###### Abstract

We cast a nonzero-temperature analysis of the jamming transition [1] into the
framework of a scaling ansatz. We show that four distinct regimes for scaling
exponents of thermodynamic derivatives of the free energy such as pressure,
bulk and shear moduli, can be consolidated by introducing a universal scaling
function with two branches. Both the original analysis and the scaling theory
assume that the system always resides in a single basis in the energy
landscape. The two branches are separated by a line $T^{*}(\Delta\phi)$ in the
$T-\Delta\phi$ plane, where $\Delta\phi=\phi-\phi_{c}^{\Lambda}$ is the
deviation of the packing fraction from its critical, jamming value,
$\phi_{c}^{\Lambda}$, for that basin. The branch for $T<T^{*}(\Delta\phi)$
reduces at $T=0$ to an earlier scaling ansatz [2] that is restricted to $T=0$,
$\Delta\phi\geq 0$, while the branch for $T>T^{*}(\Delta\phi)$ reproduces
exponents observed for thermal hard spheres. In contrast to the usual scenario
for critical phenomena, the two branches are characterized by different
exponents. We suggest that this unusual feature can be resolved by the
existence of a dangerous irrelevant variable $u$, which can appear to modify
exponents if the leading $u=0$ term is sufficiently small in the regime
described by one of the two branches of the scaling function.

##  Chapter 0 Universal scaling function ansatz for finite temperature jamming

Like crystallization, jamming provides a paradigm for how particulate systems
become rigid. For ideal spheres that increasingly repel each other as they
overlap and do not interact if they do not overlap, the onset of rigidity with
increasing packing fraction at zero temperature $T$ is discontinuous for the
most stable possible state, the perfect FCC crystal, and critical for the
least stable possible state, the jammed state at the jamming transition at a
packing fraction $\phi_{c}$ [3, 4]. While crystallization is a straightforward
first-order transition, the extent to which the jamming transition can be
described within the framework of critical phenomena is still unclear.
Certainly many quantities exhibit power-law scaling, but are those scalings
consistent with universal scaling functions with associated scaling relations
among exponents?

In contrast to normal critical transitions, in which a diverging correlation
length corresponds to fluctuations that diverge at the critical point, the
jamming transition displays diverging correlations of fluctuations that
_vanish_ at the transition [5]. Moreover, the jamming transition point depends
on the history of the material – the protocol with which one increases the
density and decreases the temperature to reach jamming. It is encouraging,
however, that a universal scaling ansatz [2] successfully described the zero-
temperature elastic properties. Moreover, a jamming-like transition in a
diluted spring network [6] has been cast into a universal scaling form (see
Supplement [6]), and since turned into a general linear response theory [7,
8].

The situation at nonzero temperature $\tilde{T}$ is more challenging because
the systems are evolving with time. Work by De Giuli, et al. [1] has bypassed
this difficulty by assuming that at $\tilde{T}>0$ the system always resides in
a given basin in the energy landscape, whose jamming transition is at
$\phi_{c}^{\Lambda}$. That is, they describe a system near jamming using an
effective contact network with contacts between particles that collide. They
use the well-studied force distribution of jammed packings to add the forces
generated by temperature-induced collisions across the gaps, analogous to
methods developed to study stresses in experiments measuring colloidal
particle positions [9]. This description is valid on sufficiently short
timescales where this contact network does not change, analogous to methods
developed to study stresses in experiments measuring colloidal particle
positions [9]. While this assumption must break down at long time scales, it
is reasonable to ask whether the scaling ansatz can be extended to nonzero
temperatures within this approximation. We note that the zero-temperature
scaling ansatz makes a similar approximation, because the protocol and sample-
to-sample fluctuations of the critical jamming packing fraction $\phi_{c}$ are
swept under the rug by constructing a scaling ansatz in terms of
$\Delta\phi=\phi-\phi_{c}^{\Lambda}$, where $\phi_{c}^{\Lambda}$ can vary
among configurations $\\{\Lambda\\}$.

The theoretical arguments of De Giuli, et al. [1] apply to systems of $N$
soft, frictionless spheres in a volume $V$ at temperature $\tilde{T}$. The
packing fraction is $\phi=\frac{1}{V}\sum_{i}V_{i}$, where $V_{i}$ is the
$d$-dimensional volume of particle $i$. The spheres interact via pairwise
harmonic repulsions when they overlap, with a spring stiffness $k$. We
introduce the dimensionless free energy, pressure, shear stress, bulk modulus
and shear modulus using $k$ with appropriate factors of $D_{\text{avg}}$, the
average particle diameter, so that $F=\textrm{free
energy}/kD_{\text{avg}}^{2}$, $p=\textrm{pressure}*D_{\text{avg}}/k$, etc..
Note that we are departing from the usual custom of using temperature $T$ to
construct a dimensionless free energy; this is important because we
specifically interested in describing behavior at $T=0$ as well as $T>0$.
Instead, we introduce the dimensionless temperature
$T=\tilde{T}/kD_{\text{avg}}^{2}$. The $T=0$ scaling ansatz [2] is expressed
in terms of these dimensionless quantities for the same reason.

The marginal stability arguments and effective medium theory of De Giuli, et
al. [1] suggest that there are four different scaling regimes in the $T-\phi$
plane [1], Fig. 1(top), only one of which is described by the existing scaling
ansatz [2]. The predicted exponents in the different regimes are consistent
with mean field solutions for soft spheres [10] and the perceptron model [11],
although these later investigations showed only a crossover, not a sharp
transition – perhaps the smearing expected by the evolving landscape at finite
temperatures.

The existence of four distinct scaling regimes all terminating at the same
critical point could suggest rather exotic criticality. Here we show that the
results of De Giuli, et al. [1] are captured by a scaling function in a manner
similar to that of ordinary criticality, with two branches (Fig. 1(bottom)
separated by a continuous phase transition. This scaling function fully
describe all of the scaling behaviors seen near the jamming transition in the
full space described by the jamming phase diagram, namely temperature $T$,
shear strain $\epsilon$, and packing fraction $\Delta\phi=\phi-\phi_{c}$,
where $\phi_{c}$ is sample dependent. Unlike in ordinary criticality, the two
branches are characterized by different exponents. We suggest that the
difference in exponents can result from the existence of a dangerous
irrelevant variable $u$, which can appear to modify exponents if the leading
$u=0$ term is small in the regime described by one of the two branches of the
scaling function.

Figure 1:  Top: The phase diagram of De Giuli et al[1]. Different colours,
separated by lines, indicate four distinct scaling regimes, which are not
necessarily separated by sharp phase transitions. Bottom: The phase diagram
corresponding to the scaling ansatz Equation 1. The $-$ branch reproduces the
scaling exponents of three of De Giuli et al’s scaling regimes. The phase
boundary lies along a constant ratio
$T^{*}/\Delta\phi^{\zeta_{+}/\delta_{\phi,+}}=C^{*}$ in the notation of Eq. 1
(see footnote on page 1).

Each state is also characterized by average number of interacting neighbors
per particle (the contact number, $Z$), which satisfies $Z\geq
Z_{\textrm{min}}$ where $Z_{\textrm{min}}=2d-(2d-2)/N$ approaches the
isostatic value $Z_{\textrm{iso}}\equiv 2d$ at the jamming transition in the
thermodynamic limit [3, 12]; we define $\Delta Z=Z-Z_{\textrm{min}}$. The
scaling ansatz of Ref. 2 was written for the elastic energy with the excess
coordination beyond isostaticity, $\Delta Z$, as the control parameter
analogous to reduced temperature $t$ in the Ising model. We will follow suit,
constructing a scaling ansatz for the dimensionless free energy $F$ instead of
the dimensionless elastic energy $E$ in order to describe behavior at $T\geq
0$:

$F\left(\Delta Z,\Delta\phi,\varepsilon,N,T\right)=\Delta
Z^{\zeta_{\pm}}\mathcal{F}_{\pm}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,\pm}}},\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,\pm}}},N\Delta Z^{\psi_{\pm}},\frac{T}{\Delta
Z^{\zeta_{\pm}}}\right),$ (1)

where $\pm$ denote the two branches of the scaling function and the scaling
exponents for the two branches, $\zeta_{\pm}$, $\delta_{\phi,\pm}$,
$\delta_{\varepsilon,\pm}$ and $\psi_{\pm}$, are to be determined. Eq. 1 is
set up so that the leading singular part of the free energy in the
thermodynamic limit is proportional to $\Delta Z^{\zeta}_{\pm}$. Note that
$\zeta_{\pm}$ also appear in the temperature scaling; this follows simply from
dimensional analysis [2].

Note that the finite-size scaling is expressed in terms of the total number of
particles, $N$, instead of the system length, $L\sim N^{1/d}$. We use $N$
instead of $L$ because of arguments [12, 2, 13, 14] that the upper critical
dimension for jamming is $d_{u}=2$. Finite-size effects for $d\geq d_{u}$
should scale with $N$ with the exponent $\psi=d_{u}\nu$ where $\nu$ is the
mean-field correlation length exponent [15].

The excess packing fraction $\Delta\phi$ and shear strain $\varepsilon$ in Eq.
1 represent components of the same strain tensor (compression and shear
respectively) but are allowed to scale differently. Note that we assume
systems are prepared isotropically so that different shear directions are
statistically equivalent; this is why we represent them with a single
$\varepsilon$. For systems prepared by shearing or by applying other loads, a
more complicated formulation of the scaling ansatz is needed. For the $T=0$,
$\phi>\phi_{c}$ case, Ref. 16 points the way towards such a formulation for
shear-jammed systems.

DeGiuli, et al. [1] identified four regimes in the $T-\phi$ plane (Fig. 1top):
(1) The regime “S0” in Ref. 1 corresponds to low $T$ at $\phi>\phi_{c}$. The
exponents in this regime are those for $T=0$ soft spheres, described by the
scaling ansatz of Ref. 2 (see Table 1 of Ref. 1 for a list of exponents in
each regime). (2) A regime at somewhat higher temperatures at $\phi>\phi_{c}$,
called “SE” in Ref. 1, or “entropic soft spheres.” In this regime the pressure
is controlled by overlaps but the vibrational properties are strongly affected
by thermal collisions. (3) An anharmonic regime at packing fractions $\phi$
close to $\phi_{c}$ at nonzero $T$, called “AH” in Ref. 1. In this regime the
pressure and other properties are purely entropic and controlled only by $T$.
(4) A regime at $\phi<\phi_{c}$ and lower $T$, extending down to $T=0$, called
“HS” in Ref. 1 because it is described by the exponents near the jamming
transition for thermal hard spheres. We will use the predicted exponents of De
Giuli, et al. [1] in each of the four regimes to determine the exponents
introduced in the scaling ansatz, and will show that they are fully consistent
with the ansatz.

At $T=0$, $\phi>\phi_{c}$, Eq. 1 reduces to the scaling ansatz already
proposed in Ref. 2, with $\mathcal{F}_{+}=\mathcal{E}_{0}$ from Eq. 1 of Ref.
2. Therefore, the exponents for the positive branch of the scaling function
can be read off from Table 1 of Ref. 2: $\zeta_{+}=4$, $\delta_{\phi,+}=2$,
$\delta_{\varepsilon,+}=3/2$, $\psi_{+}=1$. These exponents yield exponents
for the pressure, bulk and shear moduli, etc. that are in agreement with those
listed in the S0 regime in Ref. 1; the exponents in this regime have been
established for a long time [3]. As in Ref. 2, $\psi_{+}=d_{u}\nu_{T}$, where
$\nu_{T}$ is the exponent for the transverse length scale, $\ell_{T}$. This
length scale characterizes the wavelength of transverse phonons at the boson
peak frequency, $\omega^{*}$ [17, 18]: $\ell_{T}\sim c_{T}/\omega^{*}$ where
the speed of transverse sound, $c_{T}$, scales as the square root of the shear
modulus, $G$. Note that the boson peak frequency corresponds to the Ioffe-
Regel crossover frequency where the phonon wavelength becomes comparable to
the mean free path between phonon scattering events [19]. Thus, $\ell_{T}$
also corresponds to the phonon mean free path at $\omega^{*}$. Equivalently,
$\ell_{T}$ corresponds to the length scale of correlations of polarization
vectors in vibrational modes at $\omega^{*}$ [20, 1] (their
$\ell_{S}(\omega^{*})$). It is straightforward to show that
$\ell_{T}\sim\sqrt{G}/\omega^{*}$ has the same scaling as
$\ell_{S}(\omega^{*})$ in Table 1 of Ref. 1 in all four regimes S0, SE, AH and
HS.

The next step is to calculate the exponents for the negative branch of the
scaling function. This can be done by comparing to exponents in the SE, AH or
HS regimes of Ref. 1. We will use the SE exponents, and then show that the
resulting exponents for the negative branch are fully consistent with the
exponents for the AH and HS regimes, as well, so that the negative branch
describes all three regimes.

We first consider the finite-size scaling behavior. The free energy must be
continuous at all $T>0$ for any $\phi$ for any system size $N$. This implies
that we must have equality of the finite-size scaling exponents for both
branches: $\psi_{-}=\psi_{+}=1$. This implies a correlation length exponent of
$\nu=\psi/d_{u}=1/2$, just as in the S0 regime. From Table 1 in Ref. 1, we see
that the length scale $\ell_{T}$ ($\ell_{S}(\omega^{*})$ in their notation)
scales as $(T/\Delta\phi)^{\frac{a-1}{4}}$ where
$a=(1-\theta_{e})/(3+\theta_{e})$ and $\theta_{e}\approx 0.42$ is the exponent
characterizing the low force tail of the force distribution in mean field
theory [21, 22] , as well as in low dimensions when bucklers are excluded
[23]. Note that $\Delta Z$ scales as $(T/p)^{2b}$ where $p\sim\Delta\phi$ in
this regime, where $b=(1+\theta_{e})/(3+\theta_{e})$. Thus,
$\ell_{T}\sim\Delta Z^{-1/2}$, so that $\nu_{T}=1/2=\nu$. Thus, the relevant
length scale controlling finite-size effects in both the S0 and SE regimes is
the transverse length scale $\ell_{T}$.

The remaining exponents can be obtained by differentiating Eq. 1 and comparing
with the scalings of Table 1 of Ref. 1. For example, by differentiating $F$
with respect to $\Delta\phi$, we find that the pressure satisfies $p\sim\Delta
Z^{\zeta_{-}-\delta_{\phi,-}}$. But from Table 1, we see that
$p\sim\Delta\phi\sim\Delta Z^{\delta_{\phi,-}}$, so
$\zeta_{-}=2\delta_{\phi,-}$. But we also have $\Delta
Z\sim\left(T/p\right)^{2b}\sim\Delta Z^{2b\zeta_{-}}/\Delta
Z^{2b\left(\zeta_{-}-\delta_{\phi,-}\right)}\sim\Delta Z^{2b\delta_{\phi,-}}$.
Thus, $\delta_{\phi,-}=1/(2b)=(3+\theta_{e})/(1+\theta_{e})$ and
$\zeta_{-}=2(3+\theta_{e})/(1+\theta_{e})$. Similarly,
$\delta_{\varepsilon,-}=(2+\theta_{e})/(1+\theta_{e})$.

It is straightforward to show that if one adopts the exponents from Table 1 of
Ref. 1 in the hard- sphere (HS) regime, one obtains the same values of the
exponents $\psi_{-}$, $\zeta_{-}$, $\delta_{\phi,-}$ and
$\delta_{\varepsilon,-}$. Moreover, by adopting these exponents and applying
the scaling ansatz in Eq. 1 at $\Delta Z=0$, $T>0$, where the behavior must be
analytic, one recovers the exponents for the anharmonic (AH) regime of Table 1
of Ref. 1. Thus, three of the four regimes, HS, AH and SE, of Fig. 1 in Ref. 1
(the blue and purple regions) all correspond to the negative branch of the
scaling function with identical exponents (for details, see Appendix 0.B). The
remaining regime (S0) is described by the positive branch of the scaling
function. In every regime, the length scale controlling finite-size effects is
the transverse length scale $\ell_{T}$ ($\ell_{S}(\omega^{*})$ in the notation
of Ref. 1).

The free energy must be continuous everywhere, so the two branches of the
scaling function must yield the same free energy along a curve
$T^{*}(\Delta\phi,\varepsilon)$ for every system size $N$. Here we restrict
ourselves to the case where $\varepsilon=0$ and $N\rightarrow\infty$ to
compare directly to the prediction of Ref. 1. Clearly we must have

$\Delta Z^{\zeta_{+}}\mathcal{F}^{\infty}_{+}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,+}}},\frac{T^{*}}{\Delta Z^{\zeta_{+}}}\right)=\Delta
Z^{\zeta_{-}}\mathcal{F}^{\infty}_{-}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,-}}},\frac{T^{*}}{\Delta Z^{\zeta_{-}}}\right).$ (2)

Eliminating $\Delta Z$ from the equation (see Appendix 0.A), we obtain

$T^{*}\mathcal{\tilde{F}}^{\infty}_{+}\left(\frac{\Delta\phi^{\zeta_{+}/\delta_{\phi,+}}}{T^{*}}\right)=T^{*}\mathcal{\tilde{F}}^{\infty}_{-}\left(\frac{\Delta\phi^{\zeta_{-}/\delta_{\phi,-}}}{T^{*}},\right).$
(3)

Remarkably, we note that
$\zeta_{+}/\delta_{\phi,+}=\zeta_{-}/\delta_{\phi,-}=2$. If these ratios were
not equal for the two branches of the scaling function, continuity of $F$
would not be possible. Eq. 3 implies that $T^{*}\sim\Delta\phi^{2}$. This
agrees with earlier predictions [24, 25, 2] but differs from the results of
Ref. 1, who found $T^{*}\sim\Delta\phi^{(2-a)/(1-a)}$; our arguments reproduce
their scalings without introducing this anomalous temperature scale.

The resulting phase diagram in the $T-\phi$-plane is as depicted in Fig. 1.
The exponents are summarized in Table . The red regime is described by the
positive branch of the scaling function while the blue regime is described by
the negative branch. The curve $T^{*}\sim\Delta\phi$ describes the boundary
between the two branches. The free energy is continuous at $T^{*}$ but its
derivatives are not necessarily continuous in the limit $N\rightarrow\infty$,
so $T^{*}(\Delta\phi)$ is a line of phase transitions. This can be checked by
calculating derivatives of the functions $\tilde{F}_{\pm}$ for different $N$
and evaluating them at $T^{*}$.

Values of critical exponents for the two branches ($+$ and $-$) of the scaling
ansatz of Eq. 1.  + - $\zeta$ 4 $2(3+\theta_{e})/(1+\theta_{e})$
$\delta_{\phi}$ 2 $(3+\theta_{e})/(1+\theta_{e})$ $\delta_{\varepsilon}$ 3/2
$(2+\theta_{e})/(1+\theta_{e})$ $\psi$ 1 1

For jammed soft spheres, Goodrich, et al. [2] argued for a scaling relation
connecting shear stress to the pressure:

$s^{2}N/p^{2}=\mathcal{S_{+}}(Np^{1/2}).$ (4)

This holds for the positive branch of the scaling function. A consequence of
the scaling ansatz is that for the negative branch there is anomalous scaling
of shear stress fluctuations so that

$s^{2}N/p^{x}=\mathcal{S_{-}}(Np^{2b}),$ (5)

at fixed $T/\Delta\phi^{2}$ or $T/p^{2}$, where
$x=\frac{7+\theta}{3+\theta}\approx 2.17$ and
$2b=(1+\theta_{e})/(3+\theta_{e})$. Because at fixed $p,T$ the correlation
length is finite, the scaling function $\mathcal{S}_{-}{\left(X\right)}$ must
approach a constant as $X\to\infty$.

As shown in Appendix 0.C, for small $\left|\Delta\phi\right|$ the scaling
ansatz predicts the anomalous scaling

$s^{2}\propto\frac{T^{x/2}}{N}\approx\frac{T^{1.09}}{N}.$ (6)

This is a slightly stronger scaling than equipartition for a fixed spring
network, consistent with the thermal stiffening of the effective contacts.

We note that the critical exponents in our scaling function differ on the two
sides of the transition. There is a longstanding tradition in statistical
mechanics of allowing for this possibility, but almost always the exponents
are the same – as follows from the linearized flow of the renormalization
group. (The exponents are ratios of the eigenvalues of the flow at a fixed
point, and thus are equal on both sides of the fixed point.) Different sets of
exponents for the two branches would seem to imply a highly unusual form of
criticality.

It is possible, however, to generate this apparent difference of exponents
through an alternative scenario, which, though somewhat contrived, has the
advantage of being consistent with the traditional renormalization group
picture. Consider the effects of an irrelevant variable $u$, which we
incorporate by adding another argument $X_{u}=u\Delta Z^{\zeta_{-}-\zeta_{+}}$
to the new scaling function $\mathcal{F}_{u}$. Here, $u$ is irrelevant because
$\zeta_{-}>\zeta_{+}$. Since $u$ is irrelevant, near the transition we may
expand the scaling function around $u=0$. We do so in an unorthodox way,
evaluating the derivative not at zero (as in a Taylor expansion) but at the
current value $X_{u}=u\Delta Z^{\zeta_{-}-\zeta_{+}}$:

$\displaystyle F\left(\Delta Z,\Delta\phi,\varepsilon,T\right)=$
$\displaystyle\Delta
Z^{\zeta_{+}}\mathcal{F}_{u}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,+}}},\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,+}}},\frac{T}{\Delta Z^{\zeta_{+}}},u\Delta
Z^{\zeta_{-}-\zeta_{+}}\right).$ (7) $\displaystyle\approx$
$\displaystyle\Delta
Z^{\zeta_{+}}\left(\mathcal{F}_{u}\left(X^{+}_{\phi},X^{+}_{\varepsilon},X^{+}_{T},0\right)+X_{u}\mathcal{F}_{u}^{[0,0,0,1]}\left(X^{+}_{\phi},X^{+}_{\varepsilon},X^{+}_{T},X_{u}\right)\right).$

where the invariant scaling combinations $X^{+/-}_{x}$ are those valid in the
plus/minus region. The first term
$\mathcal{F}_{u}(X^{+}_{\phi},X^{+}_{\varepsilon},X^{+}_{T},0)$ equals
$\mathcal{F}_{+}(X^{+}_{\phi},X^{+}_{\varepsilon},X^{+}_{T})$, the traditional
universal scaling function dominant as $\Delta Z\to 0$. But suppose that this
dominant term is zero in the minus region111 The plus region is $X_{\phi}>0$,
$X_{T}<C^{*}X_{\phi}^{\zeta/\delta_{\phi}}=C^{*}X_{\phi}^{2}$, see Fig. 1. (as
the scaling form for the Ising magnetization is zero for $T>T_{c}$). Then the
free energy in the minus region will be given by the second term, subdominant
by a factor $\Delta Z^{\zeta_{-}-\zeta_{+}}$, as desired. To get the scaling
form of Eq. 1 for the minus region, this derivative must take the form
(Appendix 0.D)

$\displaystyle
u\mathcal{F}_{u}^{[0,0,0,1]}\left(X^{+}_{\phi},X^{+}_{\varepsilon},X^{+}_{T},X_{u}\right)$
$\displaystyle=\mathcal{F}_{-}\left(X^{-}_{\phi},X^{-}_{\varepsilon},X^{-}_{T}\right)$
(8)
$\displaystyle=\mathcal{F}_{-}\left(u^{1/2}X^{+}_{\phi}/X_{u}^{1/2},u^{1/4}X^{+}_{\varepsilon}/X_{u}^{1/4},uX^{+}_{T}/X_{u}\right).$

for the particular value of $u$ for this system. This leads to

$\displaystyle F\left(\Delta Z,\Delta\phi,\varepsilon,T\right)=$
$\displaystyle\Delta Z^{\zeta_{+}}\mathcal{F}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,+}}},\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,+}}},\frac{T}{\Delta Z^{\zeta_{+}}}\right)$ (9)
$\displaystyle+\Delta
Z^{\zeta_{-}}\mathcal{F}_{-}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,-}}},\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,-}}},\frac{T}{\Delta Z^{\zeta_{-}}}\right)$

as desired.

One prediction of this picture is that it should be possible to find an
irrelevant variable that, when tuned to a large value, induces corrections to
the ordinary zero-$T$ soft sphere scaling that realize the hard-sphere
exponents. Some changes to the potential are irrelevant. Changes to the
preparation protocol of states are also irrelevant, and it would be
interesting to explore whether they could be responsible for these predicted
corrections to scaling.

In summary, a scaling theory can be constructed for the jamming transition as
a function of the variables encapsulated by the jamming phase diagram, namely
packing fraction, shear strain and temperature. We find that the known
behaviors are captured by two branches of a scaling function for the free
energy. The two branches join together at a temperature
$T^{*}\sim\Delta\phi^{2}$.

Recall that temperature $T$ is dimensionless:
$T=\tilde{T}/kD_{\mathrm{avg}}^{2}$ where $\tilde{T}$ is the dimensional
temperature. We can interpret the $+$-branch of the scaling function,
corresponding to $T<T^{*}$, as the regime in which the behavior is dominated
by the particle stiffness $k$ or the interaction energy. On the other hand,
the $-$-branch, where $T>T^{*}$, is dominated by temperature, $\tilde{T}$, or
thermal collisions.

We note that the arguments presented here can be generalized to a family of
potentials described by the exponent $\alpha$:

$U(r_{ij})=\frac{U_{0}}{\alpha}\left(1-\frac{r_{ij}}{R_{i}+R_{j}}\right)^{\alpha}\Theta\left(1-\frac{r_{ij}}{R_{i}+R_{j}}\right),$
(10)

where $r_{ij}$ is the distance between the centers of particles $i$ and $j$
and $R_{i}$ and $R_{j}$ are the radii of the particles. Here $\Theta(x)$ is
the Heaviside step function and $U_{0}$ is the interaction energy scale.
Following earlier work [19], it is useful to define an effective spring
constant, $k_{\textrm{eff}}=U_{0}(\alpha-1)/D_{\textrm{avg}}^{2}\Delta
Z^{2(\alpha-2)}$, where $D_{\textrm{avg}}$ is the average particle diameter.
If we use $k_{\textrm{eff}}$ to construct dimensionless free energy,
temperature, etc., as discussed before Equation 1, then the results are
generally applicable for any $\alpha>1$.

The limit $\alpha=1$, where $k_{\mathrm{eff}}=0$, is singular. There the de-
dimensionalization of scaling variables using $k_{\textrm{eff}}$ clearly
fails. In this case, scaling exponents are predicted to be zero and
simulations and mean-field calculations by Sclocchi, Urbani, and Franz [26,
27, 28] show that logarithmic corrections are dominant. It would be
interesting to see if the scaling ansatz applies to these logarithmic
corrections as well.

In our formulation of the scaling ansatz, we used $\Delta Z$ as a control
variable, following the approach of Goodrich, et al. [2]. This choice is
appealing because it treats $\Delta\phi$ and $\varepsilon$–compressive and
shear strain–in the same way. However, Goodrich, et al. [2] pointed out that
we could alternatively choose $\Delta\phi$ as the control variable. This would
lead to the scaling ansatz:

$F\left(\Delta Z,\Delta\phi,\varepsilon,N,T\right)=\Delta
Z^{\zeta_{\pm}}\mathcal{F}_{\pm}\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,\pm}}},\frac{\varepsilon}{|\Delta\phi|^{\delta_{\varepsilon,\pm}/\delta_{\phi,\pm}}},N|\Delta\phi|^{\psi_{\pm}/\delta_{\phi,\pm}},\frac{T}{|\Delta\phi|^{\zeta_{\pm}/\delta_{\phi,\pm}}}\right),$
(11)

Note, however, that this form requires using $|\Delta\phi|$ to scale
$\epsilon$, $N$ and $T$, which causes problems because it introduces non-
analyticity at $\Delta\phi=0$ even at $T>0$. It would seem, therefore, that
$\Delta Z$ is a better choice for the control variable. The discovery of a
correlation length that characterizes $\Delta Z$ fluctuations and diverges at
the jamming transition [5] suggests still another choice for the control
variable, namely the pressure, $p$, which is conjugate to $\Delta\phi$. So
far, the length scale characterizing contact number fluctuations is the only
diverging correlation length that has been identified for the jamming
transition, although there are multiple diverging length scales have been
identified that are not associated with correlation functions. It could be
argued that $\ell_{T}$ is a diverging correlation length because it
corresponds to the length scale for correlations in normal modes of vibrations
at the boson peak frequency $\omega^{*}$. The length scale
$\ell_{S}{\left(\omega^{*}\right)}$, however, involves the boson peak
frequency $\omega^{*}$ and does not correspond to the correlation length of
any of the variables that appear in the scaling ansatz. We note that the
$\Delta Z$ correlation length (which is associated with $\Delta Z$, one of the
variables in the scaling ansatz) does not appear to scale the same way as
$\ell_{S}(\omega^{*})$ (which controls finite-size scaling). This lack of
correspondence between a correlation length and the length scale
characterizing finite-size effects, while unusual, appears to be
characteristic of other systems with sharp global transitions but
configuration-dependent critical densities, as originally identified in the
depinning of charge-density waves [29, 30, 31].

While it is normal for a correlation length to diverge at a critical point, it
is decidedly abnormal for the fluctuations to vanish at long length
scales–usually they diverge. Contact number fluctuations must vanish at long
length scales because the average contact number at the transition must be
$Z=2d$, as dictated by isostaticity. As a result, contact number fluctuations
are hyperuniform [5, 13], implying that there can be no square-gradient
expansion–no Ginzburg-Landau theory–for jamming. It is not clear how to
construct a renormalization group for a system that is hyperuniform at long
length scales. The existence of other systems that exhibit hyperuniformity at
transitions, such as systems with transitions that separate absorbing states
from nonabsorbing ones [32], suggest that the challenge may not be restricted
to the jamming transition. It is interesting and nontrivial that despite being
highly unusual in exhibiting hyperuniformity, the jamming transition can
nevertheless be described by a scaling ansatz.

It should be noted, however, that the vanishing of $\Delta Z$ fluctuations may
be specific to systems prepared at fixed pressure (pressure control). A recent
study at fixed $\Delta\phi$ (phi-control) suggests that that $\Delta Z$
fluctuations might diverge instead of vanish and that even the finite-size
scaling exponent is different in that case, at least for $\Delta Z$ and energy
fluctuations [33]. It would be interesting to see whether a different scaling
ansatz can be constructed for that case, and how the ansatzes for pressure-
and $\phi$-control might be related.

Finally, we must again raise the caveat that the arguments of De Giuli, et al.
[1] and our scaling ansatz are predicated on the approximation that the system
is restricted to a single basin in the energy landscape. At $T>0$, we know
that this is not true–given enough time, the system will explore multiple
basins in the landscape, each with a minimum corresponding to a different
value of the jamming critical packing fraction, $\phi_{c}$. It is likely that
thermal fluctuations that drive the system over energy barriers will destroy
the line of phase transitions at the boundary separating the two branches of
the scaling function. In other words, glass physics may well smear the phase
transition line into a crossover. However, the jamming transition and the line
that emanates from it have such unusual properties in the canon of
criticality, and are so important to our understanding of disordered solids,
that it is important to understand them whether the line is singular or only
demarcates a crossover.

### Acknowledgements

We thank G. Biroli, S. Franz, and F. Zamponi for useful discussions, the
Simons Foundation for supporting this work through the ”Cracking the glass
problem” collaboration (#454945, SAR and AJL), the NSF (DMR-1719490 to JPS),
and the 2019 Beg Rohu Summer School, where this work was primarily done, as
well as the Center for Computational Biology at the Flatiron Institute, where
the manuscript was primarily written. This chapter is dedicated to the memory
of Michael E. Fisher, the thesis advisor of Andrea J. Liu and colleague of J.
P. Sethna. Liu, in particular, is grateful for having had the opportunity to
absorb scaling theory and criticality directly from Fisher. We hope that this
contribution helps to demonstrate the lasting value of looking at the world
through the lens of critical phenomena.

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##  Appendix 0.A Converting between $\Delta Z$ and $\Delta\phi$ in the scaling
ansatz

As in the zero-temperature case, [2], the scaling ansatz has the unusual
feature that the variables in it are not all independent: at fixed
$\Delta\phi,T,N$, and preparation protocol, the coordination $\Delta Z$ is
fixed by some equation of state.

In addition to the scaling ansatz, we assume a scaling equation of state

$\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,\pm}}}=f{\left(\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,\pm}}},N\Delta Z^{\psi_{\pm}},\frac{T}{\Delta
Z^{\zeta_{\pm}}}\right)}.$ (12)

If this is assumed, then it is possible to eliminate $\Delta Z$ from the
scaling ansatz, in order to obtain Equation 3 in the main text, or make
comparisons to De Giuli et al’s scaling predictions (which are expressed in
terms of $\Delta\phi$ rather than $\Delta Z$).

This equation of state can be inverted to yield

$\Delta
Z^{\delta_{\phi,\pm}}=\left|\Delta\phi\right|g_{\pm}{\left(\frac{\varepsilon}{\left|\Delta\phi\right|^{\delta_{\varepsilon,\pm}/\delta_{\phi,\pm}}},N\left|\Delta\phi\right|^{\psi_{\pm}/\delta_{\phi,\pm}},\frac{T}{\left|\Delta\phi\right|^{\zeta_{\pm}/\delta_{\phi,\pm}}}\right)}.$
(13)

Here the branches $f_{\pm}$ correspond to the sign of $\Delta\phi$ rather than
the regions $-,+$ of the phase diagram; if we wished to completely eliminate
$\Delta Z$ from the scaling ansatz in all regions simultaneously it may
introduce a nonanlyticity at $\Delta\phi=0$. However, this is not a problem if
we only wish to eliminate $\Delta Z$ from an equation with the sign of
$\Delta\phi$ fixed.

Subject to this restriction, this allows us to eliminate $\Delta Z$ from any
equation of the original scaling variables:

$\displaystyle h{\left(\frac{\Delta\phi}{\Delta
Z^{\delta_{\phi,\pm}}},\frac{\varepsilon}{\Delta
Z^{\delta_{\varepsilon,\pm}}},N\Delta Z^{\psi_{\pm}},\frac{T}{\Delta
Z^{\zeta_{\pm}}}\right)}$
$\displaystyle=\tilde{h}_{\pm}{\left(\frac{\varepsilon}{\left|\Delta\phi\right|^{\delta_{\varepsilon,\pm}/\delta_{\phi,\pm}}},N\left|\Delta\phi\right|^{\psi_{\pm}/\delta_{\phi,\pm}},\frac{T}{\left|\Delta\phi\right|^{\zeta_{\pm}/\delta_{\phi,\pm}}}\right)}$
(14)

Restricting to $\Delta\phi>0$ we may thus eliminate $\Delta Z$ to obtain
equation 3 in the main text, and as necessary in the following section
Appendix 0.B to verify that the scaling ansatz reproduces the scaling regimes
of De Giuli et al.[1]

##  Appendix 0.B Deriving the De Giuli et al. exponents from the scaling
ansatz

Here we show in more detail how to obtain the various scalings described by De
Giuli et al. [1] from the scaling ansatz.

We begin with the scaling behaviour of the pressure. Start with the ansatz of
Equation (1), and take $N\to\infty$ and $\epsilon=0$. Taking a derivative with
respect to the first argument yields the scaling ansatz for the pressure,

$\displaystyle P$ $\displaystyle=\Delta
Z^{\zeta_{\pm}-\delta_{\phi,\pm}}\mathcal{P}_{\pm}{\left(\frac{\Delta\phi}{\Delta
Z^{\delta}_{\phi,\pm}},\frac{T}{\Delta Z^{\zeta_{\pm}}}\right)}.$ (15)

Eliminating $\Delta Z$ as above gives

$\displaystyle P$
$\displaystyle=\left|\Delta\phi\right|^{\zeta_{\pm}/\delta_{\phi,\pm}-1}\mathcal{P}_{\pm}{\left(\frac{T}{\left|\Delta\phi\right|^{\zeta_{\pm}/\delta_{\phi,\pm}}}\right)}.$
(16)

First, we study the “anharmonic” limit $T\gg\Delta\phi^{2}$, in which De Giuli
et al. predict $P\propto\sqrt{T}$. We set $\Delta\phi\to 0$ in the scaling
function at fixed $T$, and require that the behaviour of the scaling function
is not singular in $\Delta\phi$. This requires that the asymptotic behaviour
of the scaling function has the right power law to balance all powers of
$\Delta\phi$, i.e.

$\displaystyle P$
$\displaystyle\propto\left|\Delta\phi\right|^{\zeta_{-}/\delta_{\phi,-}-1}\left(\frac{T}{\left|\Delta\phi\right|^{\zeta_{-}/\delta_{\phi,-}}}\right)^{\frac{\zeta_{-}-\delta_{\phi,-}}{\zeta_{-}}}$
(17) $\displaystyle=T^{\frac{\zeta_{-}-\delta_{\phi,-}}{\zeta_{-}}}.$ (18)

Thus, consistency with De Giuli et al requires that

$\zeta_{-}=2\delta_{\phi,-}.$ (19)

Note that $\zeta_{+}$ and $\delta_{\phi,+}$ are already known to satisfy the
same requirement.

In the hard-sphere limit ($\Delta\phi<0,T\to 0$), dimensional analysis
requires that $P\propto T$. Thus we extract the asymptotic equation of state

$\displaystyle P$
$\displaystyle\propto\left|\Delta\phi\right|^{\zeta_{-}/\delta_{\phi,-}-1}\left(\frac{T}{\left|\Delta\phi\right|^{\zeta_{-}/\delta_{\phi,-}}}\right)\propto\frac{T}{\left|\Delta\phi\right|}$
(20)

Thus, the hard-sphere pressure scaling is automatically satisfied.

The pressure scaling in the regimes that De Giuli et al call ‘soft-zero-T“ and
“soft-entropic” are obtained by taking $T\to 0$ for positive $\Delta\phi$;
this recovers the previous zero-temperature scaling ansatz, giving
$P\propto\Delta\phi$ in agreement with their results.

Now we must verify the scaling of the shear modulus $G$. At zero strain, it
obeys the scaling form

$\displaystyle G$ $\displaystyle=\Delta
Z^{\zeta_{\pm}-2\delta_{\epsilon,\pm}}\mathcal{G}_{\pm}{\left(\frac{\Delta\phi}{\Delta
Z^{\delta}_{\phi,\pm}},\frac{T}{\Delta Z^{\zeta_{\pm}}}\right)}.$ (21)

The anharmonic and hard-sphere limits are treated identically to the case of
the pressure, demanding a finite value at zero $\Delta\phi$ in the anharmonic
case while requiring $G\propto T$ in the hard-sphere case. Matching the
anharmonic limit for $G$ with De Giuli et al turns out to require

$\frac{\delta_{\epsilon,-}}{\zeta_{-}}=\frac{1}{4}\frac{4+2\theta_{e}}{3+\theta_{e}},$
(22)

while the hard-sphere limit requires

$\frac{\delta_{\epsilon,-}}{\delta_{\phi,-}}=\frac{1}{2}\frac{4+2\theta_{e}}{3+\theta_{e}}.$
(23)

Note that these equations are consistent with equation 19, and are both
satisfied by the exponents in Table .

The “soft entropic” regime of De Giuli et al. is less obvious, but is still
consistent with the scaling ansatz in the $-$ branch. After eliminating
$\Delta Z$ we have

$\displaystyle G$
$\displaystyle=\left|\Delta\phi\right|^{2-2\delta_{\epsilon,-}/\delta_{\phi,-}}\mathcal{G}_{-}{\left(\frac{T}{\Delta\phi^{2}}\right)}$
(24)
$\displaystyle=T^{1-2\delta_{\epsilon,-}/\delta_{\phi,-}}\left|\Delta\phi\right|^{2\delta_{\epsilon,-}/\delta_{\phi,-}}g_{-}{\left(\frac{T}{\Delta\phi^{2}}\right)}.$
(25)

If the function $g_{-}$ approaches a finite limit, this matches the “soft
entropic” scaling as long as

$\frac{\delta_{\epsilon,-}}{\delta_{\phi,-}}=\frac{1}{2}\frac{4+2\theta_{e}}{3+\theta_{e}},$
(26)

which shows consistency with the exponents in the hard-sphere and anharmonic
regimes.

To see the “soft entropic” scaling in our picture we must move along the phase
boundary, $T=C^{*}\Delta\phi^{2}$, since our picture does not have the
anomalous temperature scale $T^{*}$ of De Giuli et al; $g_{-}$ is constant
along this trajectory, as required.

##  Appendix 0.C Shear stress fluctuations at zero $\Delta\phi$

At finite $T$, in the limit of small $\Delta\phi$ the scaling form for the
shear stress fluctuations $s^{2}$ is

$\displaystyle s^{2}$
$\displaystyle=\left|\Delta\phi\right|^{2\left(\zeta_{-}-\delta_{\epsilon,-}\right)/\delta_{\phi,-}}\mathcal{S}_{-}{\left(N\left|\Delta\phi\right|^{\psi_{-}/\delta_{\phi,-}},\frac{T}{\left|\Delta\phi\right|^{2}}\right)}.$
(27)

A finite correlation length requires $1/\sqrt{N}$ fluctuations of shear-stress
for large $N$:

$\displaystyle s^{2}$
$\displaystyle=\frac{1}{N}\left|\Delta\phi\right|^{\left(2\zeta_{-}-2\delta_{\epsilon,-}-\psi_{-}\right)/\delta_{\phi,-}}\mathcal{\tilde{S}}_{-}{\left(\frac{T}{\left|\Delta\phi\right|^{2}}\right)}.$
(28)

Finally, at finite $T$ the fluctuations cannot diverge or vanish as
$\Delta\phi\to 0$. Imposing the required power law to cancel all factors of
$\Delta\phi$ yields

$\displaystyle s^{2}N$ $\displaystyle\propto
T^{\left(2\zeta_{-}-2\delta_{\epsilon,-}-\psi_{-}\right)/2\delta_{\phi,-}}=T^{\frac{7+\theta_{e}}{6+2\theta_{e}}}$
(29)

##  Appendix 0.D The necessary form of the small-$u$ scaling function

Note that

$\displaystyle\zeta_{-}-\zeta_{+}$
$\displaystyle=\frac{2-2\theta_{e}}{1+\theta_{e}}$ (30)
$\displaystyle\delta_{\phi,-}-\delta_{\phi,+}$
$\displaystyle=\frac{1-\theta_{e}}{1+\theta_{e}}=\frac{1}{2}\left(\zeta_{-}-\zeta_{+}\right)$
(31) $\displaystyle\delta_{\varepsilon,-}-\delta_{\varepsilon,+}$
$\displaystyle=\frac{1-\theta_{e}}{2+2\theta_{e}}=\frac{1}{4}\left(\zeta_{-}-\zeta_{+}\right).$
(32)

Thus, the $-$ exponents are recovered if the correction to scaling is, for
small $u$, a function of $X_{T}/X_{u}$, $X_{\phi}/X_{u}^{1/2}$, and
$X_{\phi}/X_{u}^{1/4}$.


# Virtually expanding dynamics

Masato TSUJII  Department of Mathematics, Kyushu University, Fukuoka, 819-0395
<EMAIL_ADDRESS>
###### Abstract.

We introduce a class of discrete dynamical systems that we call _virtually
expanding_. This is an open subset of self-covering maps on a closed manifold
which contains all expanding maps and some partially hyperbolic volume-
expanding maps. We show that the Perron-Frobenius operator is quasi-compact on
a Sobolev space of positive order for such class of dynamical systems.

###### Key words and phrases:

expanding map, Perron-Frobenius operator, quasi-compactness

This work was supported by JSPS Grant-in-Aid for Scientific Research (B)
15H03627. The author thanks Hesam Rajabzadeh (IPM, Iran), Roberto Castorrini
(GSSI, Italy) and the anonymous referee for pointing out a few errors in the
previous versions of this paper.

##  1\. Introduction

Expanding maps are a class of discrete dynamical systems that exhibit
typically chaotic behavior of orbits and studied extensively as a standard
model of chaotic dynamical systems. Among others, an important fact about
expanding dynamical system is that the associated Perron-Frobenius operators
are quasi-compact on the space of functions with some smoothness. (See [9,
2].) Based on the recent understanding on expanding and hyperbolic dynamical
systems in terms of Fourier analysis [3, 6, 4, 5], this fact can be understood
as a consequence of the property of the Perron-Frobenius operators that they
reduce high-frequency components of functions by transferring them to low-
frequency ones. But actually in order to apply this argument and obtain the
conclusion of quasi-compactness, we only need to have that property “in
average with respect to preimages” and “direction-wise”. Indeed we find that
the argument about quasi-compactness of the Perron-Frobenius operators for
expanding maps can be extended to a much more general class of dynamical
systems, which we would like to call _virtually expanding maps_. Below we
present the definition and the main property of virtually expanding maps. In
the last section, we discuss briefly about a further generalization of the
class of virtually expanding maps.

##  2\. Main result

Let $M$ be a closed connected $C^{\infty}$ manifold and let $f:M\to M$ be a
$C^{\infty}$ covering map. We suppose that $M$ is equipped with some Riemann
metric and that the Jacobian determinant $Jf(p)$ does not vanish for any $p\in
M$. Let $T^{*}M$ be the cotangent bundle of $M$. The push-forward action of
$f$ on $T^{*}M$ is written as

$f^{{\dagger}}:T^{*}M\to T^{*}M,\quad(q,\xi)\mapsto
f^{{\dagger}}(q,\xi)=(f(q),((D_{q}f)^{*})^{-1}(\xi)).$

For $\mu\geq 0$ and each non-zero covector $(q,\xi)\in T^{*}M$, we set

(1)
$b^{\mu}((q,\xi);f)=\sum_{f^{{\dagger}}(p,\eta)=(q,\xi)}\frac{(|\xi|/|\eta|)^{\mu}}{|Jf(p)|}$

where the sum on the right-hand side denotes that over the points $(p,\eta)\in
T^{*}M$ satisfying $f^{{\dagger}}(p,\eta)=(q,\xi)$ and we write $|\cdot|$ for
the induced norm on the cotangent bundle from the Riemann metric. Then we
define

(2) $B^{\mu}(f)=\sup\\{\,b^{\mu}((q,\xi);f)\mid 0\neq(q,\xi)\in T^{*}M\\}.$

Note that, for another $C^{\infty}$ covering map $g:M\to M$ with non-vanishing
Jacobian determinant, we have

$B^{\mu}(f\circ g)\leq B^{\mu}(f)\cdot B^{\mu}(g).$

In particular $B^{\mu}(f^{n})$ is sub-multiplicative with respect to $n$:

(3) $B^{\mu}(f^{n+m})\leq B^{\mu}(f^{n})\cdot
B^{\mu}(f^{m})\quad\quad\mbox{for }n,m\geq 1.$

######  Definition.

For $\mu>0$, a $C^{\infty}$ covering map $f:M\to M$ is $\mu$-virtually
expanding if

$\inf_{n\geq
1}(B^{2\mu}(f^{n}))^{1/n}=\lim_{n\to\infty}(B^{2\mu}(f^{n}))^{1/n}<1,$

where the equality holds by Fekete subadditive lemma. We write
$\mathcal{VE}^{\mu}(M)$ for the set of $\mu$-virtually expanding maps on $M$.
We call an element of $\mathcal{VE}(M)=\cup_{\mu>0}\mathcal{VE}^{\mu}(M)$
virtually expanding. $\mathcal{VE}^{\mu}(M)$ for $\mu>0$ as well as
$\mathcal{VE}(M)$ are $C^{1}$ open subsets of $C^{\infty}(M,M)$.

######  Remark 1.

The definition of virtually expanding property above does not depend on the
choice of the Riemann metric on $M$, and we may even consider any continuous
Finsler metric in the place of the Riemann metric.

An expanding map $f:M\to M$ is virtually expanding. Indeed by a standard
distortion estimate we have that

$\lim_{n\to\infty}(B^{0}(f^{n}))^{1/n}\leq 1$

and hence, for any $\mu>0$,

$\lim_{n\to\infty}(B^{2\mu}(f^{n}))^{1/n}\leq\lambda^{-2\mu}<1$

where $\lambda>1$ is the minimum expansion rate of $f$.

But the converse is not true except for one dimensional cases because, in the
definition (1), we consider _each_ (direction of) non-zero covector
$(q,\xi)\in T^{*}M$ and take the _averaged value_ of the expansion rates over
the backward images of $(q,\xi)$ with weight $1/|Jf|$. There are in fact many
virtually expanding maps that are not expanding. Let us say that a
$C^{\infty}$ map $f:M\to M$ is volume-expanding if

$\lim_{n\to\infty}\left(\inf_{p\in M}|Jf^{n}(p)|\right)^{1/n}>1.$

A virtually expanding map is necessarily volume-expanding. The example below
shows that the set $\mathcal{VE}(\mathbb{T}^{2})$ of virtually expanding maps
on the $2$-dimensional torus $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$
contains a non-empty $C^{1}$ open subset of volume-expanding maps, which are
not expanding.

######  Example.

Consider a map $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ on the torus
$\mathbb{T}^{2}$ defined by

(4) $f(x,y)=(mx,y+m\cos 2\pi x)\quad\mod\mathbb{Z}^{2}$

with $m$ an integer. For this map and its small $C^{1}$ perturbations, we can
check the condition $B^{2\mu}(f)<1$ at least if we assume $m$ to be
sufficiently large depending on $\mu>0$. A proof will be provided in the
appendix.

######  Remark 2.

The argument in the proof of the claim $B^{\mu}(f)<1$ is closely related to
that for partially hyperbolic endomorphisms in [1, 8, 11] using some
transversality condition on the images of unstable cones. It will be possible
to prove that a generic $C^{\infty}$ partially hyperbolic volume-expanding
maps are virtually expanding. Further it should not be too optimistic to
expect that the same is true without the assumption of partial hyperbolicity.

The Perron-Frobenius operator $P$ expresses the push-forward action of $f$ on
densities on $M$ and is defined explicitly by

$Pu(q)=\sum_{p\in f^{-1}(q)}\frac{u(p)}{|Jf(p)|}.$

It is not difficult to see that $P$ gives a bounded operator on the Sobolev
space $H^{\mu}(M)$ of any order $\mu\in\mathbb{R}$. (See [10, §1.3, §1.5] for
the definition of $H^{\mu}(M)$.) Our main theorem is the following.

######  Theorem.

If $f$ belongs to $\mathcal{VE}^{\mu}(M)$ for some $\mu>0$, the Perron-
Frobenius operator $P$ is quasi-compact on the Sobolev space $H^{\mu}(M)$ of
order $\mu>0$, that is, the essential spectral radius of $P:H^{\mu}(M)\to
H^{\mu}(M)$ is smaller than its spectral radius $1$.

######  Remark 3.

Once we show that the essential spectral radius of $P:H^{\mu}(M)\to
H^{\mu}(M)$ is smaller than $1$, it is easy to see that the spectral radius
equals $1$ as the spectral radius is just the maximum of the moduli of the
peripheral discrete eigenvalues.

######  Remark 4.

In the conclusion of Theorem, there may be several eigenvalues of modulus $1$
and the multiplicity of the eigenvalue $1$ may be greater than $1$. It will be
possible to observe various (complicated but relatively tame) bifurcations of
those eigenvalues when we perturb the maps in the class
$\mathcal{VE}^{\mu}(M)$.

From the conclusion of the theorem, a few important dynamical properties of
the map $f$ follows immediately. For instance we have

######  Corollary.

If $f$ belongs to $\mathcal{VE}^{\mu}(M)$ for some $\mu>0$, there are finitely
many absolutely continuous ergodic measures whose densities with respect to
the smooth measure are in $H^{\mu}(M)$ and almost every point on $M$ with
respect to the smooth measure is generic for either of them.

###### Proof.

We only prove the latter claim. (The proof of the former claim should be
rather standard if we note that $P$ preserves positive and negative part of
functions.) We write $m$ for the Riemann volume on $M$. Let
$\mu_{i}=\varphi_{i}dm$, $1\leq i\leq\ell$, be the absolutely continuous
ergodic measures whose densities $\varphi_{i}$ belong to $H^{\mu}(M)$. Suppose
that the measurable subset

$X=\\{x\in M\mid\text{$x$ is not generic for either of $\mu_{i}$ for $1\leq
i\leq\ell$}\\}$

has positive measure with respect to $m$. We approximate the characteristic
function of $X$ by a $C^{\infty}$ positive-valued function $\chi$ in $L^{2}$
sense. From the quasi-compactness of $P$, the sequence of functions

$\chi_{m}:=\frac{1}{m}\sum_{k=0}^{m-1}P^{k}\chi$

converges, as $m\to\infty$, to a function in the eigenspace $E$ for the
eigenvalue $1$ in $H^{\mu}(M)$ and hence in $L^{2}(M)$. However, since
$f(X)=X$ and since $P$ preserves the $L^{1}$ norms of positive-valued
functions, the $L^{2}$-norm of the component of $\chi_{m}$ that is
$L^{2}$-orthogonal to the subspace $E$ is bounded from below by a positive
constant, provided that $\chi$ is sufficiently close to the characteristic
function of $X$ in $L^{2}$ sense. This is a contradiction and therefore $X$ is
a null subset. ∎

##  3\. Proof of Theorem

We will use a few basic theorems on pseudo-differential operators and
notations given in [10, Ch.1 and 2]. (In particular, see the remark at the end
of [10, §2.5] for the concept of a pseudo-differential operator on a manifold
$M$.) First we take a small open connected subset $U\subset M$ so that its
backward image $f^{-1}(U)$ consists of connected open subsets $V_{1}$,
$V_{2}$, …, $V_{\deg f}$, on each of which the map $f$ is a diffeomorphism
onto $U$. Here $\deg f$ denotes the mapping degree of $f$. We write
$f_{j}:V_{j}\to U$ for the restriction of $f$ to $V_{j}$. For convenience, we
suppose that each of $U$ and $V_{j}$, $1\leq j\leq\deg f$, is contained in a
single coordinate chart.

Suppose that a function $u\in C^{\infty}(M)$ is supported on
$f^{-1}(U)=\cup_{j=1}^{\deg f}V_{j}$ and we write $u_{j}$ for the restriction
of $u$ to $V_{j}$. Then we may write

$Pu=\sum_{j=1}^{\deg f}P_{j}u_{j},\quad P_{j}u_{j}=\frac{u_{j}}{Jf}\circ
f_{j}^{-1}$

where $P_{j}$ denotes the restriction of $P$ to the functions supported on
$V_{j}$. Note that $P_{j}$ viewed in the local coordinate chart is a
multiplication by a smooth function composed with an operation of change of
variables.

For $\mu\geq 0$, we define a function

$\mathcal{W}^{\mu}:T^{*}M\to\mathbb{R},\quad\mathcal{W}^{\mu}(x,\xi)=\langle|\xi|\rangle^{\mu}$

with setting $\langle s\rangle:=\sqrt{1+s^{2}}$. This function belong to a
standard symbol class $S^{\mu}_{1,0}$ (see [10, §2.1]) and we may define the
norm $\|\cdot\|_{H^{\mu}}$ on the Sobolev space $H^{\mu}(M)$ by

$\|u\|_{H^{\mu}}^{2}=\|\mathrm{Op}(\mathcal{W}^{\mu})u\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}$

where (and henceforth) we write111In the reference [10], pseudo-differential
operators are written as $F(x,D)$. Here we use a different notation for
convenience. $\mathrm{Op}(F)$ for the pseudo-differential operator with symbol
$F$. (See [10, §2.1, §2.5].) For convenience we will understand
$H^{\mu}(M)=L^{2}(M)$ when $\mu<0$.

We write $Df^{*}_{j}:T^{*}U\to T^{*}V_{j}$ for the pull-back action of $f_{j}$
on the cotangent bundle restricted $T^{*}U$. This is a bijection. Let us
consider the pseudo-differential operators

$A_{j}=\mathrm{Op}\left(\frac{\mathcal{W}^{\mu}}{\sqrt{Jf\circ
f_{j}^{-1}}\cdot\mathcal{W}^{\mu}\circ Df_{j}^{*}}\right)\quad\text{and}\quad
B_{j}=\mathrm{Op}\left(\frac{\mathcal{W}^{\mu}\circ Df_{j}^{*}}{\sqrt{Jf\circ
f_{j}^{-1}}}\right)$

and set

$v_{j}:=u_{j}\circ f_{j}^{-1}\quad\text{and}\quad b_{j}=B_{j}v_{j}$

for $j=1,2,\dots,\deg f$. Then, by the product formula ([10, §2.4]) of pseudo-
differential operator, we have

$\displaystyle\|\mathrm{Op}(\mathcal{W}^{\mu})Pu\|_{L^{2}}\leq\left\|\sum_{j=1}^{\deg
f}A_{j}B_{j}v_{j}\right\|_{L^{2}}+\sum_{j=1}^{\deg
f}\|\mathrm{Op}(\mathcal{W}^{\mu})P_{j}u_{j}-A_{j}B_{j}v_{j}\|_{L^{2}}$
$\displaystyle=\left\|\sum_{j=1}^{\deg
f}A_{j}B_{j}v_{j}\right\|_{L^{2}}+\sum_{j=1}^{\deg
f}\bigg{\|}\bigg{(}\mathrm{Op}(\mathcal{W}^{\mu})\mathrm{Op}\bigg{(}\frac{1}{Jf\circ
f_{j}^{-1}}\bigg{)}-A_{j}B_{j}\bigg{)}v_{j}\bigg{\|}_{L^{2}}$
$\displaystyle\leq\left\|\sum_{j=1}^{\deg
f}A_{j}B_{j}v_{j}\right\|_{L^{2}}+C\sum_{j=1}^{\deg f}\|v_{j}\|_{H^{\mu-1}}$

where $C$ is a constant that may depend on $f$ but not on $u$. (In the
following we use $C$ as a generic symbol for constants of this kind.) Recall
the Lagrange identity

$\sum_{i,j=1}^{d}|x_{i}|^{2}|y_{j}|^{2}-\left|\sum_{i=1}^{d}\bar{x}_{i}y_{i}\right|^{2}=\frac{1}{2}\sum_{i\neq
j}|\bar{x}_{i}y_{j}-\bar{x}_{j}y_{i}|^{2}.$

Following its proof with regarding $A_{i}$ and $b_{i}=B_{i}v_{i}$ as $x_{i}$
and $y_{i}$ respectively222Further regard $A_{i}^{*}$ as $\bar{x}_{i}$ and the
$L^{2}$ inner product as the product of complex numbers in a few places. and
using the product and adjoint formula ([10, §2.4]) to shuffle the pseudo-
differential operators, we see that

$\displaystyle\sum_{i,j=1}^{\deg f}\|A_{j}b_{i}\|_{L^{2}}^{2}$
$\displaystyle-\left\|\sum_{j=1}^{\deg f}A_{j}b_{j}\right\|_{L^{2}}^{2}$
$\displaystyle\geq\frac{1}{2}\sum_{i\neq
j}\|A_{j}b_{i}-A_{i}b_{j}\|_{L^{2}}^{2}-C\sum_{j=1}^{\deg
f}\|v_{j}\|_{H^{\mu-(1/2)}}^{2}.$

Further, by the product and adjoint formula, we see

$\sum_{i,j=1}^{\deg f}\|A_{j}b_{i}\|_{L^{2}}^{2}\leq\sum_{i=1}^{\deg
f}\mathrm{Re}\langle b_{i},Ab_{i}\rangle+C\sum_{j=1}^{\deg
f}\|v_{j}\|_{H^{\mu-(1/2)}}^{2}$

where $A$ is the pseudo-differential operator with real symbol

$A=\mathrm{Op}\left(\sum_{j=1}^{\deg g}\frac{\mathcal{W}^{2\mu}}{(Jf\circ
f_{j}^{-1})\cdot\mathcal{W}^{2\mu}\circ Df_{j}^{*}}\right)$

and we have used the estimate

$|\mathrm{Im}\langle b_{i},Ab_{i}\rangle|=\left|\left\langle
b_{i},\frac{A-A^{*}}{2}b_{i}\right\rangle\right|\leq
C\|v_{i}\|^{2}_{H^{\mu-(1/2)}}.$

Let $\varepsilon>0$ be a given small real number. By the definition of
$B^{\mu}(f)$, we have

$\sum_{j=1}^{\deg g}\frac{\mathcal{W}^{2\mu}}{(Jf\circ
f_{j}^{-1})\cdot\mathcal{W}^{2\mu}\circ Df_{j}^{*}}\leq
B^{2\mu}(f)+\varepsilon$

on the outside of a sufficiently large neighborhood of the zero section
depending on $\varepsilon>0$. For a negative number
$\kappa<\min\\{0,\mu-1/2\\}$, we can take a large constant $C>0$ so that

(5) $B^{2\mu}(f)+\varepsilon+C\cdot\mathcal{W}^{2\kappa}-\sum_{j=1}^{\deg
g}\frac{\mathcal{W}^{2\mu}}{(Jf\circ f_{j}^{-1})\cdot\mathcal{W}^{2\mu}\circ
Df_{j}^{*}}$

is non-negative. Hence, applying Gårding’s inequality [10, §2.8] to the
pseudo-differential operator with symbol (5), we obtain

$\mathrm{Re}\langle
b_{i},Ab_{i}\rangle\leq(B^{2\mu}(f)+\varepsilon)\|b_{i}\|_{L^{2}}^{2}+C\|v_{i}\|^{2}_{H^{\mu-(1/2)}}.$

From the relation $v_{j}:=u_{j}\circ f_{j}^{-1}$ and the coordinate change
formula [10, §2.5], we see that

$\displaystyle\|b_{j}\|_{L^{2}}$
$\displaystyle\leq\left\|\mathrm{Op}(\mathcal{W}^{\mu}\circ
Df_{j}^{*})\left(\frac{v_{j}}{\sqrt{Jf\circ
f_{j}^{-1}}}\right)\right\|_{L^{2}}+C\|v_{j}\|_{H^{\mu-1}}$
$\displaystyle\leq\left\|\frac{(\mathrm{Op}(\mathcal{W}^{\mu})u_{j})\circ
f_{j}^{-1}}{\sqrt{Jf\circ
f_{j}^{-1}}}\right\|_{L^{2}}+C^{\prime}\|u_{j}\|_{H^{\mu-1}}$
$\displaystyle=\|\mathrm{Op}(\mathcal{W}^{\mu})u_{j}\|_{L^{2}}+C^{\prime}\|u_{j}\|_{H^{\mu-1}}$

for some constants $C,C^{\prime}>0$. Finally note that we have

$\|v_{j}\|_{H^{\mu^{\prime}}}\leq C\|u_{j}\|_{H^{\mu^{\prime}}}$

for any $\mu^{\prime}\in\mathbb{R}$ and also that

$\|u_{j}\|_{H^{\mu^{\prime}}}\leq C\|u_{j}\|_{H^{\mu^{\prime\prime}}}$

when $\mu^{\prime}\leq\mu^{\prime\prime}$ where $C$ may depend on
$\mu^{\prime}$ and $\mu^{\prime\prime}$. Therefore, summarizing the argument
above, we have that, for any $\max\\{0,\mu-(1/2)\\}\leq\mu^{\prime}<\mu$ and
$\varepsilon>0$, there exists $C>1$ such that

$\|Pu\|_{H^{\mu}}\leq\sqrt{(B^{2\mu}(f)+\varepsilon)}\|u\|_{H^{\mu}}+C\|u\|_{H^{\mu^{\prime}}}.$

By using a partition of unity and the product formula ([10, §2.4]), we see
that the same inequality holds for arbitrary $u\in C^{\infty}(M)$. Since
$H^{\mu}(M)$ for $\mu>0$ is compactly embedded in $H^{\mu^{\prime}}(M)$ and
since $\varepsilon>0$ is arbitrary, the last inequality and Henion’s theorem
[7] tell that the essential spectral radius of $P$ on $H^{\mu}(M)$ is bounded
by $\sqrt{B^{2\mu}(f)}$. Finally noting that the argument above remains valid
when $f$ is replaced by its iterates $f^{n}$, we obtain the conclusion of the
theorem.

##  4\. A conclusive remark

It is possible to generalize our result without much effort. In the last
section, we consider the weight functions $\mathcal{W}^{\mu}$ for $\mu>0$. But
we could consider a more general class of positive function in the place of
$\mathcal{W}^{\mu}$. For instance, we may set

$\mathcal{W}^{\bm{\mu}}(x,\xi)=\langle|\xi|\rangle^{\bm{\mu}([(x,\xi)])}$

for a $C^{\infty}$ positive-valued function
$\bm{\mu}:\mathbb{P}T^{*}M\to\mathbb{R}$ on the projectivized cotangent bundle
$\mathbb{P}T^{*}M$ and prove Theorem333In the case where $\bm{\mu}$ takes
negative values, the argument in the proof of Corollary is not valid. with
setting

$B^{\bm{\mu}}(f)=\sup_{(x,\xi)\in T^{*}M}\sum_{(y,\eta)\in
T^{*}M:f^{{\dagger}}(y,\eta)=(x,\xi)}\frac{\mathcal{W}^{\bm{\mu}}(x,\xi)}{|Jf(y)|\cdot\mathcal{W}^{\bm{\mu}}(y,\eta)}.$

We refer [4, 5] for the treatment of the pseudo-differential operators
$\mathrm{Op}(\mathcal{W}^{\bm{\mu}})$. It will be an interesting subject to
find a reasonable but non-trivial class of dynamical systems for which
$\lim_{n\to\infty}B^{2\bm{\mu}}(f^{n})<1$.

##  Appendix A Virtually expanding maps that are not expanding

Let us recall the map $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ in Example, defined
by (4). Let $\mu>0$ be arbitrary. Below we prove $B^{\mu}(f)<1$ assuming that
$m$ is sufficiently large.

Take an arbitrary point $(q,\xi)\in T^{*}\mathbb{T}^{2}$ with $\xi\neq 0$ and
write $\xi=(\xi_{x},\xi_{y})$. We will show $b^{\mu}(q,\xi)<c<1$ for some
constant $c$ independent of $(q,\xi)$. If $\xi_{y}=0$, we see
$b^{\mu}(q,\xi)=m^{-\mu}<1$. Hence we may assume $\xi_{y}\neq 0$. Further,
since $b^{\mu}(q,\xi)=b^{\mu}(q,\beta\xi)$ for any $\beta\neq 0$, we may and
do assume $\xi_{y}=1$ and write $\xi=(\xi_{x},1)$.

Consider $(p,\eta)\in(f^{\dagger})^{-1}(q,\xi)$ and write $p=(x,y)$ and
$\eta=(\eta_{x},\eta_{y})$. Then we have $\eta_{y}=1$ and

$\displaystyle\begin{bmatrix}\xi_{x}\\\
1\end{bmatrix}=(D_{p}f^{-1})^{*}\begin{bmatrix}\eta_{x}\\\ 1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1/m&-2\pi\cdot\sin 2\pi x\\\
0&1\end{bmatrix}\begin{bmatrix}\eta_{x}\\\
1\end{bmatrix}=\begin{bmatrix}\eta_{x}/m-2\pi\cdot\sin 2\pi x\\\
1\end{bmatrix}$

This implies that, if

(6) $|\xi_{x}+2\pi\sin 2\pi x|\geq 5\pi/m,$

we have $|\eta_{x}|\geq 5\pi$ and

$\frac{|\xi|}{|\eta|}\leq\frac{\sqrt{1+(\eta_{x}/m+2\pi)^{2}}}{\sqrt{1+\eta_{x}^{2}}}\leq\frac{\sqrt{1+(2\pi)^{2}}}{\sqrt{1+(4\pi)^{2}}}$

provided that $m$ is sufficiently large. (Notice that the last inequality
holds obviously if $|\eta_{x}|$ is sufficiently large.)

From the observation above, the sum on the right hand side of (1) restricted
to those elements $(p,\eta)\in(f^{{\dagger}})^{-1}(q,\xi)$ satisfying the
condition (6) is bounded by $\sqrt{1+(2\pi)^{2}}/\sqrt{1+(4\pi)^{2}}$. The
cardinality of the remaining elements of $(f^{{\dagger}})^{-1}(q,\xi)$ is
bounded by $C\sqrt{m}$ with $C$ a constant independent of $m$. Hence the
corresponding partial sum is bounded by

$C\sqrt{m}\cdot\frac{\max_{w\in
M}\|(D_{w}f^{*})^{-1}\|^{\mu}}{m}\leq\frac{C_{\mu}}{\sqrt{m}}$

where $C_{\mu}>1$ is a constant depending on $\mu$. Therefore, letting $m$ be
sufficiently large, we obtain

$b^{\mu}(q,\xi)<\left|\frac{1+(2\pi)^{2}}{1+(4\pi)^{2}}\right|^{\mu/2}+\frac{C_{\mu}}{\sqrt{m}}<\left|\frac{1+(2\pi)^{2}}{1+(3\pi)^{2}}\right|^{\mu/2}<1$

uniformly for $(q,\xi)$. Thus we have obtained $B^{\mu}(f)<1$.

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