text:
\section{Introduction}

Although the hypothesis of dark matter is consistent with a wide
range of observations, it might be an artifact of our Newtonian
interpretation of experimental data if the gravitational $1/r^2$
law happens not to be valid at large distances. In 1983, Milgrom
proposed a simple phenomenological rule, called ``modified
Newtonian dynamics'' (MOND) \cite{Milgrom:1983ca}, depending on
an acceleration scale $a_0$: In the vicinity of a (baryonic) mass
$M$, any test particle undergoes the standard Newtonian
acceleration $a_N = GM/rc^2$ if $a_N > a_0$, but a slower
decreasing one $a = \sqrt{a_0 a_N} = \sqrt{GMa_0}/r$ if $a <
a_0$. This law happens to fit remarkably well galaxy rotation
curves for a universal constant~\cite{Sanders:2002pf}
\begin{equation}
a_0 \approx 1.2 \times 10^{-10}\, \text{m.s}^{-2}.
\label{a0}
\end{equation}
Moreover, it automatically recovers the Tully-Fisher
law~\cite{Tully:1977fu} and explains the correlation of dark
matter profiles with the baryonic ones \cite{McGaugh:2005er}.

However, reproducing this phenomenological law in a consistent
relativistic field theory happens to be quite difficult, as
illustrated by almost three decades of abundant literature. In
the present paper, we shall refine on one of the best models
proposed so far, in our opinion, namely the tensor-vector-scalar
(TeVeS) theory constructed by Bekenstein and
Sanders~\cite{Bekenstein:1992pj,Bekenstein:1993fs,Sanders:1996wk,
Bekenstein:2004ne,Bekenstein:2004ca,Sanders:2005vd}. We will show
that a generalized Galileon action allows us to suppress
deviations from general relativity at small distances, thanks to
an extension of the Vainshtein mechanism occurring in massive
gravity \cite{Vainshtein:1972sx,Babichev:2010jd}. In the model we
propose, the magnitude itself of the anomalous force tends
towards zero at small distances, and not only its ratio to the
gravitational one (as in the standard Vainshtein mechanism).

We consider a scalar-tensor theory of gravity defined by the
action
\begin{eqnarray}
S&=& \frac{c^3}{4 \pi G}\int d^4 x \sqrt{-g}
\biggl( \frac{R}{4}
+\mathcal{L}_\text{standard}
+\mathcal{L}_\text{MOND}\nonumber\\
&&+\mathcal{L}_\text{Galileon} \biggr)
+ S_\text{matter}\left[\psi_\text{matter};
\tilde{g}_{\mu \nu}\right],
\label{action}
\end{eqnarray}
where we use the sign convention of \cite{MTW} and notably the
mostly plus signature. The physical metric $\tilde{g}_{\mu \nu}$
will be defined in Eq.~(\ref{gtilde}) below. The scalar field
$\varphi$ is chosen dimensionless, and its kinetic term is the
sum of the following three contributions, where $s \equiv g^{\mu
\nu} \varphi_{,\mu}\varphi_{,\nu}$~:
\begin{eqnarray}
\mathcal{L}_\text{standard} &=& -\frac{\epsilon}{2}\,s
= - \frac{\epsilon}{2} (\partial_\lambda\varphi)^2,
\label{standard}\\
\mathcal{L}_\text{MOND} &=& - \frac{c^2}{3a_0} s \sqrt{|s|}\,,
\label{MOND}\\
\mathcal{L}_\text{Galileon} &=& - \frac{k}{3}
\varepsilon^{\alpha \beta \gamma \delta}
\varepsilon^{\mu \nu \rho \sigma}
\varphi_{,\alpha} \varphi_{,\mu} \varphi_{;\beta \nu}
R_{\gamma \delta \rho \sigma}.
\label{Galileon}
\end{eqnarray}
Here $\varepsilon^{\alpha \beta \gamma \delta}$ denotes the
Levi-Civita tensor, related to the fully antisymmetrical symbol
$\left[\alpha \beta \gamma \delta\right]$ (whose values are 0 or
$\pm 1$) by $\varepsilon^{\alpha \beta \gamma \delta}\equiv
(-g)^{-1/2}\left[\alpha \beta \gamma \delta\right]$. A small mass
term $-\frac{1}{2}m^2\varphi^2$, with $1/m$ greater than the
largest cluster sizes, might also be added to the above kinetic
term.

Denoting as $r_\text{MOND} \equiv \sqrt{GM/a_0}$ the scale where
MOND effects start to manifest around a galaxy of baryonic mass
$M$, and as $r_\text{V} \equiv (8kGM a_0)^{1/4}/c <
r_\text{MOND}$ the Vainshtein radius below which scalar effects
are suppressed, we will see below that
$\mathcal{L}_\text{standard}$ dominates at very large distances
$r > r_\text{MOND}/\epsilon$, $\mathcal{L}_\text{MOND}$ at
intermediate ones $r_\text{V} < r < r_\text{MOND}/\epsilon$, and
$\mathcal{L}_\text{Galileon}$ at small scales $r < r_\text{V}$.
As stated above, numerical fits of galaxy rotation curves
\cite{Sanders:2002pf} give the value (\ref{a0}) for $a_0$, while
their flatness up to about $10\, r_\text{MOND}$
\cite{Gentile:2006hv} imposes that the positive dimensionless
constant $\epsilon$ is smaller than $0.1$. The constant $k$
entering $\mathcal{L}_\text{Galileon}$ has dimension of a
[length]${}^4$, and we will show below that a numerical value
\begin{equation}
k \approx \left(4\times 10^{-6}\, \frac{c^2}{a_0}\right)^4
\approx \left(100\, \text{kpc}\right)^4
\label{k}
\end{equation}
allows the model to pass solar-system tests while predicting MOND
effects even for the lightest known dwarf galaxies.

Several ingredients of action (\ref{action}) are borrowed from
the MOND literature. In particular, the scalar kinetic term
$\mathcal{L}_\text{MOND}$ is well known to generate an extra
acceleration $\sqrt{G M a_0}/r$ on any test mass at distance $r$
{}from a source of baryonic mass $M$ \cite{Bekenstein:1984tv}.
Note that we did not write it simply in terms of $|s|^{3/2}$, but
that an absolute value is involved only within the square root,
to ensure that the scalar field carries positive energy whatever
the sign of $s$~\cite{Bruneton:2007si}. The standard kinetic term
$\mathcal{L}_\text{standard}$ has a negligible influence because
it is multiplied by the small positive constant $\epsilon$, but
it ensures that the dynamics of the scalar field is well defined
when $s$ passes through a vanishing
value~\cite{Bruneton:2007si,Babichev:2007dw}.

As in Refs.~\cite{Bekenstein:2004ne,Bekenstein:2004ca,
Sanders:1996wk,Sanders:2005vd}, the matter action
$S_\text{matter}$ assumes that all matter fields $\psi$ are
minimally coupled to a physical metric
\begin{equation}
\tilde{g}_{\mu \nu}\equiv
e^{-2\varphi}g_{\mu\nu}-2\,\text{sinh}(2\varphi)U_\mu U_\nu\,,
\label{gtilde}
\end{equation}
where $U_\mu$ is a timelike unit vector field, i.e.,
$g^{\mu\nu}U_\mu U_\nu = -1$. Light deflection by galaxies and
clusters would indeed be inconsistent with experiment if matter
was merely coupled to the scalar field via a conformal metric
$\tilde{g}_{\mu \nu} = e^{2\varphi}g_{\mu\nu}$
\cite{Bekenstein:1992pj,Bekenstein:1993fs,Bruneton:2007si}. In a
locally inertial frame where $g_{\mu\nu} =
\text{diag}(-1,1,1,1)$, and choosing the observer's velocity such
that $U_\mu = (1,0,0,0)$ lies along his proper time direction,
Eq.~(\ref{gtilde}) merely means that $\tilde g_{00} =
e^{2\varphi}g_{00}$ but $\tilde g_{ij} = e^{-2\varphi}g_{ij}$
(mimicking thus the behavior of pure general relativity in
presence of dark matter). Previous field theories attempting at
reproducing the MOND dynamics predicted large preferred-frame
effects in the solar system, inconsistent with experiment, if
$U_\mu$ was assumed to be a constant vector field
\cite{Sanders:1996wk,Bruneton:2007si}. This is why it was assumed
to be dynamical in the TeVeS model
\cite{Bekenstein:2004ne,Bekenstein:2004ca}, with the idea that it
could align with the matter's local proper time direction.
However, making the vector field dynamical by adding a kinetic
term $-F_{\mu\nu}^2$, or any function of it, causes this vector
to be unstable
\cite{Clayton:2001vy,Bruneton:2007si,EspositoFarese:2009aj}. In
the present paper, we will see that preferred-frame effects
remain negligible in the solar system even if $U_\mu$ is assumed
to be constant, thanks to the Vainshtein mechanism at small
distances. This mechanism also allows us to choose other forms
than (\ref{gtilde}), in contrast to
Refs.~\cite{Bekenstein:2004ne,
Bekenstein:2004ca,Sanders:1996wk,Sanders:2005vd}. We could also
choose the physical metric as $\tilde{g}_{\mu \nu} \approx
e^{2\varphi} g_{\mu\nu}
+B(\varphi,s)\varphi_{,\mu}\varphi_{,\nu}$, but $B$ would need to
be a fine-tuned function of both $\varphi$ and its standard
kinetic term, $s = (\varphi_{,\lambda})^2$, in order to be
consistent with the observed light deflection by galaxies and
clusters. Moreover, the conditions for consistency of the theory
within matter would be quite involved~\cite{Bruneton:2007si}.
The choice (\ref{gtilde}), borrowed from
\cite{Bekenstein:2004ne,Bekenstein:2004ca,
Sanders:1996wk,Sanders:2005vd}, is thus the most natural one in
the present framework.

The reason why a mere action $\mathcal{L}_\text{standard} +
\mathcal{L}_\text{MOND}$, Eqs.~(\ref{standard}) and (\ref{MOND})
above, does not suffice to define a consistent relativistic field
theory of MOND is that it would also predict an extra force
$\sqrt{GM_\odot a_0}/r$ within the solar system (where $M_\odot$
denotes the mass of the Sun), in addition to the Newtonian one
$GM_\odot /r^2$ and its post-Newtonian corrections. [We often use
the word ``force'' instead of ``acceleration'' in the present
paper, i.e., do not write the mass of the test particle to
simplify.] This would be ruled out by tests of Kepler's third law
and those of post-Newtonian dynamics. The literature considered
thus ``Relativistic AQUAdraric Lagrangians'' (RAQUAL), also known
as ``k-essence'' theories in the cosmological framework, i.e., a
scalar kinetic term $-\frac{1}{2}\,f(s)$ depending on a
\textit{function} of $s = (\varphi_{,\lambda})^2$. In order to
reproduce the MOND dynamics, this kinetic term was assumed to
take the form (\ref{MOND}) for small accelerations (i.e., small
values of $s$), while the scalar field behavior within the solar
system depended on the shape of $f(s)$ for large $s$. The clever
choice of the literature was to impose that $f'(s)$ tends towards
a constant value for large $s$, say $f'_\infty$, in order to
recover a Brans-Dicke behavior $\varphi = -GM_\odot/f'_\infty
rc^2$ at small distances, so that the physical metric
(\ref{gtilde}) reproduce the standard Schwarzschild solution up
to a rescaling of the gravitational constant $G_\text{eff} = G
(1+1/f'_\infty)$. The parametrized post-Newtonian (PPN)
parameters $\beta$ and $\gamma$ \cite{Will:2005va} take then
strictly their general relativistic values $\beta = \gamma = 1$,
and classical solar-system tests are passed. However,
binary-pulsar tests are directly sensitive to the matter-scalar
coupling strength $1/f'_\infty$, independently of the fact that
the physical metric (\ref{gtilde}) contains a ``disformal''
contribution proportional to $U_\mu U_\nu$, and they impose
$f'_\infty > 10^4$ \cite{EspositoFarese:2004tu}. As discussed in
\cite{Bruneton:2007si}, such a large value is difficult to
reconcile with the expression (\ref{MOND}) needed for small $s$.
One needs a fine-tuned interpolating function $f(s)$, of a shape
similar to Fig.~3 of \cite{Bruneton:2007si}, in order to predict
MOND effects while passing binary-pulsar tests. Another idea
would thus be to choose a function $f(s)$ such that the scalar
force at small distances is negligible with respect to the
Newtonian one $GM_\odot/r^2$, instead of keeping the same radial
dependence $\varphi'(r) c^2 = GM_\odot/f'_\infty r^2$. As
underlined in \cite{Babichev:2009ee}, nonlinear kinetic terms
(i.e., those of k-essence/RAQUAL models) can reduce scalar
effects at small distances, acting as a camouflage for the scalar
(hence the name ``k-mouflage''). However, RAQUAL models must
satisfy two conditions to have a Hamiltonian bounded by below and
a well-posed Cauchy problem~\cite{Bruneton:2007si}: $f'(s) > 0$
and $2 s f''(s) + f'(s) > 0$. These conditions suffice to prove
that $\varphi''(r) < 0$, i.e., that $\varphi'(r)$ is a decreasing
function of $r$, and the best we can obtain is thus an almost
constant force $\varphi'(r) c^2 \approx a_0$ within the solar
system, the value $a_0$ being imposed by the MOND regime for
$r\sim r_\text{MOND}$. But solar-system tests are precise enough
to rule out a constant anomalous acceleration even numerically as
tiny as (\ref{a0}). We must therefore look for other possible
scalar kinetic terms, and this is where generalized Galileons
enter our discussion.

Galileons were first introduced in the cosmological context in
\cite{Nicolis:2008in} (although they had actually already been
studied in \cite{Fairlie:1992} in a quite different framework).
In flat spacetime, they are theories whose field equations depend
only on second derivatives of a scalar field, but not on their
lower (0th and 1st) nor on higher derivatives. One of their
initial motivations was to generalize the key features of the
decoupling limit of the DGP brane model \cite{Dvali:2000hr},
which yields an equation of motion $\propto (\Box\varphi)^2 -
(\varphi_{;\mu\nu})^2$ for a scalar degree of freedom, playing a
crucial role in cosmology~\cite{Deffayet:2001pu,Deffayet:2000uy}.
References~\cite{Deffayet:2009wt,Deffayet:2009mn} showed how to
extend Galileon models to curved spacetime without introducing
higher derivatives (while making now first derivatives also enter
them). The same Lagrangians can be obtained in a suitable limit
of brane models including Gauss-Bonnet-Lovelock densities
\cite{deRham:2010eu}, or from dimensional reduction of such
densities \cite{VanAcoleyen:2011mj}. They can also be extended,
in any dimension, to arbitrary $p$-forms possibly coupled to each
other \cite{Deffayet:2010zh}, or to general nonlinear models
whose field equations depend on at most second derivatives
\cite{Deffayet:2011gz}. Throughout this paper, we call
Galileons this full class of models, although they go beyond
the initial ones of \cite{Nicolis:2008in}.

The new ingredient of the present paper is the third scalar
kinetic term of action (\ref{action}),
$\mathcal{L}_\text{Galileon}$, which strongly suppresses all
scalar-field effects at small distances, as we will show below.
This action (up to the factor $-8k/3$) was obtained in
Ref.~\cite{VanAcoleyen:2011mj} for the first time by dimensional
reduction of the Gauss-Bonnet-Lovelock density. It is written
here in the compact form of \cite{Deffayet:2009mn}, and is also
in the general classes considered in
\cite{Deffayet:2010zh,Deffayet:2011gz}. It is easy to check that
all field equations involve at most second derivatives, because
of the antisymmetry of the Levi-Civita tensors entering
(\ref{Galileon}).

Although it is straightforward to write the full field equations
deriving from action (\ref{action}), for both $\varphi$ and the
metric, it will suffice in the present paper to obtain the
perturbative solution for a static and spherically symmetric
$\varphi$ in a Schwarzschild background $ds^2 = -(1-r_s/r) c^2
dt^2 + dr^2/(1-r_s/r) + r^2 d\Omega^2$, where $r_s \equiv
2GM/c^2$ denotes the Schwarzschild radius. It can be checked a
posteriori that the backreaction of the scalar field on the
metric has negligible effect as compared to present experimental
bounds. We will even assume that $r\gg r_s$ to simplify the
expressions. More detail will be provided in a forthcoming
publication \cite{BDESZ}. Denoting as before $\varphi' \equiv
d\varphi/dr$, and integrating once the field equation for
$\varphi$, we find
\begin{equation}
4 k \frac{r_s}{r^2} \varphi'^2
+ \frac{r^2 c^2}{a_0}\varphi'^2
+ \epsilon\, r^2 \varphi' \approx \frac{r_s}{2},
\label{EOM}
\end{equation}
where the origin of the different terms is obvious from
(\ref{standard})--(\ref{Galileon}). The large number of field
derivatives involved in $\mathcal{L}_\text{Galileon}$,
Eq.~(\ref{Galileon}), is responsible for the negative power of
$r$ in the first term of (\ref{EOM}). This is the central idea of
the Vainshtein mechanism, since it makes this first term dominate
at small distances. Imposing now that $\varphi'\rightarrow 0$ for
$r\rightarrow\infty$ (otherwise $\varphi$ would diverge at
infinity), we can immediately write the unique solution of
(\ref{EOM}), which is a mere second-order polynomial equation for
$\varphi'$:
\begin{equation}
\varphi' = \left(\sqrt{\frac{8 k}{r^2}+\frac{r^2 c^4}{GM a_0}
+\left(\frac{\epsilon\, r^2}{r_s}\right)^2}
+\frac{\epsilon\, r^2}{r_s}\right)^{-1}.
\label{phiprimeSol}
\end{equation}
We thus easily recover the asymptotic Brans-Dicke behavior
$\varphi' \approx GM/\epsilon\, r^2 c^2$ for very large distances
$r$, the MOND regime $\varphi' \approx \sqrt{GM a_0}/rc^2$ at
intermediate scales, but a small derivative $\varphi' \approx
r/\sqrt{8k}$ at small distances. Note that this small-distance
behavior of $\varphi'$ does not depend at all on the mass $M$,
which not only generates the right-hand side (source) of
Eq.~(\ref{EOM}), but also the background Schwarzschild geometry
entering the first term of (\ref{EOM}) via the Riemann tensor of
(\ref{Galileon}). This universal small-distance behavior means
that, paradoxically, any body generates strictly the same scalar
force $\varphi' c^2$ in its vicinity. Figure~\ref{fig:PhiPrime}
illustrates the three regimes of solution (\ref{phiprimeSol}).
\begin{figure}
\includegraphics[width=8.5cm]{PhiPrime.eps}
\caption{Behavior of the scalar force $\varphi' c^2$ in the three
regimes of the theory: $\varphi' \propto r$ at small distances
(k-mouflage mechanism), $\varphi' \propto 1/r$ at intermediate
range (MOND regime), and $\varphi' \propto 1/r^2$ at large
distances (asymptotic Brans-Dicke theory).\label{fig:PhiPrime}}
\end{figure}
Neglecting its $\epsilon$ contribution, the maximum value
$\varphi'_\text{max} = (GM a_0/2k)^{1/4}/2c$ is reached at
$r_\text{V} = (8kGM a_0)^{1/4}/c$, which defines thus a
transition radius.

In order to predict the MOND phenomenology in a galaxy of
baryonic mass $M$, we need $r_\text{V} < r_\text{MOND}$, i.e., $k
< GM c^4/8a_0^3$. The lightest dwarf galaxies for which we have
evidence for dark matter or MOND effects give thus an upper bound
for the constant $k$. Draco-like dwarfs correspond to a baryonic
mass $M$ between $10^5$ and $10^6$ solar masses, but even tiny
clusters of only $10^3 M_\odot$ seem to be dominated by dark
matter~\cite{Wolf:2009tu}. Let us thus be conservative and impose
$k \approx G(10^3 M_\odot) c^4/8a_0^3$. This is the numerical
value given in Eq.~(\ref{k}) above.

We can now estimate the order of magnitude of scalar effects
within the solar system. Since we chose a physical metric of the
disformal form (\ref{gtilde}), like in TeVeS, one could naively
conclude that the PPN parameters $\beta$ and $\gamma$ keep their
general relativistic values $\beta = \gamma = 1$. However, there
is now no meaning to define such parameters, because the PPN
formalism assumes that no length scale enters the theory, and it
needs the gravitational potential to be $\propto 1/r$ at the
Newtonian order. Therefore, the consequences of our anomalous
potential $\varphi \propto r^2 + \text{const.}$ at small
distances cannot be analyzed in the standard way. Let us thus
merely compare the anomalous force $\varphi' c^2 \approx r
c^2/\sqrt{8k}$ it generates on a test particle, with the
post-Newtonian forces $\sim (GM_\odot/r^2) \times(r_s/r)$ which
have been precisely tested in the solar system. Since their ratio
is $(r^2/r_s)^2/\sqrt{2k} \approx (r/22\,\text{AU})^4$ for the
chosen value (\ref{k}) of the constant $k$, the scalar effects
are thus smaller than $10^{-5}$ times post-Newtonian ones at the
Earth's orbit, i.e., negligible with respect to the best
experimental constraints. This ratio becomes even smaller for
inner planets, and is $< 10^{-7}$ for Mercury's orbit. On the
other hand, scalar effects grow for outer planets, but they
remain $2\times 10^{-5}$ smaller than post-Newtonian forces at
the orbit of Mars, and $3\times 10^{-3}$ at Jupiter's. This is
consistent with the most precise planetary data. When considering
the Moon's orbit, the contribution of the Earth to the Riemann
tensor entering (\ref{Galileon}) dominates over that of the Sun,
and $\varphi$ is almost spherically symmetric with respect to the
Earth's center. Denoting now as $r$ the distance to this center,
we get again the universal behavior $\varphi' c^2 \approx r
c^2/\sqrt{8k}$, which must be compared to the Newtonian
accelerations caused by both the Earth and the Sun and their
post-Newtonian corrections. We find that scalar effects on the
Moon's motion are $10^{-8}$ smaller than post-Newtonian ones,
i.e., four orders of magnitude smaller than the tightest
experimental constraints derived from Lunar Laser Ranging
\cite{Will:2005va}.

Preferred-frame effects can be estimated in a similar way.
Assuming that the solar system is moving with a velocity $\bm{w}$
with respect to the preferred frame where $U_\mu = (1,0,0,0)$, we
compute the contributions proportional to $\bm{w}^2$ in $\tilde
g_{00}$ and to $w^i$ in $\tilde g_{0i}$, and their radial
derivatives give us the magnitude of the anomalous scalar forces.
Comparing them to those generated by the $\alpha_1$-term in the
PPN formalism \cite{Will:2005va} (while the terms corresponding
to $\alpha_2$ and $\alpha_3$ vanish in the present model
\cite{Sanders:1996wk}), we find that their ratio is $\alt 8
r^3/(r_s \alpha_1\sqrt{2k}) \approx (r/10^4
\text{AU})^3/\alpha_1$, giving thus scalar effects similar to
those of an $\alpha_1 \approx 2\times 10^{-12}$ at the orbit of
Mars. As above, when considering the Moon's orbit around the
Earth, the local value of $\varphi' \approx r/\sqrt{8k}$ must be
used ($r$ denoting now the Earth-Moon distance), and we find that
preferred-frame effects caused by the scalar field are similar to
those of an $\alpha_1 \approx 10^{-15}$. Since the tightest
constraint $\alpha_1 < 10^{-4}$ comes from Lunar Laser Ranging,
we conclude that the model (\ref{action})--(\ref{gtilde}) does
not predict any detectable violation of local Lorentz invariance
in the solar system.

Binary-pulsar tests are much more subtle to compute. A precise
analysis would need either to study the (time-dependent) dynamics
of a binary system at least up to order $\mathcal{O}(1/c^3)$, or
to be able to relate scalar multipoles at infinity to their local
matter sources in spite of the nonlinearities of the Vainshtein
mechanism. In the present paper, we shall only estimate the rough
order of magnitude of scalar radiation by comparing the local
scalar forces to those of the precisely studied Brans-Dicke-like
theories. Let us first note that the monopolar radiation, naively
of order $\mathcal{O}(1/c)$, is actually reduced to order
$\mathcal{O}(1/c^5)$ because the local scalar solution generated
by any body does not depend on time; this is similar to the case
of standard scalar-tensor theories \cite{Damour:1992we}. On the
other hand, the dipolar radiation starts at order
$\mathcal{O}(1/c^3)$ in spite of the fact that the local scalar
solution is the same around any body, independently of its mass.
Indeed, if the two bodies of a binary system do not have the same
mass (say, $m_A\neq m_B)$, they do not move on the same orbit
around their common center of mass, and the global scalar field
they generate defines a preferred (oriented) direction in space.
The dominant scalar radiation is thus a dipole, and we can
estimate its order of magnitude by multiplying the one predicted
in standard scalar-tensor theories (see, e.g. Eq.~(6.52b) of
Ref.~\cite{Damour:1992we}) by the square of the ratio of the
present scalar force between the two bodies ($r c^2/\sqrt{8k}$)
and the standard one. We get that the scalar field contribution
to the time derivative of the orbital period is of order $\dot P
\sim -G c P^3 (m_A-m_B)^2(m_A+m_B)/(32 \pi^2 k m_A m_B)$, and
numerically at least $10^{-32}$ smaller than the tightest
experimental uncertainties. Although this estimate might be
erroneous by some large numerical coefficient, we can anyway
conclude that the present model should easily pass all
binary-pulsar tests.

Although the Galileon field equations involve at most second
derivatives, and avoid thus the generic instability related to
higher derivatives, this does not suffice to prove that these
models are stable. One should carefully analyze both the
boundedness by below of their Hamiltonian density and the
well-posedness of their Cauchy problem. The Hamiltonian of
flat-space Galileons is straightforward to derive
\cite{Zhou:2010di,BDESZ}, but the situation is much more complex
in curved spacetime, because all field equations for $\varphi$
and $g_{\mu\nu}$ involve second derivatives of both of them.
Around a given background, one should thus diagonalize the
kinetic terms in order to test the stability of perturbations and
the hyperbolicity of their field equations. For instance, it
would have no meaning to freeze $g_{\mu\nu}$ and study the
perturbations of $\varphi$ only in the scalar field equation
(similarly to Brans-Dicke theory, which seems to contain a ghost
scalar field for $-3/2 < \omega_\text{BD} < 0$ if one freezes the
Jordan metric, whereas studying simultaneously the dynamics of
$g_{\mu\nu}$ and $\varphi$ shows that the theory is stable even
for such a slightly negative $\omega_\text{BD}$). This problem of
the consistency of Galileon field theories goes thus beyond the
scope of this paper, and we postpone it to a forthcoming
publication \cite{BDESZ}.

Let us finally comment on our choice of Lagrangian
(\ref{Galileon}) to obtain a k-mouflage mechanism reducing scalar
effects at small distances. It happens to be the most efficient
one amongst all those that we have analyzed. The highest-order
Galileon Lagrangian which is nontrivial in 4-dimensional flat
space \cite{Nicolis:2008in,Deffayet:2009wt,Deffayet:2009mn},
namely $\mathcal{L}_5 =
-k_5\, \varepsilon^{\alpha \beta \gamma \delta}
\varepsilon^{\mu \nu \rho \sigma}
\varphi_{,\alpha} \varphi_{,\mu} \varphi_{;\beta\nu}
\left[\varphi_{;\gamma\rho} \varphi_{;\delta \sigma}
-\frac{3}{4} (\varphi_{,\lambda})^2
R_{\gamma\delta \rho \sigma}\right]$,
also generates a small $\varphi'(r) \approx \sqrt{r}/(84\,
k_5)^{1/4}$ at small distances, but its slower radial dependence
makes scalar effects still marginally detectable in the solar
system (while giving again fully negligible preferred-frame
effects). Imposing as above that the MOND phenomenology should
occur in the lightest known dwarf galaxies, we get that scalar
effects in the solar system are $\sim (r/6\,\text{AU})^{7/2}$
times post-Newtonian forces, i.e., about the order of magnitude
of the most precise bounds. Similarly, scalar effects on the
Moon's motion are $10^{-4}$ smaller than post-Newtonian ones,
i.e., of the order of current limits. A full fit of planetary
data taking into account the possible presence of such a small
scalar force would thus be necessary to test whether it is
already excluded or not. If not, this opens the exciting
possibility to detect them in future higher-precision
solar-system observations. All the other known covariant Galileon
actions are either total derivatives in 4 dimensions, or yield a
vanishing scalar field equation around a Schwarzschild background
(like Eq.~(23) of Ref.~\cite{VanAcoleyen:2011mj}), or predict a
negative $\varphi''(r)$ and therefore too large scalar forces in
the solar system. On the other hand, generalized Galileon actions
\cite{Deffayet:2011gz} involving non-differentiated fields
$\varphi$ and/or negative powers of $s\equiv
(\varphi_{,\lambda})^2$ can provide alternative models
\cite{BDESZ}, but anyway less natural than (\ref{Galileon}).

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Given two Borel probability measures $\mu$ and $\nu$ on
$\mathbb{R}$, denote by $\mu\boxplus\nu$ their free additive
convolution, which is the probability distribution of $X+Y$, where
$X$ and $Y$ are free selfadjoint random variables with distributions
$\mu$ and $\nu$, respectively. Free convolution is a binary
operation on the set of Borel probability measures on $\mathbb{R}$,
which is analogue of classical convolutions. We refer to [19] for a
systematic exposition of this subject and [5] for analytic methods
for the computations of free convolutions.

One of the important properties for free convolution is
subordination. That is, the Cauchy transform (see section 2 for
definition) $G_{\mu\boxplus\nu}$ is subordinated to $G_\mu$, in the
sense that $G_{\mu\boxplus\nu}=G_\mu\circ\omega$ for some analytic
self-mapping $\omega$ of the complex upper half-plane. We refer the
reader to [7,9, and 17] for details. One of the applications of
subordination functions is to study the regularity for free
convolution. Free convolution is a highly nonlinear operation,
however, it has a stronger regularizing effect than classical
convolution. Systematic study of regularity for free convolution by
means of subordination functions can be found in [6,7,8, and 17]. In
particular, the atoms of free convolutions are determined. Moreover,
if $\gamma$ is the semicircular distribution then the density of
$\mu\boxplus\gamma$ is continuous on $\mathbb{R}$ and analytic
wherever it is positive.

For $n\in\mathbb{N}$, the $n$-fold free convolution
$\mu\boxplus\cdots\boxplus\mu$ is denoted by $\mu^{\boxplus n}$. It
is known that the discrete semigroup $\{\mu^{\boxplus
n}:n\in\mathbb{N}\}$ can be embedded in a continuous family
$\{\mu^{\boxplus t}:t\geq1\}$ such that $\mu^{\boxplus
t_1}\boxplus\mu^{\boxplus t_2}=\mu^{\boxplus(t_1+t_2)}$ for all
$t_1,t_2\geq1$. This was first proved by Bercovici and Voiculescu in
[6] for measures $\mu$ with compact support and for large values of
$t$. Later, in [14] Nica and Speicher generalized this result to
compactly supported measures and $t>1$ by exhibiting explicit random
variables with the corresponding distributions whose calculations of
the distribution are based on moments and combinatorics. The
existence of subordination functions $\omega_t$ in this context and
related consequences for the regularity for the measures
$\mu^{\boxplus t}$ were thoroughly studied by Belinschi and
Bercovici in [1] and [2]. More precisely, for $t>1$ let
$\mu^{\boxplus t}$ be the measure satisfying the requirement
$F_{\mu^{\boxplus t}}=F_\mu\circ\omega_t$, where $F_{\mu^{\boxplus
t}}=1/G_{\mu^{\boxplus t}}$ and $F_\mu=1/G_\mu$ are reciprocal
Cauchy transforms. This gives an alternate proof of the existence of
$\mu^{\boxplus t}$. With the help of $\omega_t$, it was shown that
the free convolution power $\mu^{\boxplus t}$ of $\mu$ has no
singular continuous part and its density in the absolutely
continuous part is locally analytic.

In this present paper, we use different points of views to obtain
some properties of $F_\mu$ and analyze the relation between the
supports $\mathrm{supp}(\mu)$ and $\mathrm{supp}(\mu^{\boxplus t})$
of $\mu$ and $\mu^{\boxplus t}$, $t>1$, respectively. By means of
the methods developed in this paper, we give a complete description
about how $\mathrm{supp}(\mu^{\boxplus t})$ changes when $t$
increases. Motivated by the free central limit theorem (see
[6,10,11,12, 15, and 20] for details), one of the purposes of this
paper is to study the number $n(t)$ of components in the support of
$\mu^{\boxplus t}$. We show that $n(t)$ is a decreasing function of
$t$ for $t>1$ and that $n(t)=1$ for sufficiently large $t$ if and
only if $n(t')$ is finite for some $t'>1$. We also construct a
measure $\mu$ so that $n(t)$ is infinite for all $t>1$.

The paper is organized as follows. Section 2 contains some
definitions and basic propositions from free probability theory.
Section 3 provides some properties of $F_\mu$ which are related to
the subordination function $\omega_t$, and an explicit formula for
the density of the absolutely continuous part of $\mu^{\boxplus t}$.
Section 4 contains complete investigations on the supports of the
measures $\mu^{\boxplus t}$.

\section{Preliminary}

For any complex $z$ in $\mathbb{C}$, let $\Re z$ and $\Im z$ be the
real and imaginary parts of $z$, respectively, and denote by
$\mathbb{C}^+=\{z\in\mathbb{C}:\Im z>0\}$ the complex upper
half-plane. For any Borel probability measure $\mu$ on $\mathbb{R}$,
define its Cauchy transform as
\[G_\mu(z)=\int_\mathbb{R}\frac{d\mu(s)}{z-s},\;\;\;z\in\mathbb{C}^+.\] As shown in [5], the composition
inverse $G_\mu^{-1}$ of $G_\mu$ is defined in an appropriate Stolz
angle
\[D_{\eta,\epsilon}=\{x+iy:|x|<-\eta y,\;0<-y<\epsilon\}\]
of zero in the lower half-plane. The Voiculescu transform
$\mathcal{R}_\mu$ of the measure $\mu$ defined as
\[\mathcal{R}_\mu(z)=G_\mu^{-1}(z)-z^{-1},\;\;\;z\in D_{\eta,\epsilon},\]
is the linearizing transform for the operation $\boxplus$. That is,
the identity
\[\mathcal{R}_{\mu\boxplus\nu}=\mathcal{R}_\mu+\mathcal{R}_\nu\]
holds on a domain where these three functions are defined. The
measure $\mu$ can be recovered from the Cauchy transform $G_\mu$ as
the weak$^*$-limit of the measures
\[d\mu_\epsilon(s)=-\frac{1}{\pi}\Im G_\mu(s+i\epsilon)\;ds\] as $\epsilon\to0^+$, which
we refer to as the Stieltjes inversion formula. Particularly, if the
function $\Im G_\mu(x)$ is continuous at the point $x\in\mathbb{R}$
then in the Lebesgue decomposition the density of the absolutely
continuous part of $\mu$ at $x$ is given by $-\Im G_\mu(x)/\pi$.
Therefore, this inversion formula gives a way to extract the density
function from the Cauchy transform.

The reciprocal Cauchy transform $F_\mu=1/G_\mu$ is an analytic
self-mapping of $\mathbb{C}^+$. By Nevanlinna representation, it can
be expressed as
\[F_\mu(z)=a+z+\int_\mathbb{R}\frac{1+sz}{s-z}\;d\rho(s),\leqno{(2.1)}\] where $a=\Re
F_\mu(i)$ and $\rho$ is some finite positive Borel measure on
$\mathbb{R}$ uniquely determined by $F_\mu$. Indeed, $\rho$ is the
weak$^*$-limit as $\epsilon\to0^+$ of the measures
\[d\rho_\epsilon(s)=\frac{\Im F_\mu(s+i\epsilon)}{\pi(s^2+1)}\;ds.\leqno{(2.2)}\] It was
shown in [13] that $F_\mu$ can be expressed in terms of a Cauchy
transform of some measure if $\mu$ has finite mean and unit
variance. More precisely, a measure $\mu$ satisfies
\[\int_\mathbb{R}s\;d\mu(s)=0\;\;\;\;\mathrm{and}\;\;\;\;\int_\mathbb{R}s^2\;d\mu(s)=1\] if and
only if \[F_\mu(z)=z-G_\nu(z),\;\;z\in\mathbb{C}^+,\] where $\nu$ is
some Borel probability measure on $\mathbb{R}$.

Denote by $\mathop{z\to\alpha}\limits_{\sphericalangle}$ if
$z\to\alpha$ nontangentially to $\mathbb{R}$, i.e., $(\Re
z-\alpha)/\Im z$ stays bounded. A useful criterion for locating an
atom $\alpha$ of $\mu$ is the limit
\[\lim_{\mathop{z\to\alpha}\limits_{\sphericalangle}}
(z-\alpha)G_\mu(z)=\mu(\{\alpha\}).\leqno{(2.3)}\] An equivalent
statement of the above limit is that a point $\alpha\in\mathbb{R}$
is an atom of $\mu$ if and only if $F_\mu(\alpha)=0$ and the
Julia-Carath\'{e}odoty derivative $F_\mu'(\alpha)$ (which is the
limit of
\[\frac{F_\mu(z)-F_\mu(\alpha)}{z-\alpha}\] as $z\to\alpha$ nontangentially and $z\in\mathbb{C}^+$)
is finite, in which case
\[F_\mu'(\alpha)=\frac{1}{\mu(\{\alpha\})}.\]
Similarly, there is an effective way to identify the atoms of the
measure $\rho$ in the Nevanlinna representation (2.1) for $F_\mu$.
The following lemma is well known, however, we have no recollection
of seeing it stated explicitly. We provide a full proof without
claiming any paternity.

\begin{lem} If $\alpha\in\mathbb{R}$ then
\[\lim_{\mathop{z\to\alpha}\limits_{\sphericalangle}}(z-\alpha)F_\mu(z)=-(\alpha^2+1)\rho(\{\alpha\}).\]
\end{lem}

\begin{pf} By considering the positive measure $\rho-\rho(\{\alpha\})\delta_0$, we
may assume that $\alpha$ is not an atom of $\rho$, in which case we
must show that $(z-\alpha)F_\mu(z)\to0$ as $z\to\alpha$
nontangentially. Write $z=x+iy$ and suppose
\[\frac{|x-\alpha|}{y}\leq
M\;\;\;\;\mathrm{and}\;\;\;\;|z-\alpha|<1,\] where $0<M<\infty$. Let
$N=2(|\alpha|+1)$. Note that if $s\geq N$ then
\[\left|\frac{s}{s-z}\right|=\left|\frac{\frac{s}{y}}{\frac{s-x}{y}-i}\right|\leq
\frac{s}{s-x}\leq\frac{s}{s-(\alpha+1)}\leq2,\] whence
\[|z-\alpha|\left|\frac{sz}{s-z}\right|\leq2(|\alpha|+1).\]
Similarly, the above inequality also holds for $s\leq-N$. For
$s\in(-N,N)$, we have
\[|z-\alpha|\left|\frac{sz}{s-z}\right|\leq
\frac{|z-\alpha|}{y}|sz|\leq N(|\alpha|+1)\sqrt{M^2+1}.\] Moreover,
for any $s\in\mathbb{R}$ we have
\[\frac{|z-\alpha|}{|s-z|}=\frac{\left|\frac{x-\alpha}{y}+i\right|}{\left|\frac{x-s}{y}+i\right|}\leq
\left|\frac{x-\alpha}{y}+i\right|\leq\sqrt{M^2+1}.\] Since
\[\left|(z-\alpha)\int_\mathbb{R}\frac{1+sz}{s-z}\;d\rho(s)\right|\leq
\int_\mathbb{R}\frac{|z-\alpha|+|sz||z-\alpha|}{|s-z|}\;d\rho(s),\]
and as $z\to\alpha$ nontangentially the integrand converges to zero
$\rho$-a.e. and stays bounded by the above discussions, it is easy
to see the desired result holds.
\end{pf} \qed

As in the introduction, the collection of the $t$-th convolution
power $\mu^{\boxplus t}$ of $\mu$ forms a semigroup $\{\mu^{\boxplus
t}:t>1\}$ which interpolates the discrete semigroup $\{\mu^{\boxplus
n}:n\in\mathbb{N}\}$. The construction of $\mu^{\boxplus t}$ for
arbitrary $\mu$ by using subordination functions and analytic
methods is introduced in following two paragraphs.

For any $t>1$, consider the function
\[H_t(z)=tz-(t-1)F_\mu(z),\;\;\;z\in\mathbb{C}^+.\leqno{(2.4)}\] Then we have $\Im H_t(z)\leq\Im
z$ for $z\in\mathbb{C}^+$ and
\[\lim_{y\to\infty}\frac{H_t(iy)}{iy}=1.\] Moreover, there exists
a continuous function
$\omega_t:\overline{\mathbb{C}^+}\to\overline{\mathbb{C}^+}$ such
that $\omega_t(\mathbb{C}^+)\subset\mathbb{C}^+$,
$\omega_t|\mathbb{C}^+$ is analytic, and $H_t(\omega_t(z))=z$ for
$z\in\mathbb{C}^+$. If the set $\{z\in\mathbb{C}^+:\Im H_t(z)>0\}$
is denoted by $\Omega_t$ then
\[\Omega_t=\omega_t(\mathbb{C}^+)\]
is a simply connected set whose boundary $\omega_t(\mathbb{R})$ is a
simple curve. Then the equation
$\omega_t(H_t(\omega_t(z)))=\omega_t(z)$, $z\in\mathbb{C}^+$, shows
that the equation $\omega_t(H_t(z))=z$ holds for $z\in\Omega_t$, and
hence the inverse of $\omega_t$ is $H_t|\Omega_t$. Moreover, if
$x\in\mathbb{R}$ and $\Im\omega_t(x)>0$ then $\omega_t$ can be
continued analytically to a neighborhood of $x$.

Let $\mu^{\boxplus t}$ be the measure defined by the requirement
\[F_{\mu^{\boxplus
t}}(z)=F_\mu(\omega_t(z)),\;\;\;\;z\in\mathbb{C}^+.\leqno{(2.5)}\]
It turns out that the measure $\mu^{\boxplus t}$ can be
characterized in terms of Voiculescu transform, i.e., the measure
$\mu^{\boxplus t}$ obtained in this way is the unique measure
satisfying
\[\mathcal{R}_{\mu^{\boxplus t}}(z)=t\mathcal{R}_\mu(z),\] where $z$
is in the common domain of these two functions. By using the
equation $H_t(\omega_t(z))=z$ we can rewrite (2.5) as
\[F_{\mu^{\boxplus
t}}(z)=\frac{t\omega_t(z)-z}{t-1},\;\;\;z\in\mathbb{C}^+,\] which
shows, in particular, that $F_{\mu^{\boxplus t}}$ extends
continuously to $\overline{\mathbb{C}^+}$. Also, by the equation
$\omega_t(H_t(z))=z$, $z\in\Omega_t$, we have
\[F_{\mu^{\boxplus t}}(H_t(z))=F_\mu(z),\;\;\;z\in\Omega_t.\]

Regularity properties of $\mu^{\boxplus t}$ have been studied
thoroughly in [1] and [2]. We now review some of these results which
are relevant to our investigation on the number of the components in
the support of $\mu^{\boxplus t}$. For any $t>1$, the singular part
of the measure $\mu^{\boxplus t}$ with respect to Lebesgue measure
is purely atomic and $\mu^{\boxplus t}$ has an atom $\alpha$ if and
only if $\mu(\{t\alpha\})>1-t^{-1}$ in which case $\mu^{\boxplus
t}(\{\alpha\})=t\mu(\{\alpha\})-(t-1)$. A weaker statement, i.e., a
point $\alpha$ satisfying $F_{\mu^{\boxplus t}}(t\alpha)=0$ if and
only if $\mu(\{\alpha\})\geq1-t^{-1}$, also holds. Hence for any
measure $\mu$ which is not a point mass, the atoms of $\mu^{\boxplus
t}$ disappear for sufficiently large $t$, and hence $\mu^{\boxplus
t}$ is absolutely continuous for such $t$.

\section{Decreasing number of components in the support of $\mu^{\boxplus t}$}

Throughout this section, $t$ is a parameter with $t>1$ and for any
given and fixed Borel probability measure $\mu$ on $\mathbb{R}$,
$\rho$ is the unique finite positive Borel measure in the Nevanlinna
representation (2.1) of $F_\mu$. Our analysis of the support of
$\mu^{\boxplus t}$ will be based on the following function
$g:\mathbb{R}\to\mathbb{R}_+\cup\{\infty\}$ defined as
\[g(x)=\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s),\;\;\;x\in\mathbb{R}.\]
Associated with the function $g$ and the parameter $t$ are the
following sets
\[V_t^+=\left\{x\in\mathbb{R}:g(x)>\frac{1}{t-1}\right\},\]
\[V_t=\left\{x\in\mathbb{R}:g(x)=\frac{1}{t-1}\right\},\]
\[V_t^-=\left\{x\in\mathbb{R}:g(x)<\frac{1}{t-1}\right\},\]
and the function
\[f_t(x)=\inf\left\{y\geq0:\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y^2}\;d\rho(s)\leq\frac{1}{t-1}\right\}\]
which will play important roles in the analysis.

\begin{lem} If $V_t\cup V_t^-$ contains an open interval $I$
then $\rho(I)=0$ and $g$ is strictly convex on $I$.
\end{lem}

\begin{pf} Let $(a,b)$ be any interval contained in $I$. Then for
any $x\in(a,b)$ we have
\[\frac{1}{t-1}\geq\int_a^b\frac{s^2+1}{(s-x)^2}\;d\rho(s)\geq
\int_a^b\frac{1}{(b-a)^2}\;d\rho(s)=\frac{\rho((a,b))}{(b-a)^2},\]
from which we obtain that $\rho((a,b))=0$. Since the second order
derivative of $g$ is positive on $(a,b)$, the second assertion
follows. This completes the proof.
\end{pf} \qed

By the definition of $H_t$ in (2.4), we have
\[\Im H_t(z)=(\Im
z)\left(1-(t-1)\int_\mathbb{R}\frac{s^2+1}{|z-s|^2}\;d\rho(s)\right),\;\;\;z\in\mathbb{C}^+.
\leqno{(3.1)}\]

\begin{prop} The function $H_t$ satisfies $\Im H_t(z)>0$ for all
$z\in\mathbb{C}^+$ if and only if $\mu$ is a Dirac measure in which
case $\mu^{\boxplus t}$ is a Dirac measure as well.
\end{prop}

\begin{pf} It follows from (3.1) that $\Im H_t(z)>0$ holds for all $z\in\mathbb{C}^+$
if and only if
\[\int_\mathbb{R}\frac{s^2+1}{|z-s|^2}\;d\rho(s)<\frac{1}{t-1},\;\;\;z\in\mathbb{C}^+,\]
which happens if and only if $g(x)\leq(t-1)^{-1}$ holds for all
$x\in\mathbb{R}$ by monotone convergence theorem. Then Lemma 3.1
shows that $\Im H_t(z)>0$, $z\in\mathbb{C}^+$, if and only if $\rho$
is a zero measure or, equivalently, $F_\mu=a+z$, i.e., $\mu$ is the
point mass $\delta_{-a}$. It is easy to see from (2.4) and (2.5)
that $F_{\mu^{\boxplus t}}(z)=ta+z$, whence $\mu^{\boxplus
t}=\delta_{-ta}$.
\end{pf} \qed

From the proof of the preceding proposition, $\mu$ is a point mass
if and only if $V_t^+$ is an empty set for all $t>1$. For the rest
of the paper, we confine our attention to the case of $\mu$ which is
not a Dirac measure.

The following lemmas are essentially similar to the results in [8]
and [11]. For the completeness, we provide the proofs here.

\begin{lem} The set $V_t^+$ coincides with
$\{x\in\mathbb{R}:f_t(x)>0\}$ and
$\{x\in\mathbb{R}:f_t(x)=0\}=V_t\cup V_t^-$. Moreover, for any
$x\in\mathbb{R}$ we have
\[\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+f_t^2(x)}\;d\rho(s)\leq\frac{1}{t-1},\leqno{(3.2)}\] where the equality
holds for $x\in V_t^+$.
\end{lem}

\begin{pf} First note that if $z=x+iy\in\mathbb{C}^+$, i.e., $y>0$,
then
\[\Im
F_\mu(z)=y\left(1+\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y^2}\;d\rho(s)\right),\]
which gives, particularly,
\[\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y^2}\;d\rho(s)<\infty.\]
It follows that for any fixed $x\in\mathbb{R}$, the mapping
\[y\mapsto\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y^2}\;d\rho(s)\] is
decreasing and continuous on $(0,\infty)$ by Fatou's lemma. Indeed,
for any sequence $\{y_n\}$ converging to $y\in(0,\infty)$,
\begin{align*}
\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y^2}\;d\rho(s)&\leq
\liminf_{n\to\infty}\int_\mathbb{R}\frac{s^2+1}{(x-t)^2+y_n^2}\;d\rho(s) \\
&\leq\limsup_{n\to\infty}\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+y_n^2}\;d\rho(s)\leq
\int_\mathbb{R}\frac{s^2+1}{(x-t)^2+y^2}\;d\rho(s)
\end{align*} which shows the continuity of the mapping.
If $f_t(x)>0$ then the definition of $f_t$ shows that for small
$\epsilon>0$ we have
\[\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+(1+\epsilon)f_t^2(x)}\;d\rho(s)\leq
\frac{1}{t-1}<\int_\mathbb{R}\frac{s^2+1}{(x-s)^2+(1-\epsilon)f_t^2(x)}\;d\rho(s),\]
which gives the equality in (3.2) by letting $\epsilon\to0$. If
$f_t(x)=0$ then it is easy to see from Fatou's lemma that $x\in
V_t\cup V_t^-=\mathbb{R}\setminus V_t^+$, whence
$V_t^+=\{x\in\mathbb{R}:f_t(x)>0\}$ and $V_t\cup
V_t^-=\{x\in\mathbb{R}:f_t(x)=0\}$. This completes the proof.
\end{pf} \qed

With the help of preceding lemma, we are able to provide another
description of the set $\Omega_t=\{z\in\mathbb{C}^+:\Im H_t(z)>0\}$.

\begin{lem} The set $\{x+iy\in\mathbb{C}^+:y>f_t(x)\}$ coincides with the set $\Omega_t$ and
\[\int_\mathbb{R}\frac{s^2+1}{|z-s|^2}\;d\rho(s)<\frac{1}{t-1},\;\;\;\;z\in\Omega_t.\leqno{(3.3)}\]
The function $f_t$ is continuous on $\mathbb{R}$ and the set $V_t^+$
is open. Moreover, if $1<t_1<t_2$ then
$\Omega_{t_2}\subset\Omega_{t_1}$ and $f_{t_1}(x)\leq f_{t_2}(x)$
for all $x\in\mathbb{R}$.
\end{lem}

\begin{pf} The equation (3.1) shows that $z\in\Omega_t$ if and only if (3.3) holds.
By Lemma 3.3 and the definition of $f_t$, we have
$\Omega_t=\{x+iy\in\mathbb{C}^+:y>f_t(x)\}$, and hence the boundary
$\partial\Omega_t$ is the graph of $y=f_t(x)$. Since
$\partial\Omega_t$ is simply connected and
$V_t=\{x\in\mathbb{R}:f_t(x)>0\}$, the second assertion holds. The
last assertion follows from (3.3).
\end{pf} \qed

It was shown in [8] that the Cauchy transform $G_\mu$ is Lipschitz
continuous on some subset of $\overline{\mathbb{C}^+}$. The
following proposition gives similar results for the reciprocal
Cauchy transform $F_\mu$ on $\overline{\Omega_t}$.

\begin{prop} The reciprocal Cauchy transform $F_\mu$ extends continuously to
$\overline{\Omega_t}$ and satisfies
\[|F_\mu(z_1)-F_\mu(z_2)|\leq\frac{t}{t-1}|z_1-z_2|,\;\;\;\;z_1,z_2\in\overline{\Omega_t}.\leqno{(3.4)}\]
Moreover, this continuous extension can be represented as
\[F_\mu(z)=a+z+\int_\mathbb{R}\frac{1+sz}{s-z}\;d\rho(s),\;\;\;\;z\in\overline{\Omega_t}.\leqno{(3.5)}\] If
$t>2$ then
\[\frac{t-2}{t-1}|z_1-z_2|\leq|F_\mu(z_1)-F_\mu(z_2)|,\;\;\;z_1,z_2\in\overline{\Omega_t},\]
and, consequently, $F_\mu$ is one-to-one.
\end{prop}

\begin{pf} Using (3.3) and monotone convergence theorem gives the inequality
\[\int_\mathbb{R}\frac{s^2+1}{|z-s|^2}\;d\rho(s)\leq\frac{1}{t-1},\;\;\;\;z\in\overline{\Omega_t},\]
whence the integral
\[\int_\mathbb{R}\frac{s}{|z-s|^2}\;d\rho(s)\] converges
for all $z\in\overline{\Omega_t}$ by H\"{o}lder inequality. This
implies that
\[\int_\mathbb{R}\frac{1+sz}{s-z}\;d\rho(s)=\int_\mathbb{R}
\frac{(1-|z|^2)s+x(s^2-1)}{|s-z|^2}\;d\rho(s)+iy\int_\mathbb{R}
\frac{s^2+1}{|s-z|^2}\;d\rho(s)\] converges as well for
$z=x+iy\in\overline{\Omega_t}$. This implies that (3.5) holds, and
moreover for $z_1,z_2\in\overline{\Omega_t}$ we have
\begin{align*}
\left|\frac{F_{\mu}(z_1)-F_{\mu}(z_2)}{z_1-z_2}\right|&
\leq1+\left|\int_\mathbb{R}\frac{s^2+1}{(s-z_1)(s-z_2)}\;d\rho(s)\right| \\
&\leq1+\left(\int_\mathbb{R}\frac{s^2+1}{|s-z_1|^2}\;d\rho(s)\right)^{1/2}
\left(\int_\mathbb{R}\frac{s^2+1}{|s-z_2|^2}d\rho(s)\right)^{1/2} \\
&\leq1+\frac{1}{t-1}=\frac{t}{t-1},
\end{align*} where the H\"{o}lder inequality is used in the second
inequality. Similarly, if $t>2$ then
\[\left|\frac{F_{\mu}(z_1)-F_{\mu}(z_2)}{z_1-z_2}\right|\geq1-\frac{1}{t-1},\]
which implies the desired inequality. This completes the proof.
\end{pf} \qed

For $t>1$, it could happen that the boundary $\partial\Omega_t$
contains an interval $I\subset\mathbb{R}$, i.e., $f_t$ vanishes on
$I$. The following corollary characterizes such an interval.

\begin{cor} Let $I$ be an open interval in $\mathbb{R}$. Then
$\rho(I)=0$ if and only if for any closed interval $J\subset I$
there exists some $t>1$ such that $f_t(x)=0$ for $x\in J$. Moreover,
if $\rho(I)=0$ then the expression $(3.5)$ for $F_\mu$ holds for
$x\in I$ and $F_\mu$ is strictly increasing on $I$.
\end{cor}

\begin{pf} First, assume that
$I$ is bounded, $\rho(I)=0$, and $J$ is any closed interval
contained in $I$. Since $g$ is continuous on $I$, there exists a
number $t>1$ such that $g(x)\leq(t-1)^{-1}$ or, equivalently,
$f_t(x)=0$ for all $x\in J$. If $I=(a,\infty)$ (resp.
$I=(-\infty,a)$) for some finite $a$ and $\rho(I)=0$ then $g$ is
decreasing (resp. increasing) on $I$, whence it is easy to see that
the necessity follows. The sufficiency follows from Lemma 3.1 and
3.3.

By Proposition 3.5 and the first assertion of the corollary, it is
easy to see that (3.5) holds on any interval $I$ having
$\rho$-measure zero. For such an interval $I$, taking the derivative
gives that
\[F_\mu'(x)=1+\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s),\;\;\;\;x\in I,\] and therefore
$F_\mu$ is strictly increasing on $I$.
\end{pf} \qed

From the proof of Proposition 3.5, the function $H_t$ also satisfies
similar estimates. That is,
\[|H_t(z_1)-H_t(z_2)|\leq2|z_1-z_2|,\;\;\;\;z_1,z_2\in\overline{\Omega_t},\]
from which we deduce that $H_t$ is also Lipscitz continuous and has
a continuous extension to $\overline{\Omega_t}$. If this continuous
extension is still denoted by $H_t$ then the equation
$H_t(\omega_t(z))=z$, $z\in\overline{\mathbb{C}^+}$, shows that
\[2|z_1-z_2|\leq|\omega_t(z_1)-\omega_t(z_2)|,\;\;\;z_1,z_2\in\overline{\mathbb{C}^+}.\]
The results in the following corollary are direct consequences of
(3.3) and preceding discussions. We refer the reader to [2] for
related results of the function $H_t$ and the subordination function
$\omega_t$.

\begin{cor} The function $H_t$ is a conformal mapping
from $\Omega_t$ onto $\mathbb{C}^+$ and is a homeomorphism from
$\overline{\Omega_t}$ to $\overline{\mathbb{C}^+}$.
\end{cor}

Note that the continuity of $f_t$ on $\mathbb{R}$ ensures that the
map $x\mapsto x+if_t(x)$ is a homeomorphism of $\mathbb{R}$ onto
$\partial\Omega_t$. Define the map $\psi_t:\mathbb{R}\to\mathbb{R}$
as
\[\psi_t(x)=H_t(x+if_t(x)).\] Then
$\psi_t$ is a homeomorphism as well by Corollary 3.7 and the
equation
\[\omega_t(\psi_t(x))=\omega_t(H_t(x+if_t(x)))=x+if_t(x)\;\;\;\mathrm{holds}\;\;\;\mathrm{for}
\;\;\;x\in\mathbb{R}.\] It is apparent that we have
\[F_{\mu^{\boxplus t}}(\psi_t(x))=\frac{tx-\psi_t(x)+itf_t(x)}{t-1},\;\;\;x\in\mathbb{R}.\leqno{(3.6)}\]
Since $\omega_t$ can be continued
analytically to a neighborhood of $\psi_t(x)$ if $f_t(x)>0$, it
follows that the function $f_t$ is analytic on $V_t^+$.

Now we are in a position to state the main theorem of this section.
For any measure $\nu$ on $\mathbb{R}$, let $\mathrm{supp}(\nu)$ be
the support of $\nu$ and denote by $\nu^{\mathrm{ac}}$ the
absolutely continuous part of $\nu$ with respect to Lebesgue
measure.

\begin{thm} Suppose that $\mu$ is a Borel probability measure $($not a point mass$)$ on $\mathbb{R}$ and
$t>1$. Then the following statements hold.
\\ $(i)$ The absolutely continuous part of $\mu^{\boxplus t}$ is concentrated on
the closure of $\psi_t(V_t^+)$.
\\ $(ii)$ The density of $(\mu^{\boxplus t})^{\mathrm{ac}}$ on the set $\psi_t(V_t^+)$ is given by
\[\frac{d\mu^{\boxplus t}}{dx}(\psi_t(x))=\frac{t(t-1)f_t(x)}{\pi|tx-\psi_t(x)+itf_t(x)|^2},\;\;\;x\in V_t^+.\]
$(iii)$ The density of $(\mu^{\boxplus t})^{\mathrm{ac}}$ is
analytic on the set $\psi_t(V_t^+)$ and at the point $\psi_t(x)$,
$x\in V_t^+$, it is bounded by $(t-1)/(\pi tf_t(x))$.
\\ $(iv)$ The number of the
components in $\mathrm{supp}((\mu^{\boxplus t})^{\mathrm{ac}})$ is a
decreasing function of $t$.
\end{thm}

\begin{pf} Since $F_{\mu^{\boxplus t}}$ is continuous on
$\overline{\mathbb{C}^+}$, the density of the nonatomic part of
$\mu^{\boxplus t}$ is continuous except at the points $x$ at which
$F_{\mu^{\boxplus t}}(x)=0$. In view of (3.6) and the Stieltjes
inversion formula, the density of $\mu^{\boxplus t}$ at $\psi_t(x)$
where $x\in V_t^+$ is given by
\begin{align*}
\frac{d\mu^{\boxplus t}}{dx}(\psi_t(x))&=-\frac{1}{\pi}\Im
G_{\mu^{\boxplus
t}}(\psi_t(x)) \\
&=\frac{t(t-1)f_t(x)}{\pi|tx-\psi_t(x)+itf_t(x)|^2}.
\end{align*}
This shows that the support of $(\mu^{\boxplus t})^{\mathrm{ac}}$ is
the closure of $\psi_t(V_t^+)$ and they have the same number of
components since $\psi_t$ is a homeomorphism. The assertion (iii)
follows from part (ii) and the fact that $f_t$ is analytic on
$V_t^+$. To verify the statement (iv), we need to show the the
number of the components in $V_t^+$ is nonincreasing as $t$
increases. This will hold if we show that $g$ never has a local
maximum in any open interval $(a,b)$ in $\mathbb{R}\backslash
V_t^+$. Indeed, $g$ is strictly convex on such an interval by Lemma
3.1, whence (iv) holds.
\end{pf} \qed

Before we state the next result, observe that the number of atoms of
$\mu^{\boxplus t}$ decreases as a function of $t$. Indeed, this is a
direct consequence of the fact that a point $\alpha$ is an atom of
$\mu^{\boxplus t}$ if and only $\mu(\{\alpha/t\})>1-t^{-1}$. Since
the open set $V_t^+$ can be written as a countable union of open
intervals, combining the result for atoms and Theorem 3.8 gives the
following conclusion.

\begin{cor} With the same assumption in Theorem $\mathrm{3.8}$, the measure $\mu^{\boxplus
t}$ has at most countably many components in the support, which
consists of finitely many points $($atoms$)$ and countably many
intervals. Moreover, the number of the components in
$\mathrm{supp}(\mu^{\boxplus t})$ is a decreasing function of $t$.
\end{cor}

\section{Support and Regularity for $\mu^{\boxplus t}$}

In this section, we will investigate the support and regularity
property for the free convolution power $\mu^{\boxplus t}$ of $\mu$
where $t>1$ and $\mu$ is a Borel probability measure (not a point
mass so that $V_t^+$ is nonempty) on $\mathbb{R}$. By analyzing the
formula
\[F_{\mu^{\boxplus t}}(\psi_t(x))=\frac{tx-\psi_t(x)+itf_t(x)}{t-1},\;\;\;x\in\mathbb{R},\leqno{(4.1)}\]
established in the previous section, we are able to understand the
relation between supports and regularities for the measures $\mu$
and $\mu^{\boxplus t}$. If $f_t(x)=0$ for some $x\in\mathbb{R}$ then
$H_t(x)=\psi_t(x)$ and (4.1) can be rewritten as
\[F_\mu(x)=F_{\mu^{\boxplus
t}}(\psi_t(x))=\frac{tx-\psi_t(x)}{t-1}.\leqno{(4.2)}\] The
following result gives another basic property of the homeomorphism
$\psi_t$.

\begin{lem} The derivative $\psi_t'(x)>0$
for $x\in V_t^+$ and $\psi_t$ is a strictly increasing function on
$\mathbb{R}$.
\end{lem}

\begin{pf} Let $z=x+if_t(x)$ be any point with $x\in V_t^+$. Since functions $H_t$ and $f_t$ are
analytic at points $z$ and $x$, respectively, it follows that
\begin{align*}
\psi_t'(x)&=\left(\frac{dH_t}{dz}(x+if_t(x))\right)(1+if_t'(x)) \\
&=\Re\left(\frac{dH_t}{dz}(x+if_t(x))\right)-\Im\left(\frac{dH_t}{dz}(x+if_t(x))\right)f_t'(x).
\end{align*} Observe that
\begin{align*}
\frac{dH_t}{dz}(x+if_t(x))&=1-(t-1)\int_\mathbb{R}\frac{s^2+1}{(s-z)^2}\;d\rho(s) \\
&=1-(t-1)\int_\mathbb{R}\frac{(s^2+1)[(s-x)^2-f_t^2(x)+2i(s-x)f_t(x)]}{|s-z|^4}\;d\rho(s) \\
&=1-(t-1)\int_\mathbb{R}\frac{(s^2+1)[|s-z|^2-2f_t^2(x)+2i(s-x)f_t(x)]}{|s-z|^4}\;d\rho(s) \\
&=2(t-1)\left[\int_\mathbb{R}\frac{(s^2+1)f_t^2(x)}{|s-z|^4}\;d\rho(s)-i
\int_\mathbb{R}\frac{(s^2+1)(s-x)f_t(x)}{|s-z|^4}\;d\rho(s)\right]
\end{align*} where the identity
\[\int_\mathbb{R}\frac{s^2+1}{|s-z|^2}\;d\rho(s)=\frac{1}{t-1}\leqno{(4.3)}\] is used in
the forth equality. On the other hand, differentiating (4.3) with
respect to $x$ gives
\[\int_\mathbb{R}\frac{(s^2+1)(s-x)}{|s-z|^4}\;d\rho(s)=f_t(x)f_t'(x)
\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s),\] whence
\[\frac{dH_t}{dz}(x+if_t(x))=2(t-1)f_t^2(x)\left[\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s)-if_t'(x)
\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s)\right].\] Combining
these results gives that
\begin{align*}
\psi_t'(x)&=2(t-1)f_t^2(x)\left[\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s)+f_t'^2(x)
\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s)\right] \\
&=2(t-1)f_t^2(x)(f_t'^2(x)+1)\int_\mathbb{R}\frac{s^2+1}{|s-z|^4}\;d\rho(s) \\
&>0,
\end{align*} as desired. Since $\psi_t$ is one-to-one and
continuous on $\mathbb{R}$ with nonnegative derivative on a nonempty
subset of $\mathbb{R}$, it must be strictly increasing.
\end{pf} \qed

Note that if $x\in V_t$ then $f_t(x)=0$, and Proposition 3.5 shows
that the limit $F_\mu(x)=\lim_{\epsilon\to0}F_\mu(x+i\epsilon)$
exists and is a finite number. The next result characterizes the
atoms of $\mu$ in terms of $V_t$ and $F_\mu$.

\begin{lem} A point $\alpha\in\mathbb{R}$ is an
atom of $\mu$ with mass $\mu(\{\alpha\})=1-t^{-1}$ if and only if
$\alpha\in V_t$ and $F_\mu(\alpha)=0$ in which case the
Julia-Carath\'{e}odory derivative of $F_\mu$ at $\alpha$ is
\[F_\mu'(\alpha)=\frac{1}{\mu(\{\alpha\})}=1+\int_\mathbb{R}\frac{s^2+1}{(s-\alpha)^2}\;d\rho(s).\]
\end{lem}

\begin{pf} First assume that $\alpha$ is an atom of $\mu$, i.e., the
limit in (2.3) is $1-t^{-1}>0$. This implies that
\begin{align*}
\lim_{\epsilon\to0}\frac{\Re F_\mu(\alpha+i\epsilon)}{i\epsilon}&
=\frac{1}{2}\lim_{\epsilon\to0}\left(\frac{F_\mu(\alpha+i\epsilon)}{i\epsilon}
+\frac{\overline{F_\mu(\alpha+i\epsilon)}}{i\epsilon}\right) \\
&=\frac{1}{2}\lim_{\epsilon\to0}\left(\frac{1}{i\epsilon
G_\mu(\alpha+i\epsilon)}-\frac{1}{\overline{i\epsilon
G_\mu(\alpha+i\epsilon)}}\right) \\
&=0.
\end{align*}
Using the limit shown above gives
\begin{align*}
\frac{1}{\mu(\{\alpha\})}&=\lim_{\epsilon\to0}\frac{F_\mu(\alpha+i\epsilon)}{i\epsilon} \\
&=\lim_{\epsilon\to0}\frac{\Im F_\mu(\alpha+i\epsilon)}{\epsilon} \\
&=\lim_{\epsilon\to0}\frac{\epsilon+\epsilon\int_\mathbb{R}\frac{s^2+1}{(s-\alpha)^2+\epsilon^2}\;d\rho(s)}{\epsilon} \\
&=1+\lim_{\epsilon\to0}\int_\mathbb{R}\frac{s^2+1}{(s-\alpha)^2+\epsilon^2}\;d\rho(s) \\
&=1+\int_\mathbb{R}\frac{s^2+1}{(s-\alpha)^2}\;d\rho(s),
\end{align*} where in the last equality monotone convergence theorem is used.
Hence $\alpha$ belongs to the set $V_t$, and $F_\mu(\alpha)$ is
defined and equals zero.

To show the sufficiency, by the arguments shown above it suffices to
show that
\[\lim_{\epsilon\to0}\frac{\Re F_\mu(\alpha+i\epsilon)}{i\epsilon}=0\]
or, equivalently,
\[\lim_{\epsilon\to0}\frac{\Re[F_\mu(\alpha+i\epsilon)-F_\mu(\alpha)]}{\epsilon}=0.\leqno{(4.4)}\]
First, in view of (3.5) we obtain
\[\frac{F_\mu(\alpha+i\epsilon)-F_\mu(\alpha)}{\epsilon}=i\left(1+\int_\mathbb{R}\frac{s^2+1}{(s-\alpha-i\epsilon)
(s-\alpha)}\;d\rho(s)\right).\] Since
\[\int_\mathbb{R}\left|\frac{s^2+1}{(s-\alpha-i\epsilon)
(s-\alpha)}\right|d\rho(s)\leq\int_\mathbb{R}\frac{s^2+1}{(s-\alpha)^2}\;d\rho(s)=\frac{1}{t-1},\]
it follows that
\[\lim_{\epsilon\to0}\frac{F_\mu(\alpha+i\epsilon)-F_\mu(\alpha)}{\epsilon}=i\frac{t}{t-1}\]
by dominated convergence theorem which implies (4.4). The expression
of the Julia-Carath\'{e}odory derivative of $F_\mu$ is clear.
\end{pf} \qed

It was proved in [2] that if a point $x\in\mathbb{R}$ belongs to
$\omega_t(\mathbb{R})$, i.e., $f_t(x)=0$ then the
Julia-Carath\'{e}odory derivative $H_t'(x)\geq0$, where the
existence of
\[H_t'(x)=\lim_{\epsilon\downarrow0}\frac{H(x+i\epsilon)-H(x)}{i\epsilon}\]
is guaranteed by the existence of $H_t(x)\in\mathbb{R}$. Indeed, by
a similar proof in Lemma 4.2 and Lemma 3.3 at such a point $x$ the
Julia-Carath\'{e}odory derivative of $F_\mu$ is given by
\[F_\mu'(x)=1+\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s)\leq
t/(t-1),\leqno{(4.5)}\] from which we deduce that
\[H_t'(x)=1-(t-1)\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s)\geq0.\leqno{(4.6)}\]
Note that (4.6) shows that if $f_t(x)=0$ then $H_t'(x)>0$ if and
only if $x\in V_t^-$. In the following proposition, by means of
functions $f_t$ and $\psi_t$ we are able to obtain certain
regularity results which were proved in [2].

\begin{prop} Let $\alpha$ be a point in $\mathbb{R}$ and $t>1$. Then the following conditions are
equivalent:
\\ $(i)$ $f_t(\alpha)=0$ and $F_\mu(\alpha)=0$;
\\ $(ii)$ $F_{\mu^{\boxplus t}}(t\alpha)=0$;
\\ $(iii)$ $\mu(\{\alpha\})\geq1-t^{-1}$.
\end{prop}

\begin{pf} The equivalence of (i) and (ii) follows from the bijectivity of
$\psi_t$ and (4.1). Indeed, letting $\psi_t(\alpha')=t\alpha$ for
some $\alpha'\in\mathbb{R}$ implies that $F_{\mu^{\boxplus
t}}(t\alpha)=0$ if and only if $t\alpha'=\psi_t(\alpha')$ and
$f_t(\alpha')=0$ which translate to the conditions in (i). By Lemma
4.2 and (4.5), it is easy to see that (i) implies (iii). On the
other hand, by Lemma 4.2 we see that if
$\mu(\{\alpha\})=1-t_0^{-1}\geq1-t^{-1}$ then $F_\mu(\alpha)=0$ and
$f_{t_0}(\alpha)=0$. Since $t\leq t_0$, $f_t(\alpha)=0$ by Lemma
3.4, whence (iii) implies (i).  This completes the proof.
\end{pf} \qed

Next, we investigate those intervals on which $f_t$ vanishes.

\begin{prop} If $f_t$ vanishes on some interval $I$ then
the following statements are equivalent:
\\ $(i)$ $\mu(I)=0$;
\\ $(ii)$ $F_\mu(x)\neq0$ for any $x$ in $I$.
\\ If $(i)$ or $(ii)$ is satisfied then $\mu^{\boxplus
t}(\psi_t(I))=0$.
\end{prop}

\begin{pf}  The implication that (i) implies (ii) follows from Proposition 4.3.
Indeed, if $\Re F_\mu(x)=0$ for some $x\in I$ then $x$ is an atom of
$\mu$, and hence $\mu(I)>0$. Next, we prove that (ii) implies (i).
First we consider the case that $I$ is a compact set $[a,b]$. Since
$F_\mu$ is continuous and nonzero on $[a,b]$, $G_\mu(x)$ is a
continuous and real-valued function on $[a,b]$, and consequently
$\mu([a,b])=0$ by the Stieltjes inversion formula. Next, suppose
that $I$ is bounded but not closed. If $I=(a,b]$ then we have
$\mu([a+\epsilon,b])=0$ for any small $\epsilon>0$ by the result
established above, and therefore letting $\epsilon\to0$ gives
$\mu((a,b])=0$. Similarly, the assertion (i) holds if $I=[a,b)$ or
$I=(a,b)$. Finally, if $I$ is unbounded (closed or not closed) then
by the results shown above and the countable additivity of $\mu$, it
is easy to see that $\mu(I)=0$. This completes the proof that (ii)
implies (i).

By similar arguments and (4.2), it is easy to see that the last
statement follows from (i) or (ii).
\end{pf} \qed

\begin{cor} Assume that $I\subset\mathbb{R}$ is an interval on which $f_t$ vanishes and $\mu(I)>0$.
Then the interval $I$ contains one and only one atom $\alpha$ of
$\mu$ and this only atom has mass $\mu(\{\alpha\})\geq1-t^{-1}$.
Moreover, $\mu(I\setminus\{\alpha\})=0$ and $\mu^{\boxplus
t}(\psi_t(I))=t\mu(\{\alpha\})-(t-1)$.
\end{cor}

\begin{pf} First observe that $F_\mu$ has zeros in $I$. Since $F_\mu$
is strictly increasing on $I$ by Corollary 3.6, it only has one zero
$\alpha$, i.e., $\alpha$ is the only atom of $\mu$ in $I$ by Lemma
4.2. The atom $\alpha$ with the desired mass is a direct consequence
of Proposition 4.3. It is clear that the set $I\backslash\{\alpha\}$
must have $\mu$-measure zero. Since $I\backslash\{\alpha\}\subset
V_t\cup V_t^-$ and $\mu^{\boxplus
t}(\{t\alpha\})=t\mu(\{\alpha\})-(t-1)$, $\psi_t(I)$ has the desired
$\mu^{\boxplus t}$-measure.
\end{pf} \qed

\begin{thm} Let $I$ be any component in $\mathrm{supp}(\mu)$ and $t>1$. If $I$ is an
interval then it does not contain a closed interval on which $f_t$
vanishes and the interval $\psi_t(I)$ is contained in some component
of $\mathrm{supp}(\mu^{\boxplus t})$.
\end{thm}

\begin{pf} Let $\alpha_1,\cdots,\alpha_n$ be all the atoms
of $\mu$ in $I$ with $\mu(\{\alpha_j\})\geq1-t^{-1}$,
$j=1,\cdots,n$. Then it suffices to show that there does not exist
an open interval $J\subset I\backslash\{\alpha_1,\cdots,\alpha_n\}$
such that $f_t$ vanishes on $J$. If such an interval $J$ exists then
the fact $\mu(J)>0$ gives the existence of an atom $\alpha$ of $\mu$
in $J$ with mass $\mu(\{\alpha\})\geq1-t^{-1}$ by Corollary 4.5, a
contradiction. Hence the first desired result follows. This result
also reveals that $g>(t-1)^{-1}$ on the interval $I$ except at
points $x$ such that $f_t(x)=0$. Since $\mu^{\boxplus t}$ does not
contain the singular continuous part, the last assertion follows
from Theorem 3.8(i).
\end{pf} \qed

It is known that the measure $\mu^{\boxplus t}$ has fewer atoms when
$t$ increases. The next result explains where these disappearing
atoms go.

\begin{prop} Assume that $\alpha$ is an atom of $\mu$ and
$\mu(\{\alpha\})=1-t_0^{-1}$. Then $\mu^{\boxplus
t}(\{\psi_t(\alpha)\})>0$ for $1<t<t_0$ and
$\psi_t(\alpha)\in\mathrm{supp}((\mu^{\boxplus t})^{\mathrm{ac}})$
for $t\geq t_0$.
\end{prop}

\begin{pf} For $1<t<t_0$, we have $\mu(\{\alpha\})>1-t^{-1}$, which implies
that $\psi_t(\alpha)=t\alpha$ is an atom of $\mu^{\boxplus t}$ by
(4.1) and Proposition 4.3. Next note that there does not exist an
open interval $I$ containing $\alpha$ such that $f_{t_0}(x)=0$ for
all $x\in I$. Indeed, if such an interval $I$ exists then the facts
that $g(\alpha)=(t_0-1)^{-1}$ and $g$ is strictly convex on $I$ will
lead to a contradiction. Hence $\psi_{t_0}(\alpha)$ must be in the
closure of $\psi_{t_0}(V_{t_0}^+)$. Since $\alpha\in V_t^+$ for
$t>t_0$, the desired result follows from Theorem 3.8(i).
\end{pf} \qed

Theorem 4.6 and Proposition 4.7 give the inclusion
$\psi_t(\mathrm{supp}(\mu))\subset\mathrm{supp}(\mu^{\boxplus t})$
or, equivalently, the complement
$\mathbb{R}\backslash\mathrm{supp}(\mu^{\boxplus t})$ is contained
in $\psi_t(\mathbb{R}\backslash\mathrm{supp}(\mu))$ for all $t>1$.
Since $\mathbb{R}\backslash\mathrm{supp}(\mu)$ is a countable union
of open intervals, the preceding observation leads us to investigate
open intervals $I$ which have $\mu$-measure zero. First note that
the Cauchy transform $G_\mu$ extends analytically through the
interval $I$ and takes real values on $I$. Moreover, it can be
expressed as
\[G_\mu(x)=\int_\mathbb{R}\frac{1}{x-s}\;d\mu(s),\;\;\;x\in I,\]
and $G_\mu'(x)=-\int(s-x)^{-2}d\mu(s)<0$ which shows that $G_\mu$
has at most one zero in $I$. To analyze the values of $f_t$ on the
set $\mathbb{R}\backslash\mathrm{supp}(\mu)$, we need the following
lemma.

\begin{lem} Suppose that $I$ is an open interval in $\mathbb{R}$ with $\mu(I)=0$.
\\ $(i)$ If $G_\mu$ has a zero $x_0$ in $I$ then $x_0$ is an atom of
$\rho$ with mass
\[\rho(\{x_0\})=\frac{-1}{(x_0^2+1)G_\mu'(x_0)}\] and
$\rho(I\setminus\{x_0\})=0$.
\\ $(ii)$ At any point $x\in I$, the function $g$ can be expressed as
\[g(x)=\frac{-G_\mu'(x)}{G_\mu^2(x)}-1,\] where the
equality is interpreted as $+\infty$ on both sides at the unique
zero $x_0$, if it exists, of $G_\mu$ on $I$. In addition, if
$I=(a,b)$ is bounded then
\[\inf_{x\in I}g(x)\geq\frac{\rho(\mathbb{R})}{1+\max\{a^2,b^2\}}.\]
\end{lem}

\begin{pf} First observe that $F_\mu$ is continuous and takes real values on
$I\backslash\{x_0\}$, whence the weak$^*$-limit of the measures
$d\rho_\epsilon(s)$ in (2.2) as $\epsilon\downarrow0$ is zero which
gives $\rho(I\backslash\{x_0\})=0$. By Lemma 2.1, we have
\[\lim_{\epsilon\downarrow0}i\epsilon
F_\mu(x_0+i\epsilon)=-(x_0^2+1)\rho(\{x_0\}).\] On the other hand,
\[\lim_{\epsilon\downarrow0}
\frac{G_\mu(x_0+i\epsilon)}{i\epsilon}=\lim_{\epsilon\downarrow0}
\frac{G_\mu(x_0+i\epsilon)-G_\mu(x_0)}{i\epsilon}=G_\mu'(x_0),\]
which gives the desired mass for $\rho(\{x_0\})$ in (i). To verify
the statements in (ii), by (i) we may assume that $G_\mu$ is nonzero
on $I$. Then observe that for any $x\in I$ and $\epsilon>0$ we have
\begin{align*}
\Im
F_\mu(x+i\epsilon)&=\Im\frac{\overline{G_\mu(x+i\epsilon)}}{|G_\mu(x+i\epsilon)|^2}
=\frac{-\Im
G_\mu(x+i\epsilon)}{|G_\mu(x+i\epsilon)|^2}
=\epsilon\frac{\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2+\epsilon^2}}{\left|\int_\mathbb{R}\frac{d\mu(s)}
{x-s+i\epsilon}\right|^2}.
\end{align*} On the other hand, the Nevanlinna representation for
$F_\mu$ gives
\[\Im
F_\mu(x+i\epsilon)=\epsilon\left(1+\int_\mathbb{R}\frac{s^2+1}{(s-x)^2+\epsilon^2}\;d\rho(s)\right),\]
from which we obtain
\[\int_\mathbb{R}\frac{s^2+1}{(s-x)^2+\epsilon^2}\;d\rho(s)
=\frac{\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2+\epsilon^2}}{\left|\int_\mathbb{R}\frac{d\mu(s)}
{x-s+i\epsilon}\right|^2}-1.\leqno{(4.7)}\] Since $\mu(I)=0$ and
$G_\mu$ is nonzero on $I$, it follows that
\[\lim_{\epsilon\to0}\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2+\epsilon^2}=
\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2}<\infty\] and
\[\lim_{\epsilon\to0}\int_\mathbb{R}\frac{d\mu(s)}{(x-s)+i\epsilon}=
\int_\mathbb{R}\frac{d\mu(s)}{x-s}\neq0\] for all $x\in I$.
Therefore, by the equation (4.7) we have
\begin{align*}
g(x)&=\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s) \\
&=\lim_{\epsilon\to0}\int_\mathbb{R}\frac{s^2+1}{(s-x)^2+\epsilon^2}\;d\rho(s) \\
&=\lim_{\epsilon\to0}\left(\frac{\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2+\epsilon^2}}{\left|\int_\mathbb{R}\frac{d\mu(s)}
{x-s+i\epsilon}\right|^2}-1\right) \\
&=\frac{\int_\mathbb{R}\frac{d\mu(s)}{(x-s)^2}}{\left(\int_\mathbb{R}\frac{d\mu(s)}
{x-s}\right)^2}-1 \\
&=\frac{-G_\mu'(x)}{G_\mu^2(x)}-1.
\end{align*} Finally, for any $x$ in the bounded interval $(a,b)$ and
$s\in\mathbb{R}\backslash(a,b)$ we have
\[\frac{s^2+1}{(s-x)^2}\geq\frac{1}{1+x^2}\geq\frac{1}{1+\max\{a^2,b^2\}},\]
from which we deduce that
\begin{align*}
g(x)&=\int_{-\infty}^a\frac{s^2+1}{(s-x)^2}\;d\rho(s)+
\int_b^\infty\frac{s^2+1}{(s-x)^2}\;d\rho(s) \\
&\geq\frac{\rho(\mathbb{R})}{1+\max\{a^2,b^2\}},
\end{align*} as desired. This finishes the proof.
\end{pf} \qed

\begin{prop} If $I$ is a bounded component in
$\mathbb{R}\backslash\mathrm{supp}(\mu)$ then for each $t>1$ the
interval $\psi_t(I)$ contains at most two components of
$\mathbb{R}\backslash\mathrm{supp}(\mu^{\boxplus t})$.
\end{prop}

\begin{pf} If $G_\mu$ has no zero in $I$ then the strict convexity of $g$ on $I$
shows that there exists at most one subinterval $J$ of $I$ such that
$f_t$ vanishes on $J$ when $t$ varies from one to infinity. If
$G_\mu$ has a zero $x_0$ in $I=(a,b)$ then applying the preceding
argument to the intervals $(a,x_0)$ and $(x_0,b)$ gives that two
subintervals at most. Hence the desired conclusion follows from
Theorem 3.8(i).
\end{pf} \qed

For any $t>1$, denote by $n(t)$ the number of components in
$\mathrm{supp}(\mu^{\boxplus t})$.

\begin{thm} If $\mu$ is any Borel probability measure on
$\mathbb{R}$ then the following statements are equivalent:
\\ $(i)$ $n(t)=1$ for sufficiently large $t$;
\\ $(ii)$ $n(t)<\infty$ for some $t>1$;
\\ $(iii)$ the infimum $m$ of $g$ on the set of all bounded
components of $\mathbb{R}\setminus\mathrm{supp}(\mu)$ is nonzero.
\\ Moreover, if $m>0$ then $n(t)=1$ for $t>t_0$, where
\[t_0=\max\left\{1+\frac{1}{m},\frac{1}{1-\mu(\{\alpha\})}\right\}\] and $\alpha$ is one of the atoms of
$\mu$ with the largest mass.
\end{thm}

\begin{pf} It is clear that (i) implies (ii). Next, suppose that
$n(t_1)<\infty$ for some $t_1>1$ and $\mu(\{\alpha\})<1-t_1^{-1}$
for any atom $\alpha$ of $\mu$. Let $S_{t_1}$ be the set of all
components which are bounded intervals in
$\{x\in\mathbb{R}:f_{t_1}(x)=0\}$. Then $S_{t_1}$ is a finite set
and by Corollary 4.5 the set $S_{t_1}$ is contained in the union of
some components $I_1,\cdots,I_n$ of
$\mathbb{R}\backslash\mathrm{supp}(\mu)$. Note that each $I_k$ is
bounded. Indeed, if $\mu((M,\infty))=0$ for some finite number $M$
then $G_\mu>0$ on $(M,\infty)$, whence $\rho((M,\infty))=0$ and $g$
is strictly decreasing on $(M,\infty)$. This implies that
$g(x)\leq(t_1-1)^{-1}$ for sufficiently large $x$, and so $f_{t_1}$
vanishes on $(c,\infty)$ for some finite number $c$. Similarly, no
$I_k$ is of the form $(-\infty,M)$. Let $[a,b]$ be the smallest
closed interval containing these $I_k$'s. Then
\[\inf\{g(x):x\in(\mathbb{R}\backslash\mathrm{supp}(\mu))\cap[a,b]\}\geq\frac{\rho(\mathbb{R})}{1+\max\{a^2,b^2\}}\]
by Lemma 4.8(ii). If $I$ is the intersection of any bounded
component in $\mathbb{R}\backslash\mathrm{supp}(\mu)$ with
$(b,\infty)$ and $I\neq\emptyset$ then $g(x)\geq(t_1-1)^{-1}$ on
$I$. Indeed, if $g(x)<(t_1-1)^{-1}$ for some $x\in I$ then there
exists some interval $J\subset I$ such that $g<(t_1-1)^{-1}$ on $J$,
i.e., $f_{t_1}=0$ on $J$, which violates the definition of
$S_{t_1}$. Similarly, $g\geq(t_1-1)^{-1}$ on any bounded component
in $(\mathbb{R}\backslash\mathrm{supp}(\mu))\cap(-\infty,a)$, and
hence (ii) implies (iii). Finally, assume that the number $m$ in the
statement (iii) is nonzero and let $t>t_0$, where $t_0$ is defined
as in theorem. Then with the help of Theorem 4.6, we see that
$\{x\in\mathbb{R}:f_{t}(x)=0\}$ contains at most two components
which are intervals, in which case these two intervals must be
unbounded. In other words, the closure of $V_t^+$ is an interval,
whence $n(t)=1$ and (iii) implies (i).
\end{pf} \qed

In the proof of the preceding theorem, the set
$\{x\in\mathbb{R}:f_t(x)=0\}$ might contain two unbounded components
for any $t>1$. Actually, this happens when $\mu$ is compactly
supported which was proved in [6] and [14]. In the following, we
provide an analytic way to prove this fact.

\begin{cor} Let $t>1$. Then the following statements hold.
\\ $(i)$ The function $f_t$ vanishes on $(a,\infty)$ $($resp.
$(-\infty,a)$$)$ for some finite number $a$ if and only if
$\mu((M,\infty))=0$ $($resp. $(-\infty,M)$$)$ for some finite number
$M$.
\\ $(ii)$ The measure $\mu$ has compact support if and only if so
does $\mu^{\boxplus t}$.
\\ $(iii)$ If $\mu$ is compactly supported then $n(t)=1$
for large $t$.
\end{cor}

\begin{pf} If $f_t(x)=0$ for $x\in(a,\infty)$ then by Corollary 4.5 there exists a point
$M\geq a$ such that $\mu((M,\infty))=0$. Similarly, if $f_t$
vanishes on $(-\infty,a)$ then $\mu((-\infty,M))=0$ for some finite
number $M\leq a$. Conversely, if $\mu((b,\infty))=0$ for some $b$
then as shown in the preceding theorem we have $f_t(x)=0$ for
sufficiently large $x$. Similarly, if $\mu((-\infty,b))=0$ then
$f_t(x)=0$, $x\leq a$, for some finite number $a$, and so (i)
follows. The assertion (ii) follows from (i), while (iii) is a
direct consequence of Lemma 4.8(ii) and Theorem 4.10.
\end{pf} \qed

Now we are able to give a complete description of
$\mathrm{supp}(\mu^{\boxplus t})$ in terms of $\mathrm{supp}(\mu)$.
Consider a component $I=(a,b)$ in
$\mathbb{R}\backslash\mathrm{supp}(\mu)$. The case $b=\infty$ is
discussed in Corollary 4.11. If $b<\infty$ then one of the following
cases holds.
\\ $(i)$ The set $\mathbb{R}\backslash\mathrm{supp}(\mu)$ contains a component $(b,c)$ in
which case $b$ is an atom.
\\ $(ii)$ There exists a point $c>b$ such that $(b,c)$ is contained in a
component of $\mathrm{supp}(\mu)$.
\\ $(iii)$ There exists a sequence of components $(a_n,b_n)$ in $\mathbb{R}\backslash\mathrm{supp}(\mu)$
such that $b_{n+1}\leq a_n$ for all $n$ and $a_n\downarrow b$ as
$n\to\infty$.
\\ Cases (i) and (ii) are
analyzed in Proposition 4.3, Theorem 4.6, and Proposition 4.9.
Applying Corollary 4.5 to case (iii), it is easy to see that for
each $t>1$ there exists a point $c>b$ such that there is no interval
contained in $(b,c)$ on which $f_t$ vanishes, whence
$g\geq(t-1)^{-1}$ on $(b,c)$ and $\psi_t((b,c))$ is contained in
some component of $\mathrm{supp}(\mu^{\boxplus t})$. Similar
statements also hold for $a$.

It was proved in [1] that $\mu^{\boxplus t}((t\alpha,t\beta))>0$ if
$t\alpha,t\beta$ are atoms of $\mu^{\boxplus t}$. Using functions
$f_t$ and $\psi_t$, we can prove a somewhat stronger result.

\begin{prop} If $\mu$ has atoms $\alpha<\beta$ such that
$\mu(\{\alpha\}),\mu(\{\beta\})\geq1-t^{-1}$ then $\mu^{\boxplus
t}((t\alpha,t\beta))>0$.
\end{prop}

\begin{pf} First observe that $f_t(\alpha)=f_t(\beta)=F_\mu(\alpha)=F_\mu(\beta)=0$
and $\psi_t(\alpha)=t\alpha$, $\psi_t(\beta)=t\beta$. Without loss
of generality, we assume that $F_{\mu^{\boxplus t}}(x)\neq0$ for all
$x$ in $(t\alpha,t\beta)$ or, equivalently, $F_{\mu^{\boxplus
t}}(\psi_t(x))\neq0$ for all $x\in(\alpha,\beta)$. By Theorem
3.8(i), it suffices to show that $f_t$ is positive at some point in
$[\alpha,\beta]$. If $f_t$ vanishes on $[\alpha,\beta]$ then $F_\mu$
is continuous and strictly increasing on $[\alpha,\beta]$ by
Proposition 3.5 and Corollary 3.6, a contradiction. This completes
the proof.
\end{pf} \qed

By the work of Maassen [13], for any probability measure $\nu$ on
$\mathbb{R}$ there exists a unique probability measure $\mu$ on
$\mathbb{R}$ with mean zero and unit variance such that
\[F_\mu(z)=z-G_\nu(z),\;\;\;z\in\mathbb{C}^+.\leqno{(4.8)}\] If $\rho$ denotes the finite positive
Borel measure in the Nevanlinna representation of $F_\mu$ as before
then
\[(s^2+1)\;d\rho(s)=d\nu(s)\] by the Stieltjes inversion formula and (2.2). In
this case,
\[H_t(z)=z+(t-1)G_\nu(z),\;\;\;\;z\in\mathbb{C}^+,\] and hence $z\in\Omega_t$ if and only if
\[\int_\mathbb{R}\frac{d\nu(s)}{|z-s|^2}<\frac{1}{t-1}.\] Moreover, the function
$g$ defined in section 3 is given by the formula
\[g(x)=\int_\mathbb{R}\frac{d\nu(s)}{(s-x)^2},\;\;\;x\in\mathbb{R}.\leqno{(4.9)}\]

\begin{prop} There exists a Borel probability measure $\mu$ on $\mathbb{R}$ such that the number of
components in $\mathrm{supp}(\mu^{\boxplus t})$ is infinite for any
$t>1$.
\end{prop}

\begin{pf} Define the probability measure $\nu$ on $\mathbb{R}$ as
\[\nu=\sum_{n=1}^\infty2^{-n}\delta_{2^n},\] and let $\mu$ be the
unique measure satisfying the requirement (4.8). Then (4.9) shows
that the function $g$ is continuous and strictly convex on the
intervals $I_n=(2^n,2^{n+1})$, $n\geq1$, and
\[\lim_{x\downarrow2^n}g(x)=\lim_{x\uparrow2^{n+1}}g(x)=\infty,\leqno{(4.10)}\] from which
we see that $g$ reaches its minimum $m_n$ on each $I_n$. If $x_n$ is
the middle point of $I_n$ then for each $s\in\mathbb{R}\backslash
I_n$,
\[(s-x_n)^2\geq\left(\frac{2^{n+1}-2^n}{2}\right)^2=2^{2n-2},\] from
which we deduce that
\[m_n\leq g(x_n)=\int_{\mathbb{R}\backslash
I_n}\frac{d\nu(s)}{(s-x_n)^2}\leq2^{-2n+2}.\] This shows that for
any $t>1$, there are infinitely many intervals
$\{I_{n_k}\}_{k=1}^\infty$ each of which contains a closed
subinterval $J_{n_k}$ such that $f_t$ vanishes on $J_{n_k}$. Note
that (4.10) shows that these intervals $J_{n_k}$ are separated by
intervals on which $f_t$ is positive. Hence we conclude that the
closure of the set
\[V_t^+=\{x\in\mathbb{R}:g(x)>(t-1)^{-1}\}\] contains infinitely
many components, whence the desired result follows.
\end{pf} \qed

We conclude this section with an example. For any $0<\epsilon<1$,
consider the measure
\[\mu_\epsilon=\frac{\epsilon}{2}(\delta_{-1}+\delta_1)+(1-\epsilon)\delta_0.\]
Since
\[G_{\mu_\epsilon}(z)=\frac{z^2-1+\epsilon}{z(z^2-1)}\] has zeros at
$\pm\sqrt{1-\epsilon}$, by Lemma 4.8 and some simple manipulations
we obtain
\[\rho_\epsilon=\frac{\epsilon}{2(2-\epsilon)}\left(\delta_{-\sqrt{1-\epsilon}}+\delta_{\sqrt{1-\epsilon}}\right)\]
and
\[g_\epsilon(x)=\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho_\epsilon(s)=\frac{\epsilon x^2+
\epsilon(1-\epsilon)}{(x^2-1+\epsilon)^2}.\] Note that
$g_\epsilon(x)=\epsilon(1-\epsilon)^{-1}$ at points
$x=0,\pm\sqrt{3(1-\epsilon)}$ which implies that for
$1<t<\epsilon^{-1}$ the set $V_t^+=\{x:g_\epsilon(x)>(t-1)^{-1}\}$
consists of two components which are contained in intervals
$(-\sqrt{3(1-\epsilon)},0)$ and $(0,\sqrt{3(1-\epsilon)})$,
respectively. Moreover, when $t=\epsilon^{-1}$ the closure of the
union of these two components is the interval
$[-\sqrt{3(1-\epsilon)},\sqrt{3(1-\epsilon)}]$. Hence the support
$\mathrm{supp}((\mu_\epsilon^{\boxplus t})^{\mathrm{ac}})$ consists
of two components for $1<t<\epsilon^{-1}$ and these two components
merge into one piece when $t\geq\epsilon^{-1}$. On the other hand,
if $\epsilon<2/3$ then the measure $\mu_\epsilon^{\boxplus t}$ has
three atoms $0,\pm t$ for $1<t<2(2-\epsilon)^{-1}$, has one atom $0$
for $2(2-\epsilon)^{-1}\leq t<\epsilon^{-1}$, and has no atom for
$t\geq\epsilon^{-1}$. If $\epsilon=2/3$ then the measure
$\mu_\epsilon^{\boxplus t}$ has three atoms $0,\pm t$ for $1<t<3/2$
and has no atom for $t\geq3/2$. If $2/3<\epsilon$ then the measure
$\mu_\epsilon^{\boxplus t}$ has three atoms $0,\pm t$ for
$1<t<\epsilon^{-1}$, has two atoms $\pm t$ for $\epsilon^{-1}\leq
t<2(2-\epsilon)^{-1}$, and has no atom for
$t\geq2(2-\epsilon)^{-1}$. By Proposition 4.7 we see that
\[0\in\mathrm{supp}((\mu_\epsilon^{\boxplus
t})^{\mathrm{ac}})\;\;\;\;\;\mathrm{for}\;\;\;\;\;t\geq\epsilon^{-1}\]
and
\[\pm t\in\mathrm{supp}((\mu_\epsilon^{\boxplus
t})^{\mathrm{ac}})\;\;\;\;\;\mathrm{for}\;\;\;\;\;t\geq2(2-\epsilon)^{-1}.\]
From the above discussions, we conclude that the numbers of
components in $\mathrm{supp}(\mu_\epsilon^{\boxplus t})$ as follows.
\\ If $0<\epsilon<2/3$ then
\[n(t)=\left\{
\begin{array}{lll}
5       ,              &\hbox{for\;\;$t\in(1,2(2-\epsilon)^{-1})$} \\
3       ,              &\hbox{for\;\;$t\in[2(2-\epsilon)^{-1},\epsilon^{-1})$} \\
1 ,                    &\hbox{for\;\;$t\in[\epsilon^{-1},\infty)$}. \\
\end{array}
\right.\] If $\epsilon=2/3$ then
\[n(t)=\left\{
\begin{array}{ll}
5       ,              &\hbox{for\;\;$t\in(1,3/2)$} \\
1       ,              &\hbox{for\;\;$t\in[3/2,\infty)$}. \\
\end{array}
\right.\] Similarly, if $2/3<\epsilon<1$ then $n(t)$ begins with
$5$, reduces to $3$, and then becomes $1$ for large $t$.
\\ $\mathbf{Acknowledgments}$ The author wishes to thank his
advisor, Professor Hari Bercovici, for his generosity,
encouragement, and invaluable discussion during the course of the
investigation.