{\mathbb{R}}.$

Furthermore, if $h\in L^{\infty}({\mathbb{R}})$, then there exists a bounded
operator $h(T)\colon L^{2}(\mu)\to L^{2}(\mu)$ that commutes with $T^{t}$,
$t\in{\mathbb{R}}$, and satisfies

$\int h(T)f\cdot\overline{g}\,d\mu=\int h\,d\sigma_{f,g}$

for all $f,g\in L^{2}(\mu)$. We then have

(61) $(h_{1}+h_{2})(T)=h_{1}(T)+h_{2}(T),\quad h_{1},h_{2}\in
L^{\infty}(\mu),$

and

(62)
$d\sigma_{h(T)f}=h\,d\sigma_{f},\quad\left\|h(T)f\right\|_{L^{2}(\mu)}=\left\|h\right\|_{L^{2}(\sigma_{f})}.$

We will use the following fact:

######  Lemma 7.3.

Let $(X,\mu,T^{t})$, $t\in{\mathbb{R}}_{+}$, be a measure preserving flow.
Then the set

$\mathcal{G}:=\\{f\in L^{2}(\mu)\colon\sigma_{f}\text{ has compact
support}\\}$

is dense in $L^{2}(\mu)$.

###### Proof.

Let $f\in L^{2}(\mu)$. Using the previous notation we have by (62) that for
$n\in{\mathbb{N}}$ the spectral measure of the function $f_{n}={\bf
1}_{[-n,n]}(T)f$ is supported on the interval $[-n,n]$, hence
$f_{n}\in\mathcal{G}$. Moreover, since ${\bf 1}_{\mathbb{R}}(T)f=f$, using
(61) and (62) we get

$\left\|f-f_{n}\right\|_{L^{2}(\mu)}=\left\|{\bf
1}_{{\mathbb{R}}\setminus[-n,n]}(T)f\right\|_{L^{2}(\mu)}=\left\|{{\bf
1}_{{\mathbb{R}}\setminus[-n,n]}}\right\|_{L^{2}(\sigma_{f})}\to 0$

as $n\to\infty$, since $\sigma_{f}$ is a bounded measure. ∎

######  Lemma 7.4.

Let $b\colon{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ be a function from a Hardy
field that satisfies $t^{\delta}\prec b(t)\prec t^{1-\delta}$ for some
$\delta>0$, and $(X,\mu,T^{t})$, $t\in{\mathbb{R}}_{+}$, be a measure
preserving flow. Then for every $f\in L^{\infty}(\mu)$ and $c\in{\mathbb{R}}$
we have

(63)
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}|f(T^{b(t+c)}x)-f(T^{b(t)}x)|^{2}\,dt=0$

for $\mu$-almost every $x\in X$.

######  Remark.

More generally, our argument works if the function $b$ satisfies the
properties mentioned in the remark following Lemma 7.2 and also $b(t)\prec
t^{1-\delta}$ and $b^{\prime}$ monotonically decreases to $0$ as
$t\to+\infty$.

###### Proof.

We can assume that $c\geq 0$. Our assumptions imply that there exists
$t_{0}>0$ such that for $t\geq t_{0}$ we have $0\leq b^{\prime}(t)\leq 1$,
$b^{\prime}(t)$ is decreasing, and $b(t)\leq t^{1-\delta}$. Since $f\in
L^{\infty}(\mu)$ it suffices to show that

(64)
$\lim_{y\to+\infty}\frac{1}{y}\int_{t_{0}}^{y}|f(T^{b(t+c)}x)-f(T^{b(t)}x)|^{2}\,dt=0$

for $\mu$-almost every $x\in X$.

Let $\varepsilon>0$. If $\mathcal{G}$ is the dense subset of $L^{2}(\mu)$
given by Lemma 7.3, we have a decomposition

$f=f_{1}+f_{2}$

where $f_{1}\in\mathcal{G}$ and
$\left\|f_{2}\right\|_{L^{2}(\mu)}\leq\varepsilon$.

For $g\in L^{2}(\mu)$ let

$A_{y}(g):=\frac{1}{y}\int_{t_{0}}^{y}|g(T^{b(t+c)}x)-g(T^{b(t)}x)|^{2}\,dt,\quad
y\geq t_{0}.$

We first deal with the contribution of $f_{2}$. We clearly have

$\limsup_{y\to+\infty}A_{y}(f_{2})\leq
4\lim_{y\to+\infty}\frac{1}{y}\int_{t_{0}}^{y}|f_{2}(T^{b(t)}x)|^{2}\,dt.$

Hence, using Lemma 7.2 and the pointwise ergodic theorem for flows we deduce
that

(65) $\int\limsup_{y\to+\infty}A_{y}(f_{2})\,d\mu\leq
4\int\Big{(}\lim_{y\to+\infty}\frac{1}{y}\int_{t_{0}}^{y}|f_{2}(T^{t}x)|^{2}\,dt\Big{)}d\mu=4\int|f_{2}|^{2}\,d\mu\leq
4\varepsilon^{2}.$

Next, we deal with the contribution of $f_{1}$. Since $f_{1}\in\mathcal{G}$,
there exists $M>0$ such that the spectral measure $\sigma_{f_{1}}$ of $f_{1}$
is supported on the set $[-M,M]$. Using the Fubini-Tonelli theorem we get

$\left\|A_{y}(f_{1})\right\|_{L^{1}(\mu)}=\frac{1}{y}\int_{t_{0}}^{y}\int_{X}|f_{1}(T^{b(t+c)}x)-f_{1}(T^{b(t)}x)|^{2}\,d\mu\,dt.$

Using (60) and the Fubini-Tonelli theorem again, we get that the last
expression is equal to

$\int_{-M}^{M}\frac{1}{y}\int_{t_{0}}^{y}|e(sb(t+c))-e(sb(t))|^{2}\,dt\,d\sigma_{f_{1}}(s).$

After bounding the integrant pointwise, using the mean value theorem, the fact
that $b^{\prime}$ is non-negative, and decreasing for $t\geq t_{0}$, and that
$c\geq 0$, we get that the last expression is bounded by a constant multiple
of

$\int_{-M}^{M}\frac{1}{y}\int_{t_{0}}^{y}\big{(}scb^{\prime}(t))^{2}\,dt\,d\sigma_{f_{1}}(s)\leq
M^{2}c^{2}\left\|f_{1}\right\|^{2}_{L^{2}(\mu)}\frac{1}{y}\int_{t_{0}}^{y}(b^{\prime}(t))^{2}\,dt.$

Since $0\leq b^{\prime}(t)\leq 1$ for $t\geq t_{0}$ we have for $y\geq t_{0}$
that

$\int_{t_{0}}^{y}\big{(}b^{\prime}(t))^{2}dt\leq\int_{t_{0}}^{y}b^{\prime}(t)\,dt=b(y)-b(t_{0}).$

Since $b(y)\leq y^{1-\delta}$ for $y>t_{0}$, combining the above we get for
$C:=Mc^{2}\lvert\\!|\\!|f_{2}|\\!|\\!\rvert^{2}_{L^{2}(\mu)}$ that

$\left\|A_{y}(f_{1})(x)\right\|_{L^{1}(\mu)}\leq\frac{C}{y^{\delta}}$

for all $y>t_{0}$. Using the Borel-Cantelli lemma, we get that for
$a>1/\delta$ we have $\lim_{N\to\infty}A_{N^{a}}(f_{1})(x)=0$ for $\mu$-almost
every $x\in X$. From this we deduce that

(66) $\lim_{y\to+\infty}A_{y}(f_{1})(x)=0$

for $\mu$-almost every $x\in X$. Indeed, if $N_{y}\in{\mathbb{Z}}_{+}$ is such
that $N_{y}^{a}\leq y<(N_{y}+1)^{a}$, then
$\left\|A_{y}(f_{1})-A_{N_{y}^{a}}(f_{1})\right\|_{L^{\infty}(\mu)}\leq
2\,\left\|f_{1}\right\|_{L^{\infty}(\mu)}|1-N_{y}^{a}/y|\to 0$ as
$y\to+\infty$.

Combining (65) and (66) we get that

$\int\limsup_{y\to+\infty}A_{y}(f)\,d\mu\leq 8\varepsilon^{2}.$

Since $\varepsilon$ is arbitrary, we get that (64) holds for $\mu$-almost
every $x\in X$, completing the proof. ∎

###### Proof of Theorem 1.11.

We prove the statement by induction on $\ell$. For $\ell=1$ the result follows
from Lemma 7.2 and the pointwise ergodic theorem for flows. Suppose that the
statement holds for $\ell-1$, we shall show that it holds for $\ell$.

Without loss of generality we can assume that
$\left\|f_{j}\right\|_{L^{\infty}(\mu)}\leq 1$ for $j=1,\ldots,\ell$. We let

$A_{y}(f_{\ell})(x):=\frac{1}{y}\int_{0}^{y}f_{1}(T_{1}^{a_{1}(t)}x)\cdot\ldots\cdot
f_{\ell}(T_{\ell}^{a_{\ell}(t)}x)\,dt,\quad y\in{\mathbb{R}}_{+}.$

Let $\varepsilon>0$. Using a standard Hilbert space argument we get a
decomposition

$f_{\ell}=f_{\ell,1}+f_{\ell,2}+f_{\ell,3},$

where $f_{\ell,1}=\tilde{f_{\ell}}$,
$\left\|f_{\ell,2}\right\|_{L^{2}(\mu)}\leq\varepsilon$, and $f_{\ell,3}$
belongs to the linear subspace spanned by the functions $T_{\ell}^{c}h-h$,
$c\in{\mathbb{R}}$, $h\in L^{2}(\mu)$. After approximating $h$ in $L^{2}(\mu)$
by functions in $L^{\infty}(\mu)$ and incorporating the error in $f_{\ell,2}$,
we can assume that $h\in L^{\infty}(\mu)$. Furthermore, we have
$\left\|f_{\ell,1}\right\|_{L^{\infty}(\mu)}\leq\left\|f_{\ell}\right\|_{L^{\infty}(\mu)}\leq
1$ and $\left\|f_{\ell,3}\right\|_{L^{2}(\mu)}\leq 2+\varepsilon$. An
application of the Cauchy-Schwarz inequality and the Fubini-Tonelli theorem
shows that for $\mu$-almost every $x\in X$ the quantities $A_{y}(f_{\ell,j})$,
$j=1,2,3$, are well defined finite numbers.

Our goal is to show (74) below. We first deal with the contribution of the
term $f_{\ell,1}$. Since $T_{\ell}^{t}f_{\ell,1}=f_{\ell,1}=\tilde{f_{\ell}}$
for every $t\in{\mathbb{R}}_{+}$, the induction hypothesis gives

(67)
$\lim_{y\to+\infty}A_{y}(f_{\ell,1})=\tilde{f_{1}}\cdots\tilde{f_{\ell}}.$

for $\mu$-almost every $x\in X$.

Next we deal with the contribution of the term $f_{\ell,2}$. For every $y>0$,
using the Cauchy-Schwarz inequality we have

$|A_{y}(f_{\ell,2})(x)|^{2}\leq\frac{1}{y}\int_{0}^{y}|f_{\ell,2}(T_{\ell}^{a_{\ell}(t)}x)|^{2}\,dt.$

Using Lemma 7.2, the pointwise ergodic theorem for flows, and the Cauchy-
Schwarz inequality, we get (note that in the ergodic case there is no need for
the integral on the left hand side)

(68)
$\int\limsup_{y\to+\infty}|A_{y}(f_{\ell,2})|\,d\mu\leq\left\|f_{\ell,2}\right\|_{L^{2}(\mu)}\leq\varepsilon.$

It remains to deal with the contribution of the term $f_{\ell,3}$. We claim
that

(69) $\lim_{y\to+\infty}|A_{y}(f_{\ell,3})(x)|=0$

for $\mu$-almost every $x\in X$. By Lemma 7.2,161616We remark that since not
all Hardy fields are closed under composition and compositional inversion, we
cannot assume that the functions $b_{j}$ belong to some Hardy field.
Nevertheless, one can easily verify that these functions satisfy the necessary
assumptions mentioned on the remark following Lemma 7.2, so we are entitled to
apply this lemma.  it suffices to show that for $\mu$-almost every $x\in X$ we
have

(70)
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}\prod_{j=1}^{\ell-1}f_{j}(T_{j}^{b_{j}(t)}x)\cdot
f_{\ell,3}(T_{\ell}^{t}x)\,dt=0$

where $b_{j}:=a_{j}\circ a^{-1}_{\ell}$ for $j=1,\ldots,\ell-1$.

In order to establish (70) it suffices to verify that if

$f_{\ell,3}:=T_{\ell}^{c}h-h,$

for some $c\in{\mathbb{R}}$ and $h\in L^{\infty}(\mu)$, then (70) holds for
$\mu$-almost every $x\in X$. So let $c>0$ and $h\in L^{\infty}(\mu)$. After
inserting $f_{\ell,3}=T_{\ell}^{c}h-h$ in (70) and using the change of
variables $t\mapsto t-c$ in the first of the two integrals, we get that it
suffices to show that

(71)
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}\Big{(}\prod_{j=1}^{\ell-1}f_{j}(T_{j}^{b_{j}(t-c)}x)-\prod_{j=1}^{\ell-1}f_{j}(T_{j}^{b_{j}(t)}x)\Big{)}\cdot
h(T_{\ell}^{t}x)\,dt=0$

for $\mu$-almost every $x\in X$. It suffices to show that for
$d=1,\ldots,\ell-1$ we have

(72)
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}\Big{(}\prod_{j=1}^{d}f_{j}(T_{j}^{b_{j}(t-c)}x)\prod_{j=d+1}^{\ell-1}f_{j}(T_{j}^{b_{j}(t)}x)-\\\
\prod_{j=1}^{d-1}f_{j}(T_{j}^{b_{j}(t-c)}x)\prod_{j=d}^{\ell-1}f_{j}(T_{j}^{b_{j}(t)}x)\Big{)}\cdot
h(T_{\ell}^{t}x)\,dt=0$

for $\mu$-almost every $x\in X$. Finally, notice that our growth assumptions
give that $t^{\delta^{\prime}}\prec b_{d}(t)\prec t^{1-\delta^{\prime}}$ for
some $\delta^{\prime}>0$. Hence, by Lemma 7.4,171717Again here, in order to
avoid the assumption that the function $b_{d}$ belongs to some Hardy field, we
can verify that the assumptions mentioned on the remark following Lemma 7.4
are satisfied. for $d=1,\ldots,\ell-1$, we have

(73)
$\lim_{y\to+\infty}\frac{1}{y}\int_{0}^{y}|f_{d}(T_{d}^{b_{d}(t-c)}x)-f_{d}(T_{d}^{b_{d}(t)}x)|^{2}\,dt=0$

for $\mu$-almost every $x\in X$. Hence, using the Cauchy-Schwarz inequality,
equation (73), and the fact that all the functions $f_{j}$ and $h$ are
bounded, we get that (72) holds for $\mu$-almost every $x\in X$. Combning the
above we deduce that (69) holds.

From (67), (68), (69), we deduce that

(74)
$\int\limsup_{y\to+\infty}|A_{y}(f_{\ell})-\tilde{f_{1}}\cdots\tilde{f_{\ell}}|\,d\mu\leq\varepsilon.$

Since $\varepsilon$ was arbitrary, we deduce that

$\lim_{y\to+\infty}A_{y}(f_{\ell})(x)=\tilde{f_{1}}\cdots\tilde{f_{\ell}}$

for $\mu$-almost every $x\in X$, completing the proof. ∎

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