# A Numerical Method for Generating Rapidly Rotating Bipolytropic Structures
in Equilibrium

Kundan Kadam,1 Patrick M. Motl,2 Juhan Frank,1 Geoffrey C. Clayton1 and
Dominic C. Marcello1
1Department of Physics & Astronomy, Louisiana State University, Baton Rouge,
LA 70803
2The School of Sciences, Indiana University Kokomo, Kokomo, Indiana, 46904
E-mail<EMAIL_ADDRESS>
(Accepted 2016 July 21. Received 2016 July 18; in original form 2016 April 12)

###### Abstract

We demonstrate that rapidly rotating bipolytropic (composite polytropic) stars
and toroidal disks can be obtained using Hachisu’s self consistent field
technique. The core and the envelope in such a structure can have different
polytropic indices and also different average molecular weights. The models
converge for high $T/|W|$ cases, where T is the kinetic energy and W is the
gravitational energy of the system. The agreement between our numerical
solutions with known analytical as well as previously calculated numerical
results is excellent. We show that the uniform rotation lowers the maximum
core mass fraction or the Sch$\ddot{\rm{o}}$nberg-Chandrasekhar limit for a
bipolytropic sequence. We also discuss the applications of this method to
magnetic braking in low mass stars with convective envelopes.

###### keywords:

equation of state – methods: numerical – stars: rotation

††pubyear: 2016††pagerange: A Numerical Method for Generating Rapidly Rotating
Bipolytropic Structures in Equilibrium–A Numerical Method for Generating
Rapidly Rotating Bipolytropic Structures in Equilibrium

##  1 Introduction

Stellar models which use a polytropic equation of state (EoS), when combined
with the structural and thermal equations, give us a reasonable understanding
of the interiors of stars (Chandrasekhar, 1939). Even though the pressure is
decoupled from the temperature, the self gravitating structures obtained by
solving these equations show a remarkable similarity to the observations.
Polytropes have been used to represent gaseous planets, main sequence stars,
fully convective stars and even compact objects like white dwarfs and neutron
stars (Hansen et al., 2004). But for the stars which have a distinct core and
envelope structure, this approximation starts to break down. These stars with
a well defined core (terminal age main sequence, subgiant, red giant, AGB
etc.) behave in fundamentally different ways and we need a core-envelope
structure in order to represent them faithfully. The stars on the main
sequence also show a radiative core and convective envelope, or vice versa
depending on their mass (Padmanabhan, 2001).

One way to achieve a better approximation is to use bipolytropes (sometimes
also referred to as composite polytropes), where the core and the envelope of
a star are allowed to have different polytropic indices. This idea was first
proposed by Milne (1930) who suggested that the temperature, pressure and mass
enclosed would be continuous at the core-envelope interface. Henrich &
Chandrasekhar (1941) implemented this condition for a star with an isothermal
core and polytropic envelope to infer an upper limit in the core mass
fraction. Schönberg & Chandrasekhar (1942) introduced a discontinuous jump in
the molecular weights at the core-envelope boundary to deduce, the now famous,
Sch$\ddot{\rm{o}}$nberg-Chandrasekhar (SC) limit. Analytical solutions of
single polytropic spheres are possible only in the cases where the polytropic
index ($n$) equals zero (constant density), one or five, and only in the non-
rotating and spherically symmetric scenario. Analytical solutions for
bipolytropes are thus possible only in certain combinations of these three
indices. For example, see Murphy (1983) and Eggleton et al. (1998).

Stellar rotation is the most important parameter affecting the behavior of a
star after the initial mass and the metallicity (Meynet & Maeder, 2002).
Rotation has a profound effect on stellar structure equations, mass loss rate,
angular momentum loss, mixing and stellar wind. The effects are also observed
in the non-uniform surface temperature of stars and anisotropic structures in
nebulae. This is especially true for high mass stars as most are fast rotators
with equatorial velocities only a factor of two less than breakup (Kawaler,
1987).

In this paper we present the first computational method to determine the
equilibrium structures of rotating bipolytropes. This is achieved by modifying
Hachisu’s self-consistent field (HSCF) method (Hachisu, 1986); we call this
technique the bipolytropic self-consistent field or the BSCF method for
convenience. The core and envelope can have different polytropic indices and
the composition difference between them can be represented as a ratio of the
average molecular weights. We confirm the validity of this method by comparing
the results with known analytical as well as numerical results. The
bipolytropic models obtained can have a high degree of flatness or a high
$T/|W|$, where $T$ is the kinetic energy and $W$ is the gravitational energy
of the system. Although real stars rotate differentially (Zeipel, 1925), we
consider only uniform rotation because this is a good first order estimate of
the rotational properties.

We show that when the core has sufficiently high average molecular weight as
compared to rest of the star, a uniformly rotating star represented using a
bipolytrope has a lower SC limit as compared to its non-rotating counterpart.
We then discuss applications of the BSCF method for the convective magnetic
braking of low-mass stars. The bipolytropic models can also be used to probe
internal structure and estimate rotational parameters of fast spinning gaseous
planets and exoplanets. The source code for the BSCF method is publicly
available via GitHub, so that others in the astrophysics community can make
similar models for their own use. (Link to the GitHub repository:
https://github.com/kkadam/BSCF.)

##  2 Bipolytropes

The polytropic EoS assumes a power law relationship between the pressure of
the gas and the density

$P=\kappa\rho^{1+\frac{1}{n}}={\rm\kappa}\rho^{\gamma}$ (1)

where $n$ is the polytropic index, ${\rm\kappa}$ is the polytropic constant
and $\gamma$ is the ratio of specific heats. A bipolytropic star has one such
polytropic index for the core ($n_{\rm c}$) and one for the envelope ($n_{\rm
e}$). (The subscripts ${\rm c}$ and ${\rm e}$ correspond to the core and the
envelope, and the subscript ${\rm i}$ corresponds to the core-envelope
interface throughout this paper.) In general, the polytropic constant is also
different for the two regions. The physical quantities which are continuous
across the core-envelope interface are pressure, temperature and the mass
enclosed. While the pressure in the core and the envelope is represented by
the respective polytropic EoS, at any point the total pressure must also be
given by the sum of gas pressure and radiation pressure. Hence

$P=\frac{{\rm k_{B}}}{\mu{\rm m_{H}}}\rho T+\frac{1}{3}{\rm a}T^{4}$ (2)

where ${\rm k_{B}}$ is the Boltzmann constant, $\mu$ is molecular weight,
${\rm m_{H}}$ is the mass of the hydrogen atom, ${\rm a}$ is the radiation
constant and $T$ is the temperature of the gas. For an evolved star the core
composition is different from that of the envelope, hence they have different
molecular weights. If we assume temperature and pressure to be continuous
across the boundary, then the density across this interface is not continuous,
but is proportional to the ratio of molecular weights,

$\frac{\rho_{\rm ci}}{\rho_{\rm ei}}=\frac{\mu_{\rm c}}{\mu_{\rm e}}$ (3)

where $\rho_{\rm ci}$, $\rho_{\rm ei}$ are densities at the interface and
$\mu_{\rm c}$ and $\mu_{\rm e}$ are the average molecular weights. These are
the boundary conditions at the core-envelope interface of a bipolytrope. The
ratio of molecular weights,

$\alpha=\frac{\mu_{\rm c}}{\mu_{\rm e}}$ (4)

represents the discontinuity at the core-envelope interface because of the
difference in the composition between the core and the envelope.

At this point we introduce two key parameters that are used extensively in
subsequent discussions of sequences of bipolytropic models. The fractional
core mass of a bipolytrope is defined as

$\nu=\mathcal{M}_{\rm c}/\mathcal{M}$ (5)

where $\mathcal{M}_{\rm c}$ is the mass of the core and $\mathcal{M}$ is the
total mass. The fractional core radius is

$q=\mathcal{R}_{\rm c}/\mathcal{R}$ (6)

where $\mathcal{R}_{\rm c}$ is equatorial radius of the core and $\mathcal{R}$
is equatorial radius of the whole configuration. The fractional core mass and
radius are key to the discussions of model comparisons as well as any
astrophysically relevant outcomes, as we shall see in sections 5 and 6.

##  3 The BSCF Method

We can construct rapidly rotating single bipolytropic spheroidal as well as
toroidal structures with the BSCF method. We make the following two
assumptions while using the BSCF technique. The star (spheroid) or torus is
assumed to be rotating axisymmetrically and the angular velocity ($\Omega$)
throughout the structure is uniform.

###  3.1 Equations

In a frame of reference that is spinning with the star, the vector equation of
motion of a self-gravitating body in equilibrium is given by,

$\frac{1}{\rho}\nabla P+\nabla(\Phi+\Omega^{2}\Psi)=0$ (7)

where $\Psi$ is the coordinate-dependent part of the centrifugal potential and
$\Phi$ is the gravitational potential. With the specific enthalpy of the gas
given by

$H=\int_{0}^{\rm P}{\frac{\mathop{}\\!\mathrm{d}{P}}{\rho}},$ (8)

Eq. 7 can be integrated to obtain

$H+\Phi+\Omega^{2}\Psi={\rm C_{B}}$ (9)

with ${\rm C_{B}}$ as the integration constant – analogous to the Bernoulli
constant in classical fluid flows. The last term is the centrifugal potential,

$\Omega^{2}\Psi=-\Omega^{2}\frac{R^{2}}{2}$ (10)

where $R$ is the distance from the axis of rotation.

The outer boundary condition is that the density is zero at the stellar
surface. Integrating Eq. 8 with a polytropic EoS gives the enthalpy as a
function of density

$H=(1+n){\rm\kappa}\rho^{\frac{1}{n}}{\rm,}$ (11)

hence the enthalpy is also zero at the outer boundary.

###  3.2 Implementation

In order to determine a bipolytropic equilibrium structure uniquely, a
possible pair of parameters are the axis ratio (the ratio of the equatorial to
the polar radius) and core radius fraction (the ratio of the equatorial radius
of the core to the radius of the whole star). Fig. 1 shows a schematic diagram
of a meridional cross section of a star (top panel) and of a toroidal
configuration (bottom panel). Just like the HSCF method, we cannot specify the
rotation rate (or the core mass fraction, in the case of a bipolytrope) a
priori. We can only specify outer boundary points A and B, and as implemented
here, the location of the core-envelope interface C. The lower the axis ratio,
the faster the resulting configuration rotates, hence the angular velocity is
controlled by means of points A and B. The core mass fraction is set via the
location of point C. For a torus the points A and B specify an inner and an
outer radius in the equatorial plane.

Figure 1: A schematic of the meridional cross section of rotating figures in
cylindrical coordinates, showing the boundary points are A, B and the core
interface C, in the case of a spheroidal and toroidal bipolytropic structure
respectively.

In order to get an equilibrium solution, we solve Poisson’s equation

$\nabla^{2}\Phi=-4{\rm\pi G}\rho$ (12)

and the hydrostatic equation (Eq. 9) iteratively as follows. We have to assume
or “guess" an initial density distribution satisfying the outer boundary
conditions. The BSCF method is not sensitive to this initial distribution. A
typical choice for the initial guess for a spheroid is a constant density
cone, which would look like a right triangle joining points $\rm A$ and $\rm
B$ in a meridional cross section. For a toroid, the cross section of initial
density would look like a box. The potential field ($\Phi$) throughout the
grid is calculated from this density distribution by solving Poisson’s
equation.

We expect the integration constant for the core (${\rm C_{c}}$) to be
different from that for the envelope (${\rm C_{e}}$) in general. Hence for the
stellar core Eq.9 becomes

$H+\Phi+\Omega^{2}\Psi={\rm C_{c}}\quad{\rm when}\quad\rho\geq\rho_{\rm ci}$
(13)

and the same equation for the envelope is

$H+\Phi+\Omega^{2}\Psi={\rm C_{e}}\quad{\rm when}\quad\rho<\rho_{\rm ci}.$
(14)

Since $H=0$ at points A and B, we get

$H_{\rm A}=-\Phi_{\rm A}-\Omega^{2}\Psi_{\rm A}+{\rm C_{e}}=0,$ (15) $H_{\rm
B}=-\Phi_{\rm B}-\Omega^{2}\Psi_{\rm B}+{\rm C_{e}}=0.$ (16)

Hence we can obtain the angular velocity of the system

$\Omega^{2}=-\frac{\Phi_{\rm A}-\Phi_{\rm B}}{\Psi_{\rm A}-\Psi_{\rm B}}.$
(17)

Once we have $\Omega$, we can obtain the integration constant for the envelope

${\rm C_{e}}=\Phi_{\rm A}+\Omega^{2}\Psi_{\rm A}=\frac{\Phi_{\rm A}\Psi_{\rm
B}-\Phi_{\rm B}\Psi_{\rm A}}{\Psi_{\rm B}-\Psi_{\rm A}}.$ (18)

Now from the continuity of pressure at the core-envelope interface, we can
derive

$H_{\rm ci}=H_{\rm ei}\frac{(n_{\rm c}+1)}{(n_{\rm e}+1)}\frac{\rho_{\rm
ei}}{\rho_{\rm ci}},$ (19)

where $H_{\rm ei}$ can be obtained from ${\rm C_{e}}$

$H_{\rm ei}=-\Phi_{\rm i}-\Omega^{2}\Psi_{\rm i}+{\rm C_{e}}.$ (20)

The integration constant for the core can be obtained from

${\rm C_{c}}=H_{\rm ci}+\Phi_{\rm ci}+\Omega^{2}\Psi_{\rm i}.$ (21)

Now we can calculate the enthalpy throughout the star from Eqs. 13 and 14.
Once we have the enthalpy, we can calculate new density distributions of the
core and the envelope from Eq. 11 as

$\frac{\rho_{\rm c}}{\rho_{\rm c-max}}=\frac{\rho_{\rm c}}{\rho_{\rm
0}}=\left(\frac{H_{\rm c}}{H_{\rm c-max}}\right)^{n_{\rm c}}$ (22)

and

$\frac{\rho_{\rm e}}{\rho_{\rm e-max}}=\frac{\rho_{\rm e}}{\rho_{\rm
ei}}=\left(\frac{H_{\rm e}}{H_{\rm e-max}}\right)^{n_{\rm e}}.$ (23)

We choose $\rho_{\rm c-max}$, or the central density $\rho_{\rm 0}$, as unity
and the envelope density can be normalized with respect to this central
density,

$\frac{\rho_{\rm e}}{\rho_{\rm 0}}=\frac{\rho_{\rm ei}}{\rho_{\rm
0}}\left(\frac{H_{\rm e}}{H_{\rm e-max}}\right)^{n_{\rm e}}=\frac{\rho_{\rm
ci}}{\rho_{\rm 0}}\frac{\mu_{\rm e}}{\mu_{\rm c}}\left(\frac{H_{\rm e}}{H_{\rm
e-max}}\right)^{n_{\rm e}},$ (24)

and

$\frac{\rho_{\rm e}}{\rho_{\rm 0}}=\frac{\mu_{\rm e}}{\mu_{\rm
c}}\left(\frac{H_{\rm ci}}{H_{\rm c-max}}\right)^{n_{\rm c}}\left(\frac{H_{\rm
e}}{H_{\rm e-max}}\right)^{n_{\rm e}}.$ (25)

Thus we obtain a new density distribution for the next iteration cycle.

The iteration process is repeated until the relative changes in $\rm C_{c}$,
$\rm C_{e}$ and $\Omega^{2}$ are smaller than the prescribed convergence
criterion, $\delta$. Here, we use $\delta=1\times 10^{-3}$ for all our
calculations, except for the models used for the Richardson extrapolation
where $\delta=1\times 10^{-4}$, in order to achieve better accuracy (see
section 5.2).

Using the BSCF method as described above, we essentially choose the fractional
core radius, $q$ as the control parameter. The analytical relationship between
fractional core mass, $\nu$ and fractional core radius, $q$ becomes double
valued as the ratio of molecular weights, $\alpha$ increases and eventually at
$\alpha=3$, there cannot be equilibrium solutions above a certain value of
$\nu$. This is the aforementioned SC limit for the core mass fraction (see
Fig. 5). Since we can only specify point $\rm C$ and hence the value of $q$
with the BSCF method, we cannot directly control the fractional core mass,
$\nu$. For a given value of $q$, if there exist two equilibrium values of
$\nu$, we observe that the method always converges to the solution with the
lower $\nu$. In order to overcome this drawback, we can modify the BSCF method
to use the ratio of the density at the core side of the interface and the
central density, $\rho_{ci}/\rho_{0}$, as the control parameter instead of
point $\rm C$, which uniquely determines the configuration. The equations
involved are exactly the same as mentioned so far and the radius of the core
($q$) is calculated from the converged model. We discuss the implications of
this modification of the BSCF method in sections 5.2 and 6.1.

###  3.3 Normalization

We use the same normalization convention as used in Hachisu (1986), mainly for
the ease of comparison. The basis set of dimensionless quantities is maximum
density (${\rm\rho_{0}}$), equatorial radius of the star ($\mathcal{R}$), and
the gravitational constant (${\rm G}$). All computations are performed in the
dimensionless form of the variables.

##  4 Numerical Methods

We use cylindrical geometry for our computational grid because a rotating body
is naturally described in cylindrical coordinates. The Poisson solver with
this geometry was already available to us from our previous numerical work
(Motl et al., 2002). This not only minimized the time for development of the
BSCF method significantly, but also provided a well-tested, reliable and
accurate method for our investigations.

###  4.1 Poisson’s equation

The most compute-intensive section in the BSCF method is the Poisson solver,
which is used for obtaining the gravitational potential from the specified
density distribution at each integration time step. For an assumed isolated
density distribution, the boundary condition for the potential is that it goes
to zero at infinity. But in practice, since the grid is always finite, we have
to calculate the potential at the edge of the grid which is outside the mass
distribution. We construct this boundary potential using a compact expression
for the Green’s function in cylindrical coordinates. As described in Cohl &
Tohline (1999), we get an analytical expression for the integral in terms of
half-integer degree Legendre functions of the second kind. This means that we
can obtain a very accurate solution for our discretized mass distribution.
This technique is particularly useful for studying systems which conform well
to azimuthal symmetry because it can be applied to very flattened or elongated
bodies without suffering penalties in either accuracy or computation time.

We use an effective two dimensional version of the Poisson solver by setting
the number of azimuthal modes supported in the Green’s functions to one,
thereby enforcing axial symmetry. In order to obtain the interior solution for
the potential, we use an alternating direction implicit (ADI) scheme (Peaceman
& Rachford, 1955).

###  4.2 Discretization and Quality of Solutions

The computational grid geometry is cylindrical with azimuthal symmetry. The
resolution of the grid is denoted by ${\rm NUMR}\times{\rm NUMZ}$, where ${\rm
NUMR}$ is the number of cells in the $R$-direction and ${\rm NUMZ}$ is the
number of cells in the vertical or $z$-direction.

All physical quantities are calculated at the cell centers. The first cell in
both $R$ and $z$-direction is a boundary cell and the unit distance between
consecutive cells in both the directions is equal, i.e.
$\mathop{}\\!\mathrm{d}{z}=\mathop{}\\!\mathrm{d}{R}$.

The quality of a converged solution is quantified using the virial error
($VE$), which is a global measure of the accuracy of the equilibrium of a non-
linear dynamical system.

$VE=\frac{\mid 2T+W+3\Pi\mid}{\mid W\mid}$ (26)

where $\Pi$ is the volume integral of the pressure,

$\Pi=\int_{V}P\mathop{}\\!\mathrm{d}{V}$ (27)

This expression is derived from the scalar virial equation, hence the virial
error should ideally be zero for a system in equilibrium. A typical value for
virial error with a resolution of $130\times 130$ is $10^{-4}$. This value can
be pushed lower by increasing the resolution, while simultaneously decreasing
$\delta$ by an appropriate amount.

##  5 Numerical Tests

In this section we compare the equilibrium configurations obtained through the
BSCF technique with some known analytical as well as numerical results. We use
a grid size of $130\times 130$ for all our calculations, unless stated
otherwise. The equatorial radius (location of the point A in Fig. 1) of the
star for all configurations is 120 cells. In the case of tori, this is the
outer radius.

###  5.1 Polytropes, Maclaurin, and One-Ring Sequences

We construct continuous polytropic structures by setting the polytropic index
of the core equal to that of the envelope and the ratio of molecular weights
to unity. These models should be identical to single polytropes and satisfy
all the associated properties.

Figure 2: Dimensionless squared angular velocity ($\Omega^{2}/{\rm 4\pi
G}\bar{\rho}$), and rough stability indicator ($T/|W|$) versus dimensionless
squared angular momentum ($J^{2}/{\rm 4\pi
G}\mathcal{M}^{10/3}\bar{\rho}^{-1/2}$) for the Maclaurin, Eriguchi-Sugimoto
and Dyson-Wong sequences. The solid lines correspond to the models obtained
using the BSCF method and the points are previously calculated values
(Hachisu, 1986).

Maclaurin spheroids (MS) correspond to the equilibrium sequence of self-
gravitating and uniformly rotating incompressible spheroids (Tassoul, 1978).
This sequence transitions in a continuous fashion into Dyson-Wong (DW) toroids
(or the one-ring sequence) through the Eriguchi-Sugimoto (ES) sequence, as the
configurations rotate with increasing angular momentum (Eriguchi & Sugimoto,
1981). As a numerical test of the BSCF method, we reproduce the MS, ES and DW
sequences for an incompressible fluid with $n=0$. The results are plotted in
Fig. 2. The smooth curves are obtained with the BSCF method as the point B
moves along the z-axis (generating flatter spheroids by decreasing the axis
ratio) and then along the R-axis (generating thinner toroidal structures) as
depicted in Fig. 1.

We constructed similar sequences for other polytropic indices and these curves
are also in excellent agreement with the calculations already performed (see
Figs. 10 and 11 in Hachisu (1986)).

###  5.2 Spherically Symmetric Bipolytropes

As an additional test of the validity of the BSCF method, we constructed non-
rotating bipolytropic models by setting the axis ratio to one. The properties
of these structures were compared to analytical solutions.

Figure 3: Fractional core mass ($\nu$) versus logarithm of core density
contrast ($\rho_{\rm 0}/\rho_{\rm ci}$) for a $(n_{\rm c},n_{\rm e})=(5,1)$
bipolytrope with $\alpha=2$, $2.9$, $3$, $3.1$ and $4$. Points are BSCF
results and the smooth curves are analytically determined (Eggleton et al.,
1998). The cross denotes termination of the sequence in an SC limit (see Fig.
5). Figure 4:  Logarithm of the inverse of fractional core radius ($q$) versus
logarithm of core density contrast ($\rho_{\rm 0}/\rho_{\rm ci}$) for a
$(n_{\rm c},n_{\rm e})=(5,1)$ bipolytrope with $\alpha=2$, $2.9$, $3$, $3.1$
and $4$. Points are BSCF results and the smooth curves are analytically
determined (Eggleton et al., 1998). The cross denotes termination of the
sequence in an SC limit (see Fig. 5).

The analytical case of spherically symmetric $(n_{\rm c},n_{\rm e})=(5,1)$
bipolytropes was first investigated by Eggleton et al. (1998). They
demonstrate that these bipolytropes exhibit an upper core mass fraction limit
analogous to the SC limit. We reproduce the analytical solutions with the BSCF
method. In Figs. 3 and 4, respectively, we plot the core mass fraction ($\nu$)
and the logarithm of the inverse of core radius fraction ($q$) as a function
of the logarithm of the core density contrast, defined as $\rho_{\rm
0}/\rho_{\rm ci}$. The points which are obtained from the BSCF models match
the analytical sequences very well.

Figure 5: Plot of fractional core mass $\nu$ versus fractional core radius $q$
for a bipolytrope with $(n_{\rm c},n_{\rm e})=(5,1)$ for different values of
$\alpha$. The smooth curves are analytically determined and the points denote
the sequences obtained using the BSCF method. The crosses are SC limits for a
given $\alpha$ and open circles denote the switch from point $\rm C$ to
$\rho_{ci}/\rho_{0}$ as a free parameter in determining the mass of the core.

.

The aforementioned SC limit is better understood with the help of the $\nu-q$
curve. As depicted in Fig. 5, for a high enough ratio of average molecular
weight ($\alpha\geq 3$), the fractional mass shows an upper limit above which
no equilibrium configurations exist. We can obtain all configurations up to
this SC limit with the BSCF method. The open circles in Fig. 5 denote the
switch in the BSCF method from using point $\rm C$ to using the ratio of the
value of the density at the core side of the interface and the central density
($\rho_{ci}/\rho_{0}$, also equal to the inverse of the density contrast) as
the control parameter, in determining the size of the core. The exact location
of the core interface or $q$ is now calculated using the enthalpy field. The
correction to the fractional core mass $\nu$ is obtained by adding the
contribution from the corresponding shell to the core mass.

As the density contrast increases or $\rho_{\rm ci}/\rho_{\rm 0}$ gets
progressively smaller, the core gets more centrally condensed. This results in
an equilibrium structure with a tenuous envelope and a core resembling a delta
function, which contains fewer and fewer points. In order to calculate
parameters $\nu$ and $q$ accurately, we need to resolve this core with a high
resolution model. We implemented the Richardson extrapolation (Richardson,
1911) at two different resolutions (total equatorial resolution of 120 and 240
cells) in order to get better estimates of $\nu$ and $q$ in Figs. 3, 4 and 5.
This is because higher resolution models which could resolve the core were
prohibitive in computing time without parallelizing the code. This approach is
also ultimately limited by the resolution. This can be seen in Figs. 3 and 5
for $\alpha=2.9$, $3$ and $3.1$. For these curves, the BSCF results match the
analytical sequences only up to certain points. Obtaining these full curves
may require higher resolutions and higher order Richardson extrapolations.

##  6 Numerical Results

In this section we consider two applications of the BSCF technique. The first
pertains to the effects of rotation on the SC limit. We next consider the
effects of rotation on the radius of gyration of low mass stars, which in turn
affects the timescale for magnetic braking.

###  6.1 Sch$\ddot{\rm{o}}$nberg-Chandrasekhar Limit

As a star evolves along the main sequence, it burns the hydrogen in the
central region into helium. Near the end of the main sequence phase, an inert
helium core is formed at the center and the hydrogen continues to burn around
this core in a shell. This core is essentially isothermal in order for it to
be in thermal equilibrium. If the gas is assumed ideal, because of the
interplay between internal pressure and self gravity, the surface pressure of
such a core decreases strongly with the core mass fraction. Schönberg &
Chandrasekhar (1942) calculated that for a star with an isothermal core, a
radiative envelope and a large enough molecular weight difference between the
two, this core mass fraction has an upper limit above which the core cannot
support the envelope. When this SC limit is reached, the star must respond by
readjusting itself on a Kelvin-Helmholtz timescale by contraction of the core.
The resultant liberation of the gravitational energy causes an increase in the
luminosity. Applegate (1988) argues that this exceeds the maximum luminosity
that the radiative envelope can carry and the envelope density is forced down.
This causes the envelope to expand and we get evolution along the red giant
branch. This would be a first order approximate answer to the question “Why
stars become red giants". In real stars the process is much more complicated
and non-linear, with nuclear burning, opacities, convection, chemical
composition and various other quantities affecting the structure of the star
(e.g. see Sugimoto & Fujimoto (2000)).

Maeder (1971) was the first to discuss the effects of rotation on the SC
limit. He used first order perturbation analysis on the virial expression of a
star with an isothermal core and a radiative envelope. In conclusion, the
rotation could increase or decrease the SC limit, depending on the actual
rotational profile in the stellar interior. For a uniform rotation, the SC
limit decreases and even at the maximum rotation the decrease is only about
3%.

Figure 6: Plot of fractional core mass $\nu$ versus fractional core radius $q$
for a bipolytrope with $(n_{\rm c},n_{\rm e})=(5,1)$ and $\mu=4$ for different
values of axis ratios. The points denote the sequences obtained using the BSCF
method. A cross denotes the Sch$\ddot{\rm{o}}$nberg-Chandrasekhar limit and
the dashed lines represent the trend for the corresponding sequence. The solid
line is obtained from the non-rotating analytical solution.

As we saw in the section 5.2, analytical bipolytropes also show an analogous
SC limit above a certain jump in the average molecular weight. Here we observe
the effect of rotation on the SC limit on a bipolytrope with an $(n_{\rm
c},n_{\rm e})=(5,1)$ and a molecular weight ratio of $4$. Fig. 6 shows the
$\nu-q$ sequences of these bipolytropes at different axis ratios. We see that
as a bipolytrope rotates more rapidly, the maximum mass limit (denoted by a
cross) decreases. A bipolytropic star rotating with an axis ratio of $0.6$
would encounter the SC limit at a mass fraction more than $20\%$ lower than
its non-rotating counterpart. Note that these values are quantitatively true
only in the case of a bipolytropic structure with $(n_{\rm c},n_{\rm
e})=(5,1)$, $\alpha=4$ and uniform rotation. This decrease is about an order
of magnitude larger than that predicted by Maeder (1971) for an isothermal
core and a radiative envelope. An SCF technique incorporating more realistic
EoS and rotational profiles may be necessary in order to obtain a better
estimate of the effects of rotation on the SC limit of real stars.

The change in the SC limit with rotation can have significant consequences for
our understanding of stellar evolution, particularly for high mass stars as
they do not have a convective envelope to brake their rotation efficiently.
These stars also live on the main sequence for much shorter time so that they
can still be rotating rapidly as they evolve off the main sequence. The SC
limit determines the combustible stellar mass fraction of a star, thus
affecting its lifetime spent on the main sequence. The effect due to rotation
would be the next leading order of correction to the basic structure and
evolution as would be dictated by the star’s mass and chemical composition.

Dependance of the SC limit on rotation may also result in an observational
test of the rotational profiles in the stellar interiors. The slight changes
in the frequency or position of stars on the HR diagram due to the shift in
the SC limit can relate to the rotation law. The SC limit should also be
different in a close binary system as compared to an isolated star because of
the rotational flattening. Sophisticated stellar evolution codes like MESA
(Paxton et al., 2011) are required to estimate the secular consequences of
these results.

###  6.2 Radius of Gyration

Main sequence stars having masses between about $0.3-1.3M_{\odot}$ have a
radiative core and a convective envelope. These low mass stars can be
represented as a bipolytrope with $(n_{\rm c},n_{\rm e})=(3,3/2)$ (Rappaport
et al., 1983; Beech, 1988). Rucinski (1988) computed sequences of fractional
radii of gyration of such spherically symmetric and rigidly rotating
bipolytropic models. The fractional radii of gyration for the core ($k_{\rm
c}$) and envelope ($k_{\rm e}$) are defined such that the total angular
momentum $J$ is

$J=\mathcal{M}\mathcal{R}^{2}\Omega({k_{\rm c}}^{2}+{k_{\rm e}}^{2})$ (28)

where $\mathcal{M}$ and $\mathcal{R}$ are the total mass and equatorial radius
of the star respectively. This also means that the total radius of gyration is
${k_{\rm t}}^{2}={k_{\rm c}}^{2}+{k_{\rm e}}^{2}$. For an arbitrary stellar
density distribution with the core-envelope interface located at $i$, we can
calculate radii of gyration by integrating the moments of inertia,

${k_{\rm c}}^{2}=\frac{1}{\mathcal{M}\mathcal{R}^{2}}\int_{0}^{\rm
i}R^{2}\mathop{}\\!\mathrm{d}{m},$ (29)

and

${k_{\rm e}}^{2}=\frac{1}{\mathcal{M}\mathcal{R}^{2}}\int_{\rm
i}^{\mathcal{R}}R^{2}\mathop{}\\!\mathrm{d}{m}.$ (30)

We recreate the sequences of radii of gyration in Rucinski (1988) for the case
of $(n_{\rm c},n_{\rm e})=(3,3/2)$ as a function of the core radius fraction.
The original sequences correspond to numerical integrations of the expressions
for $k_{\rm c}$ and $k_{\rm e}$, which are derived from a bipolytropic
treatment of the spherically symmetric Lane-Emden equation without a
discontinuity. The BSCF models constructed also have no discontinuity in
molecular weight and the axis ratio is unity, hence they are spherically
symmetric. As depicted in Fig. 7 the results of the BSCF method are
indistinguishable from the previously calculated curves.

Figure 7: Fractional radii of gyration for a non-rotating polytrope with
$(n_{\rm c},n_{\rm e})=(3,3/2)$ and $\alpha=1$. The points are BSCF results
and the smooth curves are calculated by Rucinski (1988). ${k_{\rm t}}^{2}$,
${k_{\rm c}}^{2}$ and ${k_{\rm e}}^{2}$ denote the curves for total, core, and
envelope respectively.

With the BSCF method we can explore the effect of rotation on the radii of
gyration. Figs. 8, 9 and 10 summarize the results as we progressively decrease
the axis ratio. As a star rotates faster at a given fractional core radius,
the corresponding radius of gyration steadily decreases. For the core, $k_{\rm
c}^{2}$ shows a shift in the maximum value as the rotation increases. The
sequence with an axis ratio of 0.62 terminates at $q=0.87$ (open circle in
Figs. 8, 10) because this is the mass shedding limit. This means that the star
is rotating at the critical velocity and no equilibrium configurations are
possible above this value of $q$.

Figure 8: Total radii of gyration for rotating bipolytropes with $(n_{\rm
c},n_{\rm e})=(3,3/2)$ and $\alpha=1$. The numbers denote the axis ratio of
the models. The open circle denotes critical rotation. Figure 9: Fractional
radii of gyration of the envelope for rotating bipolytropes with $(n_{\rm
c},n_{\rm e})=(3,3/2)$ and $\alpha=1$. The numbers denote the axis ratio of
the models. Figure 10: Fractional radii of gyration of the core for rotating
bipolytropes with $(n_{\rm c},n_{\rm e})=(3,3/2)$ and $\alpha=1$. The numbers
denote the axis ratio of the models. The open circle denotes critical
rotation.

The dependence of the fractional radii of gyration on rotation can have
implications for the rotational evolution of low-mass main sequence stars. A
star can lose angular momentum through the process of magnetic braking as its
wind is magnetically coupled to the rotation (Mestel, 1968). This is an
efficient mechanism for the loss of angular momentum for a star possessing a
magnetic field. Skumanich (1972) observed that G-dwarf stars in open clusters
spin down following the empirical relation

$v_{\rm e}\propto t^{-0.5}$ (31)

where $v_{\rm e}$ is equatorial velocity and $t$ is the age. Vilhu (1982)
gives the semi-empirical law for magnetic braking

$\frac{\mathop{}\\!\mathrm{d}{J}}{\mathop{}\\!\mathrm{d}{t}}\propto\Omega^{N}$
(32)

where ${N}$ is the angular momentum loss parameter (${N}=3$ for the Skumanich
Law). Here the underlying assumption is that the braking is not caused by the
aging itself but the mechanism directly relates to the rotation of the star.
Combining Eqs. 28 and 32, the characteristic timescale for magnetic braking
can be given by

$T_{\rm b}\approx J/\dot{J}\propto{k_{\rm
t}}^{2}\mathcal{M}\mathcal{R}^{2}\Omega^{1-{N}}.$ (33)

For low-mass stars it is possible that the convective envelope decouples from
the radiative core and the measurements of the magnetic braking apply to the
envelope only (Vilhu & Moss, 1986). The characteristic timescale for magnetic
braking then depends on the radius of gyration of the envelope alone, hence

$T_{\rm b}\ \propto{k_{\rm e}}^{2}\mathcal{M}\mathcal{R}^{2}\Omega^{1-{N}}.$
(34)

In order to calculate the timescale of braking of the convective envelope as a
function of stellar mass, we follow the same recipe as in Rucinski (1988). He
used VandenBerg stellar models which have been calculated for a solar
composition of ${\rm Y}=0.27$ and ${\rm Z}=0.0169$. The only input from these
models that we use, apart from the overall mass-radius relationship, is the
value of core mass fraction ($\nu$) for a given stellar mass. The value of the
angular momentum loss parameter decreases for more rapidly spinning stars
(Vilhu, 1982) and we assume ${N}=1$ for our calculations. Thus using the
$k_{e}$ values obtained from the BSCF method and Table V in Rucinski (1988) we
can estimate the timescale for braking for the sequence of stars in the mass
range $\mathcal{M}=0.5-1.2M_{\odot}$.

Figure 11: The timescale for braking of the convective envelope ($T_{\rm
b(norm)}$) as a function of stellar mass ($\mathcal{M}$). The solid line
corresponds to the calculations performed by Rucinski (1988) using non-
rotating VandenBerg models. The dashed lines correspond to the non-rotating
and rotating bipolytropic sequences generated using the BSCF method for the
same models. $T_{\rm b(norm)}$ is normalized to the value at $1M_{\odot}$ for
the Rucinski sequence.

Fig. 11 summarizes the effects of rotation on the braking timescale of the
convective envelope. For ease of comparison, the braking timescale is
normalized to the value at $1M_{\odot}$ for the Rucinski (1988) sequence
(solid black line in Fig. 11). The timescales for non-rotating bipolytropes
calculated using the BSCF method are consistent with those estimated by
Rucinski. Notice that the timescale plateaus and shows little variation below
$0.9M_{\odot}$ but in the mass range of $0.9-1.2M_{\odot}$, it drops
precipitously. For rotating stars with an axis ratio of 0.7, this plateau
below $0.9M_{\odot}$ occurs at a significantly (about 25%) lower
characteristic braking timescale. We chose an axis ratio of 0.7 in order to
demonstrate the most extreme difference that rotation could reasonably make,
since this axis ratio puts the rotational velocity of a star close to breakup.
The sequences of braking timescales for axis ratios other than 0.7 will show
trends similar to Fig. 9.

This analysis may exaggerate the difference in braking time because as the
stars slow down, the evolution of the fractional radius of gyration is not
considered ($k_{\rm e}$ is assumed constant throughout the above exercise). We
show that the convective envelopes of these low-mass stars contain less
initial angular momentum than previously estimated, when we take the
distortion due to the rotation into account, irrespective of subsequent
evolution. This should be reflected in the estimations of the braking time and
all previous such calculations can be considered as upper limits. These
results affect the observability of rapidly rotating stars in a given cluster
as fast rotating stars brake on a shorter timescale. The results could also
have implications for the evolution of short period binary systems, because
the same braking scheme is generally used for estimating angular momentum loss
in close binary systems (Vilhu, 1982).

##  7 Conclusions

We present a numerical method to obtain self- consistent rapidly rotating
bipolytropic models in equilibrium. This is achieved using the BSCF method.
The sequences generated with this method agree with previously calculated and
well established analytical as well as numerical results. These models can be
used to investigate the effects of rotation on the internal structures and
properties of stars. We show that the structural changes accompanying rotation
lower the Sch$\ddot{\rm{o}}$nberg-Chandrasekhar mass limit in the case of
uniform rotation. The rotational evolution of low-mass main sequence stars is
also affected when we consider self-consistent rotating stellar models.

Rotating bipolytropes can also prove useful in the investigations into the
internal structure of gaseous planets. Using a bipolytropic model similar to
Criss & Hofmeister (2015), we calculated the orbital period, normalized moment
of inertia and the zonal harmonic coefficients for planet Jupiter with
reasonable accuracy. Similar rotating models can be used to narrow down the
internal structures of fast spinning exoplanets (for example, $\beta$ Pictoris
b (Snellen et al., 2014)).

In order to further improve the models generated using the BSCF method,
differentially rotating configurations with constant rotational velocity
($v$-constant) or constant angular momentum per unit mass ($j$-constant) could
be obtained by following the recipe in Hachisu (1986). This technique may also
be used in order to generate rotating multi-polytropic models with an onion-
like structure. We believe that the BSCF method is a step towards more
realistic, yet relatively simple astrophysical models which minimize the
complexity of the input physics and help us gain an intuitive understanding of
the results obtained.

## Acknowledgements

We have greatly benefitted from discussions of this topic with Prof. Joel E.
Tohline as well as from technical derivations related to the structure and
stability of bipolytropes that he has provided in online documents (search,
for example, “Tohline Bipolytropes"). We would also like to thank the
anonymous referee whose insightful comments led to the use of
$\rho_{ci}/\rho_{0}$ as a free parameter in the BSCF method.

We wish to acknowledge the support from the National Science Foundation
through CREATIV grant AST-1240655. The numerical work was carried out using
the computational resources of the Louisiana Optical Network Initiative
(LONI).

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# Mixing and ergodicity of compositions of inner functions

Gustavo R. Ferreira  Centre de Recerca Matemàtica, Barcelona, Spain
<EMAIL_ADDRESS>and  Artur Nicolau  Departament de Matemàtiques,
Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica, 08193,
Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

We study ergodic and mixing properties of non-autonomous dynamics on the unit
circle generated by inner functions fixing the origin.

The first author thanks Phil Rippon and Gwyneth Stallard for many fruitful
discussions, and especially for pointing out reference [BS01]. The second
author is supported in part by the Generalitat de Catalunya (grant 2021 SGR
00071) and the Spanish Ministerio de Ciencia e Innovación (project
PID2021-123151NB-I00). Both authors acknowledge financial support from the
Spanish Research Agency through the María de Maeztu Program (CEX2020-001084-M)

##  1\. Introduction

Let $\mathbb{D}$ be the open unit disc of the complex plane and let
$g\colon\mathbb{D}\to\mathbb{D}$ be an analytic mapping with $g(0)=0$ which is
not a rotation. On one hand, the classical Denjoy-Wolff Theorem tells us that
the iterates $g^{n}=g\circ\ldots\circ g$ converge to $0$ uniformly on compact
subsets of $\mathbb{D}$ (see e.g. [CG93, p. 79]). On the other hand, let $m$
denote normalised Lebesgue measure on the unit circle $\partial\mathbb{D}$. An
analytic self-mapping $g$ of $\mathbb{D}$ is called _inner_ if

$\hat{g}(e^{i\theta})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{r\nearrow
1}g(re^{i\theta})$

exists and has modulus one for $m$-almost every point
$e^{i\theta}\in\partial\mathbb{D}$. In this case, one can investigate the
dynamics of the measurable boundary self-map
$\hat{g}\colon\partial\mathbb{D}\to\partial\mathbb{D}$, which is actually
defined at almost every point of $\partial\mathbb{D}$. If $g$ is an inner
function fixing the origin, Lowner’s Lemma tells us that $m$ is invariant
under $\hat{g}$, that is, $m({\hat{g}}^{-1}(E))=m(E)$ for any measurable set
$E\subset\partial\mathbb{D}$. If furthermore $g$ is not a rotation, it is well
known that the mapping $\hat{g}$ is exact and hence, mixing and ergodic. In
fact, dynamical properties of the boundary map of an inner function have been
extensively studied after the pioneering papers of Aaranson, Pommerenke,
Crazier, and Doering and Mañé [Aar78, Pom81, Cra91, DM91]. Mapping and
distortion properties of inner functions have been studied in [Ale86, Ale87,
FP92, FPR96, FMP07] and the surveys [PS06, Sak07], and several stochastic
properties can be found in [NSiG22, Nic22, IU23, AN23]. In many ways, these
papers highlight the beautiful interplay between the dynamical properties of
an inner function as a self-mapping of $\partial\mathbb{D}$ and those as a
self-mapping of $\mathbb{D}$.

The main purpose of this paper is to study ergodic and mixing properties of
non-autonomous dynamics of inner functions fixing the origin. This concerns
compositions of the form
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ
g_{n-1}\circ\cdots\circ g_{1}$, where each $g_{n}$, $n\in\mathbb{N}$, is an
inner function with $g_{n}(0)=0$. Non-autonomous dynamics of inner functions
appear naturally in complex dynamics when studying simply connected wandering
domains of entire functions – see e.g. [BEF+22, Section 2] or [Fer24, Lemma
3.2]. As introduced in [BEF+24], a sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ of
inner functions fixing the origin is called _contracting_ if $G_{n}\to 0$
uniformly on compact subsets of $\mathbb{D}$ as $n\to\infty$. The behaviour of
$G_{n}$ in $\mathbb{D}$ has been studied in [BEF+22, Section 2] and [Fer23],
where the following dichotomy has been proved.

######  Theorem A ([BEF+22] and [Fer23]).

Let $g_{n}\colon\mathbb{D}\to\mathbb{D}$ be inner functions fixing the origin,
and let
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$, $n\in\mathbb{N}$.

  1. (a)

The sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ is contracting if and only if
$\sum_{n\geq 1}(1-|g_{n}^{\prime}(0)|)=\infty$.

  2. (b)

Assume $\\{g_{n}\\}_{n\in\mathbb{N}}$ is not contracting. Then any (pointwise)
limit function in $\mathbb{D}$ of a subsequence of
$\\{G_{n}\\}_{n\in\mathbb{N}}$ is a non-constant inner function fixing the
origin, and any two limit functions $H_{1}$ and $H_{2}$ satisfy
$H_{1}=\lambda\cdot H_{2}$ for some $\lambda\in\partial\mathbb{D}$.

Much less has been said about non-autonomous dynamics of inner functions in
the unit circle. In this paper, we focus on ergodic and mixing properties of
non-autonomous dynamics of inner functions fixing the origin (for the
“opposite” case where $G_{n}(0)$ tends to $\partial\mathbb{D}$, see [BEF+24]
for recurrence properties of such non-autonomous systems). In the non-
autonomous setting, there exist measure-preserving sequences of
transformations which are mixing in the usual sense, but such that time
averages do not converge to the space average almost everywhere (see e.g.
[BS01]). In order to preserve the “mixing implies ergodicity” and the “mixing
is equivalent to ergodicity of any subsequence” maxims and properly generalise
the relevant concepts, we use the following definitions due to Berend and
Bergelson [BB84]. For $(X,\mathcal{A},\mu)$ a probability space, we (here and
henceforth) denote by $L^{2}(\mu)$ the standard Hilbert space of complex
valued measurable functions $\varphi$ defined on $X$ such that

$\|\varphi\|_{2}^{2}=\int_{X}|\varphi|^{2}d\mu<\infty.$

######  Definition 1.1.

Let $(X,\mathcal{A},\mu)$ be a probability space. Let $f_{n}\colon X\to X$ be
measurable, measure-preserving transformations and let
$T_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f_{n}\circ\cdots\circ
f_{1}$, $n\in\mathbb{N}$. We say that the sequence
$\\{T_{n}\\}_{n\in\mathbb{N}}$ is ergodic if

$\lim_{N\to\infty}\left\|\frac{1}{N}\sum_{n=1}^{N}\varphi\circ
T_{n}-\int_{X}\varphi\,d\mu\right\|_{2}=0,$

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

The sequence $\\{T_{n}\\}_{n\in\mathbb{N}}$ is called mixing if all its
subsequences are ergodic.

It is not hard to show that if $\\{T_{n}\\}_{n\in\mathbb{N}}$ is ergodic in
this sense, then there exist no non-trivial completely invariant sets.
Similarly, if $\\{T_{n}\\}_{n\in\mathbb{N}}$ is mixing in the sense of the
above definition, then it is also mixing in the usual sense, that is,
$\mu\left(A\cap T_{n}^{-1}(B)\right)\to\mu(A)\mu(B)$ as $n\to+\infty$, for any
pair of measurable sets $A,B\subset X$. Notice that the converse statements do
not hold, as evidenced by the examples in [BS01]. Moreover, the conditions
given in Definition 1.1 properly generalise the usual ones – more
specifically, if $T_{n}=f^{n}$ for some measure-preserving map $f\colon X\to
X$, then the sequence $\\{T_{n}\\}_{n\in\mathbb{N}}$ is ergodic (resp. mixing)
if and only if $f$ is ergodic (resp. mixing); see [BB84].

We now fix some notation. Given inner functions $g_{n}$ fixing the origin,
$n\geq 1$, we consider the inner functions

$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1},\qquad
G_{m}^{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{m+1},\,n>m\geq 0.$ (1)

Notice that $G_{0}^{n}=G_{n}$ and $G_{n-1}^{n}=g_{n}$ for any $n\geq 1$. We
say that the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic
(respectively mixing) if the corresponding condition in Definition 1.1 holds.
We can now state our first result.

######  Theorem 1.2.

Let $g_{n}$, $n\in\mathbb{N}$, be inner functions fixing the origin and let
$G_{n}$, $G_{m}^{n}$ be given by (1). The sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if and only if

$\lim_{N\to+\infty}\Re\left(\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right)=0,$
(2)

for any $\ell\in\mathbb{N}$.

Since $((G_{m}^{n})^{\prime}(0))^{\ell}$ are complex numbers, the double sum
in Condition (2) may have cancellations, so that (2) does not depend solely on
the modulus of $(G_{m}^{n})^{\prime}(0)$. With that in mind, in Section 4 we
give a necessary and a sufficient condition for ergodicity. Using these
conditions, in the extreme cases when $g_{n}^{\prime}(0)>0$ for every
$n\in\mathbb{N}$ or when $\\{g_{n}\\}_{n\in\mathbb{N}}$ is not contracting, we
obtain the following descriptions.

######  Theorem 1.3.

Let $g_{n}$, $n\in\mathbb{N}$, be inner functions fixing the origin and let
$G_{n}$, $G_{m}^{n}$ be given by (1).

  1. (i)

Assume $g_{n}^{\prime}(0)>0$ for all $n\geq 1$. Then, the sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if and only if

$\lim_{N\to\infty}\prod_{k=\left\lfloor
N(1-\varepsilon)\right\rfloor}^{N}g_{k}^{\prime}(0)=0,$ (3)

for any $0<\varepsilon<1$.

  2. (ii)

Assume $g_{n}^{\prime}(0)>0$ for all $n\geq 1$. Then, the sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing if and only if for every
$\epsilon>0$ there exists $M_{0}$ such that, for any $N>M>M_{0}$, we have

$\prod_{k=N-M}^{N}g_{k}^{\prime}(0)<\epsilon.$ (4)

  3. (iii)

Assume the sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ is not contracting. Then
the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if and only
if the sequence $\\{e^{i\arg G_{n}^{\prime}(0)}\\}_{n\in\mathbb{N}}$ is
equidistributed on $\partial\mathbb{D}$.

Theorem 1.3 deserves some comments.

First, when $g_{n}(z)=e^{i\theta}z$, $n\geq 1$, the statement (iii) tells us
that $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if and only if
$\theta$ is irrational – in other words, we recover the classical result of
ergodic theory that a rotation of the circle is ergodic if and only if it is
irrational. Condition (3) in (i) is related to the speed of convergence of
$G_{n}(z)$, $z\in\mathbb{D}$, to $0$. Roughly speaking, the faster $G_{n}$
tends to $0$ on $\mathbb{D}$, the more expanding $\widehat{G}_{n}$ is on
$\partial\mathbb{D}$; see [Mas13, Theorem 4.8]. Hence, condition (i) is
related to the classical existence of ergodic measures for expanding maps; for
related ideas and results, see e.g. [VO16, Chapter 11], [GS09], and even
[TPvS19] for a non-autonomous approach. Thus, roughly speaking, Theorem 1.3
says that in our setting there are only two mechanisms for ergodicity, and
they are the classical ones corresponding to “expanding maps” and “irrational
rotations”.

Second, Pommerenke [Pom81] showed that contracting sequences of inner
functions fixing the origin are mixing in the usual sense on
$\partial\mathbb{D}$, that is, $m\left(A\cap G_{n}^{-1}(B)\right)\to m(A)m(B)$
as $n\to+\infty$, for any pair of measurable sets
$A,B\subset\partial\mathbb{D}$. We discuss a converse to Pommerenke’s result
in Section 5.

Third, note that, by Theorem A, $\\{g_{n}\\}_{n\in\mathbb{N}}$ is contracting
if and only if

$\prod_{j=k}^{\infty}g_{j}^{\prime}(0)=0\text{ for all $k\in\mathbb{N}$.}$

Hence conditions (3) and (4) can be understood as quantitative versions of
contractibility. In particular, condition (3) outlined in Theorem 1.3(i)
implies that the sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ is contracting, but
is strictly stronger; in Section 6, we give an example of a contracting
sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ which does not satisfy (2), and is
therefore not ergodic in the sense of Definition 1.1. Notice that, by the
results of Pommerenke [Pom81] mentioned above, this sequence is also mixing in
the usual sense; we see that (as in [BS01]) a sequence can be mixing in the
usual sense, but not ergodic. From Theorem 1.3 we derive the following more
straightforward sufficient conditions for ergodicity and mixing.

######  Corollary 1.4.

Let $g_{n}$, $n\in\mathbb{N}$, be inner functions fixing the origin and let
$G_{n}$ be given by (1).

  1. (a)

Assume that for any $0<\varepsilon<1$ we have

$\lim_{N\to\infty}\prod_{k=\left\lfloor
N(1-\varepsilon)\right\rfloor}^{N}g_{k}^{\prime}(0)=0.$ (5)

Then the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic.

  2. (b)

Assume that for every $\epsilon>0$ there exists $M_{0}$ such that, for
$N>M>M_{0}$,

$\prod_{k=N-M}^{N}|g_{k}^{\prime}(0)|<\epsilon.$ (6)

Then the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing.

Combining this result and part (iii) of Theorem 1.3 we obtain that contracting
sequences are precisely those which have a mixing subsequence. Notice the
similarity to Theorem 5.2.

######  Corollary 1.5.

Let $g_{n}$, $n\in\mathbb{N}$, be inner functions fixing the origin and let
$G_{n}$ be given by (1). Then $\\{g_{n}\\}_{n\in\mathbb{N}}$ is contracting if
and only if $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ has a mixing subsequence.

Finally, we wish to discuss what our results mean for the recurrence of the
sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$. Let $(X,\mathcal{A},\mu)$
be a measure space. A sequence of measurable, measure-preserving
transformations $T_{n}\colon X\to X$ is recurrent if for any measurable set
$A\subset X$ we have that $T_{n}(x)\in A$ for infinitely many $n\geq 1$, for
$\mu$-almost every point $x\in A$. In autonomous dynamics, that is when
$T_{n}=T^{n}$, a classical theorem of Poincaré (see e.g. [VO16, Theorem
1.2.1]) tells us that if $\mu(X)<\infty$, then the sequence $\\{T^{n}\\}$ is
recurrent for any measure-preserving transformation $T$. However, this may
fail if the preserved measure is infinite, or if the system is non-autonomous;
see [DM91] and [BEF+24]. It turns out, however, that ergodicity in the sense
of Definition 1.1 is sufficient for recurrence. In fact, the following
stronger result, which is well-known in the autonomous case, holds.

######  Theorem 1.6.

Let $(X,\mathcal{A},\mu)$ be a probability space. Let $T_{n}\colon X\to X$,
$n\geq 1$, be measurable, measure-preserving transformations. If
$\\{T_{n}\\}_{n\in\mathbb{N}}$ has an ergodic subsequence, then:

  1. (i)

$\\{T_{n}\\}_{n\in\mathbb{N}}$ is recurrent;

  2. (ii)

If, furthermore, $\mathrm{supp}(\mu)=X$ and $X$ is a second countable
topological space, then $\mu$-almost every point has a dense orbit in $X$.

For other results on recurrence and topological transitivity of boundary
extensions of inner functions and compositions thereof, see [DM91] and
[BEF+24].

The paper is organized as follows. In Section 2 we collect some standard
definitions and classical results which will be used later. Section 3 is
devoted to the proof of Theorem 1.2. In Section 4 we present one necessary and
one sufficient condition for ergodicity, and use them to prove Theorem 1.3 and
Corollary 1.4. Section 5 is devoted to prove the converse of Pommerenke’s
result on mixing sequences of inner functions. Section 6 contains some
relevant examples and the proof of Corollary 1.5. Finally, Theorem 1.6 is
proved in Section 7.

##  2\. Preliminaries

###  2.1. Inner functions and the space $L^{2}(\partial\mathbb{D})$

Let $L^{2}(\partial\mathbb{D})$ be the usual Hilbert space of measurable
complex-valued functions $\varphi\colon\partial\mathbb{D}\to\mathbb{C}$ such
that

$\left\|\varphi\right\|_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\int_{\partial\mathbb{D}}|\varphi|^{2}\,dm\right)^{1/2}<+\infty,$

armed with the corresponding inner product

$\left\langle\varphi,\psi\right\rangle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\partial\mathbb{D}}\varphi\cdot\overline{\psi}\,dm=\int_{0}^{2\pi}\varphi(e^{i\theta})\cdot\overline{\psi(e^{i\theta})}\,\frac{d\theta}{2\pi}.$

Finite linear combinations of the trigonometric monomials

$e_{n}(e^{i\theta})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{in\theta},\quad
n\in\mathbb{Z},$

are dense in $L^{2}(\partial\mathbb{D})$. The boundary extension $\hat{g}$ of
an inner function $g\colon\mathbb{D}\to\mathbb{D}$ satisfies
$|\hat{g}(\xi)|=1$ for almost every $\xi\in\partial\mathbb{D}$. Cauchy’s
Formula tells us that the Fourier coefficients of $\hat{g}$ are precisely the
coefficients of the power series expansion of $g$ at the origin, that is,

$\left\langle\hat{g},e_{n}\right\rangle=\int_{0}^{2\pi}\hat{g}(e^{i\theta})e^{-ni\theta}\,\frac{d\theta}{2\pi}=\begin{cases}0,&n<0,\\\
\frac{g^{(n)}(0)}{n!},&n\geq 0.\end{cases}$ (7)

Let $\omega_{z}$ denote the harmonic measure on $\partial\mathbb{D}$ with
respect to the point $z\in\mathbb{D}$, defined as

$\omega_{z}(E)=\int_{E}\frac{1-|z|^{2}}{|\xi-z|^{2}}dm(\xi),\quad
E\subset\partial\mathbb{D}.$

We recall the following classical result (see [DM91, Corollary 1.5]):

######  Lemma 2.1 (Lowner’s Lemma).

Let $g\colon\mathbb{D}\to\mathbb{D}$ be an inner function. Then for any
$z\in\mathbb{D}$,

$\omega_{g(z)}(E)=\omega_{z}(g^{-1}(E)),\quad E\subset\partial\mathbb{D}.$

In particular, if $g(0)=0$ and $\varphi$ is an integrable function on
$\partial\mathbb{D}$, we have

$\int_{\partial\mathbb{D}}(\varphi\circ g)dm=\int_{\partial\mathbb{D}}\varphi
dm.$

###  2.2. Equidistribution of sequences on $\partial\mathbb{D}$

This is a much-studied notion with many connections. We recall the basic
definition:

######  Definition 2.2.

We say that a sequence $\\{z_{n}\\}_{n\in\mathbb{N}}\subset\partial\mathbb{D}$
is equidistributed on $\partial\mathbb{D}$ if for any arc
$S\subset\partial\mathbb{D}$, we have

$\lim_{n\to+\infty}\frac{\\#(\\{z_{1},z_{2},\ldots,z_{n}\\}\cap S)}{n}=m(S).$

The following classical characterisation, due to Weyl (see, for instance,
[KN74, Theorem 2.1]) will allow us to discuss equidistribution in our setting.

######  Lemma 2.3 (Weyl’s Criterion).

A sequence $\\{z_{n}\\}_{n\in\mathbb{N}}\subset\partial\mathbb{D}$ is
equidistributed in $\partial\mathbb{D}$ if and only if

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}(z_{n})^{\ell}=0,$

for any integer $\ell\geq 1$.

##  3\. A general characterisation of ergodicity

In this section, we prove Theorem 1.2. We start with the following elementary
lemma.

######  Lemma 3.1.

Let $g\colon\mathbb{D}\to\mathbb{D}$ be holomorphic with $g(0)=0$. Then, for
all $n\in\mathbb{N}$,

$\left(g^{n}\right)^{(n)}(0)=n!\left(g^{\prime}(0)\right)^{n}.$

###### Proof.

For $n>0$, we use the notation $O(|z|^{n})$ to denote a function $h$ defined
on $\mathbb{D}$ for which there exists a constant $C=C(h,n)>0$ such that
$|h(z)|\leq C|z|^{n}$ for any $|z|<1/2$. Since
$g(z)=g^{\prime}(0)z+O(|z|^{2})$, we have
$g(z)^{n}=g^{\prime}(0)^{n}z^{n}+O(|z|^{n+1})$ and the result follows by
uniqueness of the Taylor series.

∎

Next we show that suitable inner products on the circle can be written as
derivatives at the origin. As before, for $\ell\in\mathbb{Z}$, let $e_{\ell}$
denote the monomial $e_{\ell}(\xi)={\xi}^{\ell}$, $\xi\in\partial\mathbb{D}$.

######  Lemma 3.2.

Let $g_{n}$, $n\geq 1$, be inner functions fixing the origin and $G_{n}$,
$G_{m}^{n}$ be given by (1). Then, for any integers $n>m\geq 0$ and
$\ell\in\mathbb{N}$,

$\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle=\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}.$

###### Proof.

Fix the integers $n>m\geq 0$ and $\ell\in\mathbb{N}$. By Lemma 2.1, we have

$\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle=\left\langle
e_{\ell}\circ\widehat{G}_{m}^{n},e_{\ell}\right\rangle.$

The right-hand side can be read as the $\ell$-th Fourier coefficient of the
boundary map of the holomorphic self-map of $\mathbb{D}$ given by
$z\mapsto\left(G_{m}^{n}(z)\right)^{\ell}$, and so by (7) we get

$\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle=\frac{\left((G_{m}^{n})^{\ell}\right)^{(\ell)}(0)}{\ell!}.$

The conclusion follows by applying Lemma 3.1 to the right-hand side. ∎

We are ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Since trigonometric polynomials are dense in $L^{2}(\partial\mathbb{D})$, it
is sufficient to show that

$\lim_{N\to\infty}\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}^{2}=0,\quad\ell\in\mathbb{Z}\setminus\\{0\\}.$

Since $e_{-\ell}=\overline{e_{\ell}}$, we can assume $\ell\geq 1$. Fix
$\ell\geq 1$. We have

$\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}^{2}=\frac{1}{N^{2}}\sum_{n=1}^{N}\left\|e_{\ell}\circ\widehat{G}_{n}\right\|_{2}^{2}+2\Re\left(\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle\right).$

Since $|e_{\ell}\circ\widehat{G}_{n}|=1$ $m$-almost everywhere on
$\partial\mathbb{D}$ for any $n\geq 1$, we obtain

$\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}^{2}=\frac{1}{N}+2\Re\left(\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle\right).$

Applying Lemma 3.2, we have $\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle=\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}$,
which finishes the proof. ∎

##  4\. Ergodicity in different scenarios

The proof of Theorem 1.3 is based on the following result.

######  Theorem 4.1.

With the same notation as Theorem 1.2, the following hold:

  1. (a)

The sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if, for all
$\ell\in\mathbb{N}$,

$\lim_{N\to+\infty}\frac{1}{N}\sum_{m=1}^{N-1}\left((G_{m}^{N})^{\prime}(0)\right)^{\ell}=0.$
(8)

  2. (b)

If the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic, then

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}=0,$
(9)

for every pair of integers $m\geq 0$ and $\ell\geq 1$.

###### Proof of Theorem 4.1.

We will prove (a) using Theorem 1.2. We start by noting that

$\Re\left(\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right)\leq\left|\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right|.$

We rewrite the sum on the right-hand side as

$\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}=\frac{1}{N^{2}}\sum_{n=2}^{N}\sum_{m=1}^{n-1}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell},$

and applying the triangle inequality yields

$\left|\frac{1}{N^{2}}\sum_{m=1}^{N}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right|\leq\frac{1}{N^{2}}\sum_{n=2}^{N}\left|\sum_{m=1}^{n-1}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right|\leq\frac{1}{N}\sum_{n=2}^{N}\left|\frac{1}{n}\sum_{m=1}^{n-1}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\right|.$

The right-hand side is now the Cesàro sum of a sequence that, by hypothesis,
goes to zero. Therefore condition (2) of Theorem 1.2 is satisfied. This
completes the proof of (a).

We now prove (b), which is actually independent of Theorem 1.2. Fix integers
$m\geq 0$ and $\ell\geq 1$. We have by the Cauchy–Schwarz inequality that

$\left\langle\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle\leq\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}\left\|e_{\ell}\circ\widehat{G}_{m}\right\|_{2},$

and since $|e_{\ell}\circ\widehat{G}_{m}|=1$ at $m$-almost every point of the
unit circle, this becomes

$\left\langle\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle\leq\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}.$
(10)

For $0\leq m<N$, we have

$\left\langle\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle=\frac{1}{N}\left(\sum_{n=1}^{m}\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle+\sum_{n=m+1}^{N}\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle\right).$

Applying Lemma 3.2 and plugging everything back into (10), we obtain

$\frac{1}{N}\sum_{n=1}^{m}\left\langle
e_{\ell}\circ\widehat{G}_{n},e_{\ell}\circ\widehat{G}_{m}\right\rangle+\frac{1}{N}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}\leq\left\|\frac{1}{N}\sum_{n=1}^{N}e_{\ell}\circ\widehat{G}_{n}\right\|_{2}.$

If we assume that $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic, the
right-hand side now goes to zero as $N\to+\infty$; the first sum on the left-
hand side depends only on $m$ and $\ell$, and thus when divided by $N$ goes to
zero as $N\to+\infty$. It follows that

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=m+1}^{N}\left((G_{m}^{n})^{\prime}(0)\right)^{\ell}=0.$

∎

We will also need the following observation about Cesàro sums.

######  Lemma 4.2.

Let $\\{z_{n}\\}_{n\in\mathbb{N}}$ and $\\{w_{n}\\}_{n\in\mathbb{N}}$ be two
sequences of complex numbers. Assume that $\sup_{n}|z_{n}|<\infty$ and that
$\lim_{n\to\infty}w_{n}=w\in\mathbb{C}\setminus\\{0\\}$. Then,

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}z_{n}w_{n}=w\cdot\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}z_{n}.$

The equality is strong in the sense that either both limits exist and the
identity holds, or neither limit exists.

###### Proof.

The proof easily follows from the observation

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}z_{n}(w_{n}-w)=0.$

∎

We are ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

We assume now that $g_{n}^{\prime}(0)>0$ for all $n\geq 1$. To show that (3)
implies ergodicity, we will show that it implies the sufficient condition (8)
given in Theorem 4.1(a). First, notice that since now
$(G_{m}^{N})^{\prime}(0)\in(0,1)$, it suffices to show that the condition (8)
is satisfied for $\ell=1$. Fixed $\varepsilon>0$, we decompose the sum in
question as

$\frac{1}{N}\sum_{m=1}^{N-1}(G_{m}^{N})^{\prime}(0)=\frac{1}{N}\sum_{m=1}^{\left\lfloor
N(1-\varepsilon)\right\rfloor}\prod_{k=m+1}^{N}g_{k}^{\prime}(0)+\frac{1}{N}\sum_{m=\left\lfloor
N(1-\varepsilon)\right\rfloor+1}^{N-1}(G_{m}^{N})^{\prime}(0).$

For the first sum on the right-hand side, note that, since $m\leq\left\lfloor
N(1-\varepsilon)\right\rfloor$ and $g_{n}^{\prime}(0)\in(0,1)$, we have

$\prod_{k=m+1}^{N}g_{k}^{\prime}(0)\leq\prod_{k=\left\lfloor
N(1-\varepsilon)\right\rfloor+1}^{N}g_{k}^{\prime}(0).$

For the second sum, note that each $(G_{m}^{N})^{\prime}(0)$ is less than one,
and the sum itself has at most $N-1-(\left\lfloor
N(1-\varepsilon)\right\rfloor+1)\leq\varepsilon N$ terms. Thus,

$\frac{1}{N}\sum_{m=1}^{N-1}(G_{m}^{N})^{\prime}(0)\leq(1-\varepsilon)\prod_{k=\left\lfloor
N(1-\varepsilon)\right\rfloor+1}^{N}g_{k}^{\prime}(0)+\varepsilon.$

By assumption, the first term on the right-hand side goes to zero as
$N\to+\infty$, and since $\varepsilon>0$ was arbitrary it follows that

$\lim_{N\to+\infty}\frac{1}{N}\sum_{m=1}^{N-1}(G_{m}^{N})^{\prime}(0)=0.$

Ergodicity follows by Theorem 4.1(a). To show the necessity of condition (3),
we use the characterisation of ergodicity given in Theorem 1.2. Assume that
there exist constants $\varepsilon>0$, $c>0$ and a subsequence
$\\{N_{k}\\}_{k\in\mathbb{N}}$ of positive integers such that

$\prod_{j=\left\lfloor
N_{k}(1-\epsilon)\right\rfloor+1}^{N_{k}}g_{j}^{\prime}(0)\geq c>0,\quad
k\in\mathbb{N}.$

We will show that condition (2) of Theorem 1.2 fails for $\ell=1$. Since all
the derivatives are positive and
$(G_{m}^{n})^{\prime}(0)\geq(G_{m}^{N_{k}})^{\prime}(0)$ for any $0\leq
m<n\leq N_{k}$, we get

$\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}\sum_{n=m+1}^{N_{k}}(G_{m}^{n})^{\prime}(0)\geq\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}\sum_{n=m+1}^{N_{k}}(G_{m}^{N_{k}})^{\prime}(0)=\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}(N_{k}-m)(G_{m}^{N_{k}})^{\prime}(0).$

Since all terms in the sum are positive, we have

$\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}\sum_{n=m+1}^{N_{k}}(G_{m}^{n})^{\prime}(0)\geq\frac{1}{N_{k}^{2}}\sum_{m=\left\lfloor
N_{k}(1-\varepsilon)\right\rfloor+1}^{N_{k}-1}(N_{k}-m)(G_{m}^{N_{k}})^{\prime}(0).$

Now, for $N_{k}>m>\left\lfloor N_{k}(1-\epsilon)\right\rfloor$, we once again
have

$(G_{m}^{N_{k}})^{\prime}(0)\geq(G_{\left\lfloor
N_{k}(1-\varepsilon)\right\rfloor}^{N_{k}})^{\prime}(0)=\prod_{j=\left\lfloor
N_{k}(1-\varepsilon)\right\rfloor+1}^{N_{k}}g_{j}^{\prime}(0)\geq c>0,$

and therefore

$\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}\sum_{n=m+1}^{N_{k}}(G_{m}^{n})^{\prime}(0)\geq\frac{c}{N_{k}^{2}}\sum_{m=\left\lfloor
N_{k}(1-\varepsilon)\right\rfloor+1}^{N_{k}-1}(N_{k}-m).$

The sum on the right-hand side is of order $N_{k}^{2}$. We conclude that

$\liminf_{k\to+\infty}\frac{1}{N_{k}^{2}}\sum_{m=1}^{N_{k}-1}\sum_{n=m+1}^{N_{k}}(G_{m}^{n})^{\prime}(0)>0,$

meaning that condition (2) fails. This completes the proof of (i).

To prove (ii), note that $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing if
and only if $\\{\widehat{G}_{n_{k}}\\}_{k\in\mathbb{N}}$ is ergodic for any
subsequence of positive integers $\\{n_{k}\\}_{k\in\mathbb{N}}$. Since
$G_{n_{k}}$ corresponds to the non-autonomous dynamics of the inner functions
$\tilde{g_{k}}=g_{n_{k}}\circ\ldots\circ g_{n_{k-1}+1}$, part (i) yields that
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing if and only if for any
$0<\varepsilon<1$, we have

$\prod_{k=\left\lfloor
N(1-\varepsilon)\right\rfloor}^{N}\prod_{j=n_{k-1}+1}^{n_{k}}g_{j}^{\prime}(0)\to
0,$

as $N\to\infty$, for any subsequence of positive integers
$\\{n_{k}\\}_{k\in\mathbb{N}}$. This last statement is equivalent to (4).

To prove (iii), we assume that $\sum(1-|g_{n}^{\prime}(0)|)<\infty$ (which, by
Theorem A, is equivalent to the sequence $\\{g_{n}\\}_{n\in\mathbb{N}}$ not
being contracting). In this case, we can assume (by discarding finitely many
$g_{n}$) that $g_{n}^{\prime}(0)\neq 0$ for all $n\in\mathbb{N}$. Since the
sequence $\\{|G_{n}^{\prime}(0)|\\}_{n\in\mathbb{N}}$ is decreasing and by
assumption is bounded away from $0$, it has a positive limit $c>0$ as
$n\to+\infty$. We now assume that $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is
ergodic and invoke Theorem 4.1(b) with $m=0$, obtaining

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}(G_{n}^{\prime}(0))^{\ell}=0$

for every $\ell\in\mathbb{N}$. Lemma 4.2 now says that

$0=\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}|G_{n}^{\prime}(0)|^{\ell}(e^{i\arg
G_{n}^{\prime}(0)})^{\ell}=c^{\ell}\cdot\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}e^{i\ell\arg
G_{n}^{\prime}(0)},\quad\ell\in\mathbb{N},$

and it follows from Weyl’s criterion that the sequence $\\{e^{i\arg
G_{n}^{\prime}(0)}\\}_{n\in\mathbb{N}}$ is equidistributed in
$\partial\mathbb{D}$.

Conversely, assume that $\\{e^{i\arg G_{n}^{\prime}(0)}\\}_{n\in\mathbb{N}}$
is equidistributed in $\partial\mathbb{D}$. Using the chain rule, we rewrite
$(G_{m}^{n})^{\prime}(0)$ as
$(G_{m}^{n})^{\prime}(0)=G_{n}^{\prime}(0)\cdot(G_{m}^{\prime}(0))^{-1}$, so
that the sum in equation (8) of Theorem 4.1(a) becomes

$\frac{1}{N}\sum_{m=1}^{N}((G_{m}^{N})^{\prime}(0))^{\ell}=\frac{G_{N}^{\prime}(0)^{\ell}}{N}\sum_{m=1}^{N}(G_{m}^{\prime}(0))^{-\ell}=\frac{G_{N}^{\prime}(0)^{\ell}}{N}\sum_{m=1}^{N}\frac{1}{|G_{m}^{\prime}(0)|^{\ell}}e^{-i\ell\arg
G_{m}^{\prime}(0)},\quad\ell\in\mathbb{N}.$

By Lemma 4.2, we get

$\lim_{N\to+\infty}\frac{1}{N}\sum_{m=1}^{N}((G_{m}^{N})^{\prime}(0))^{\ell}=\lim_{N\to+\infty}\frac{1}{N}\sum_{m=1}^{N}e^{-i\ell\arg
G_{m}^{\prime}(0)},\quad\ell\in\mathbb{N}$

Since $\\{e^{i\arg G_{n}^{\prime}(0)}\\}_{n\in\mathbb{N}}$ is equidistributed
on $\partial\mathbb{D}$, then by Weyl’s criterion this limit is zero, whence
ergodicity follows by Theorem 4.1(a). ∎

###### Proof of Corollary 1.4.

The proof mimics the previous argument and we only sketch it. For part (a) we
need to show that (5) implies the sufficient condition (8) given in Theorem
4.1(a). This follows as in the proof of part (a) of Theorem 1.3 once
triangular inequality is applied. Part (b) follows similarly. ∎

##  5\. Mixing in the usual sense

Recall that a sequence $\\{T_{n}\\}_{n\in\mathbb{N}}$ of transformations of
the measure space $(X,\mathcal{A},\mu)$ is mixing (in the usual sense) if, for
all measurable sets $A,B\subset X$,

$\mu\left(A\cap T_{n}^{-1}(B)\right)\to\mu(A)\mu(B)\text{ as $n\to+\infty$.}$

If $T_{n}=T^{n}$ and $\mu$ is finite, this implies that $\\{T^{n}\\}$ is
ergodic (see e.g. [VO16, Proposition 4.1.3]). However, if the system is non-
autonomous, this implication can fail drastically – see e.g. [BS01].
Nevertheless, it can be interesting (and useful; see [BEF+24, Theorem 7.4]) to
study “classical mixing” for compositions of inner functions.

In this vein, as previously mentioned, Pommerenke [Pom81] already showed that,
if $g_{n}\colon\mathbb{D}\to\mathbb{D}$ are inner functions fixing the origin
and the composition
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$ tends to zero locally uniformly in $\mathbb{D}$ as $n\to+\infty$, then
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing in the usual sense (in
fact, Pommerenke showed the stronger fact that
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is “exact in the usual sense”). Here,
we give a converse to his result for non-autonomous dynamics. The proof relies
on the following consequence of mixing (see [VO16, Corollary 7.1.14]), whose
short proof is included for completeness.

######  Lemma 5.1.

Let $(X,\mathcal{A},\mu)$ be a probability space. Let $f_{n}\colon X\to X$ be
measurable, measure-preserving transformations. Assume that the sequence
$F_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f_{n}\circ\cdots\circ
f_{1}$ is mixing in the usual sense. Let $\nu$ be a probability measure on $X$
which is absolutely continuous with respect to $\mu$. Then,

$\lim_{n\to\infty}\nu\left(F_{n}^{-1}(B)\right)=\mu(B)$

for any measurable set $B\subset X$.

###### Proof.

Let $\varphi$ denote the Radon-Nykodim derivative of $\nu$ relative to $\mu$,
and let $\mathbf{1}_{B}$ denote the indicator function of the measurable set
$B\subset X$. Since $\varphi$ can be approximated in $L^{1}(\mu)$ by linear
combinations of characteristic functions, the assumption that
$\\{F_{n}\\}_{n\in\mathbb{N}}$ is mixing gives that

$\int_{X}(\mathbf{1}_{B}\circ
F_{n})\cdot\varphi\,d\mu\to\int_{X}\mathbf{1}_{B}\,d\mu\int_{X}\varphi\,d\mu$

as $n\to+\infty$. The left-hand side is, by the Radon-Nykodim Theorem, equal
to $\nu(F_{n}^{-1}(B))$, while the right-hand side is equal to $\mu(B)$ since
$\nu$ is a probability measure. ∎

######  Theorem 5.2.

Let $g_{n}\colon\mathbb{D}\to\mathbb{D}$ be inner functions fixing the origin,
and let
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$, $n\geq 1$. Then, the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$
is mixing in the usual sense if and only if $\\{g_{n}\\}_{n\in\mathbb{N}}$ is
contracting.

###### Proof of Theorem 5.2.

As mentioned before, Pommerenke ([Pom81]) proved that
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing if $G_{n}$ tend to $0$
uniformly on compacts of $\mathbb{D}$. Conversely, assume that
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing but at the same time
$G_{n}\to G$ pointwise in $\mathbb{D}$, where $G$ is a non-constant inner
function. Now, take $z\in\mathbb{D}\setminus G^{-1}(0)$. Since the harmonic
measure $\omega_{z}$ is absolutely continuous with respect to Lebesgue
measure, Lemma 5.1 gives that

$\lim_{n\to\infty}\omega_{z}\left(\widehat{G}_{n}^{-1}(B)\right)=m(B)$

for every measurable set $B\subset\partial\mathbb{D}$. However, by Lemma 2.1,
we have

$\omega_{z}\left(\widehat{G}_{n}^{-1}(B)\right)=\omega_{G_{n}(z)}(B),$

for any measurable set $B\subset\partial\mathbb{D}$. Since $G_{n}(z)\to G(z)$
we have $\omega_{z}\left(\widehat{G}_{n}^{-1}(B)\right)\to\omega_{G(z)}(B)$ as
$n\to\infty$, for any measurable set $B\subset\partial\mathbb{D}$. Since
$G(z)\neq 0$, there exists a measurable set $B\subset\partial\mathbb{D}$ with
$w_{G(z)}(B)\neq m(B)$ and we obtain a contradiction, concluding the proof. ∎

##  6\. Examples and counterexamples

In this section, we apply the various necessary and sufficient conditions
obtained above to illustrate what ergodic and non-ergodic compositions of
inner functions may look like. We start with the example promised in Section 1
of a sequence that is contracting but not ergodic. This example also serves to
show that the necessary condition given in Theorem 4.1(b) cannot be
sufficient.

######  Proposition 6.1.

There exists a sequence $g_{n}\colon\mathbb{D}\to\mathbb{D}$ of inner
functions fixing the origin such that the sequence
$\\{G_{n}\\}_{n\in\mathbb{N}}$ generated by
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$ satisfies the following conditions:

  1. (1)

$G_{n}\to 0$ locally uniformly in $\mathbb{D}$;

  2. (2)

$\\{G_{n}\\}_{n\in\mathbb{N}}$ satisfies the necessary condition (9) in
Theorem 4.1(b);

  3. (3)

$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is not ergodic.

###### Proof.

Let $g_{n}\colon\mathbb{D}\to\mathbb{D}$ be the Blaschke product of degree $2$
given by

$g_{n}(z)=z\cdot\frac{z+a_{n}}{1+a_{n}z},$

with $a_{n}=n/(n+1)$. An immediate calculation shows that
$g_{n}^{\prime}(0)=a_{n}$, and so

$\sum_{n\geq 1}(1-|g_{n}^{\prime}(0)|)=\sum_{n\geq 1}\frac{1}{n+1}=\infty,$

whence $G_{n}=g_{n}\circ\cdots\circ g_{1}$ converges locally uniformly to zero
by Theorem A. Furthermore, by the chain rule, we have

$(G_{m}^{n})^{\prime}(0)=\prod_{k=m+1}^{n}\frac{k}{k+1}=\frac{m+1}{n+1},$ (11)

and so for any fixed natural numbers $\ell$ and $m$, the sequence
$((G_{m}^{n})^{\prime}(0))^{\ell}$ goes to zero as $n\to+\infty$. The
necessary condition (9) in Theorem 4.1(b) is now satisfied, since it becomes
the Cesàro sum of a sequence going to zero. We finally show that
$\\{\widehat{G}_{n}\\}$ is not ergodic. Note that $m+1\geq(n+1)/2$ if $N/2\leq
m<n\leq N$. Hence (11) gives

$\frac{1}{N^{2}}\sum_{m=1}^{N-1}\sum_{n=m+1}^{N}(G_{m}^{n})^{\prime}(0)\geq\frac{1}{2N^{2}}\sum_{m=\left\lfloor
N/2\right\rfloor}^{N-1}(N-m),$

which does not tend to $0$ as $N\to\infty$. Hence condition (2) for $\ell=1$
in Theorem 1.2 is not satisfied and consequently
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is not ergodic. ∎

Next, we use Theorem 1.3 to provide several explicit examples of mixing and
ergodic compositions of inner functions.

######  Corollary 6.2.

Let $g_{n}\colon\mathbb{D}\to\mathbb{D}$ be inner functions fixing the origin
and let
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$, $n\geq 1$.

  1. (i)

Assume $\sum(1-|g_{n}^{\prime}(0)|)=\infty$. Then
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ has a mixing subsequence.

  2. (ii)

If, furthermore, there exist constants $0<\lambda<1$, $0<c<1$ and $M_{0}>0$
such that for any $a,b\in\mathbb{N}$ with $b-a>M_{0}$ one has

$\\#\\{n\in[a,b]:|g_{n}^{\prime}(0)|\leq\lambda\\}\geq c(b-a),$ (12)

then the sequence $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing.

  3. (iii)

Assume $\sum(1-|g_{n}^{\prime}(0)|)<\infty$. Then:

    1. (a)

If $\arg g_{n}^{\prime}(0)\to\theta$ as $n\to+\infty$ for some
$\theta\in\mathbb{R}\setminus\mathbb{Q}$, then the sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic.

    2. (b)

If the arguments $\theta_{n}=\arg g_{n}^{\prime}(0)$ are independently and
identically distributed according to some non-atomic distribution on
$\partial\mathbb{D}$, then $\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic
with probability 1.

###### Proof.

Assume $\sum(1-|g_{n}^{\prime}(0)|)=\infty$. Then there exists an increasing
sequence $\\{N_{k}\\}_{k\in\mathbb{N}}$ of positive integers such that

$\prod_{j=N_{k}+1}^{N_{k+1}}|g_{j}^{\prime}(0)|\leq 1/2,\quad k=1,2,\ldots.$

Since

$\prod_{k=M}^{N}\prod_{j=N_{k}+1}^{N_{k+1}}|g_{j}^{\prime}(0)|\leq 1/2^{N-M},$

part (b) of Corollary 1.4 gives that the subsequence
$\\{\widehat{G}_{N_{k}}\\}_{k\in\mathbb{N}}$ is mixing.

Assume now that condition (12) holds. Then

$\prod_{j=M}^{N}|g_{j}^{\prime}(0)|\leq\lambda^{c(N-M)}$

if $N-M\geq M_{0}$, whence by part (b) of Corollary 1.4 the sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is mixing.

Now, assume that $G_{n}\not\to 0$ locally uniformly on $\mathbb{D}$ (which,
recall, is equivalent to $\sum_{n\geq 1}(1-|g_{n}^{\prime}(0)|)<\infty$ by
Theorem A), whence by Theorem 1.3(iii) the sequence
$\\{\widehat{G}_{n}\\}_{n\in\mathbb{N}}$ is ergodic if and only if
$\\{e^{i\arg G_{n}^{\prime}(0)}\\}_{n\in\mathbb{N}}$ is equidistributed on
$\partial\mathbb{D}$. Thus, we only need to check that the conditions outlined
in (a) and (b) imply equidistribution. That (a) does is an immediate
consequence of a theorem by van der Corput (see [KN74, Theorem 3.3]). On the
other hand, (b) implies equidistribution almost surely by a result of Robbins
[Rob53, Theorem 2], which says that sums of independent and identically
distributed random variables drawn from a non-atomic distribution are
equidistributed with probability one. ∎

We can now prove Corollary 1.5.

###### Proof of Corollary 1.5.

That contracting implies having a mixing subsequence is part (a) of the
previous result, and the converse follows from part (iii) of Theorem 1.3. ∎

##  7\. Recurrence

In this section, we prove Theorem 1.6.

###### Proof of Theorem 1.6.

We can assume without loss of generality that the sequence
$\\{T_{n}\\}_{n\in\mathbb{N}}$ is ergodic.

To show recurrence, let $A\subset X$ be a measurable set with $\mu(A)>0$. If
$\mathbf{1}_{A}$ is the characteristic function of $A$, then by ergodicity the
functions

$x\mapsto\frac{1}{N}\sum_{n=1}^{N}\mathbf{1}_{A}\circ T_{n}(x)$

converge to the constant function $\mu(A)$ in $L^{2}(\mu)$. Since a sequence
converging in $L^{2}(\mu)$ admits a subsequence converging $\mu$-almost
everywhere, recurrence follows.

Next, assume that $\mathrm{supp}(\mu)=X$, and let
$\\{U_{k}\\}_{k\in\mathbb{N}}$ be a countable basis for the topology of $X$.
For $k\in\mathbb{N}$, denote by $\tilde{U}_{k}$ the set of points $x\in X$
such that $T_{n}(x)\in U_{k}$ for only finitely many $n\geq 1$. Then, clearly,

$\frac{1}{N}\sum_{n=1}^{N}\mathbf{1}_{U_{k}}\circ T_{n}(x)\to 0\text{ as
$N\to+\infty$}$

for all $x\in\tilde{U}_{k}$. Hence, no subsequence of the time averages at
$x\in\tilde{U}_{k}$ converge to $\mu(U_{k})>0$, and so (by ergodicity)
$\tilde{U}_{k}$ must have measure zero. The set

$\tilde{U}=\bigcup_{k\in\mathbb{N}}\tilde{U}_{k}$

also has measure zero, and it is clear that any point $x\in
X\setminus\tilde{U}$ has a dense orbit. ∎

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