# Information geometry and synchronization phase transition in Kuramoto model

Artem Alexandrov<EMAIL_ADDRESS>Moscow Institute of Physics and
Technology, Dolgoprudny, 141700, Russia  Laboratory of complex networks, Brain
and Consciousness Research Center, Moscow, Russia  Alexander Gorsky  Moscow
Institute of Physics and Technology, Dolgoprudny, 141700, Russia  Institute
for Information Transmission Problems, Moscow, 127994, Russia  Laboratory of
complex networks, Brain and Consciousness Research Center, Moscow, Russia

###### Abstract

We discuss how the synchronization in Kuramoto model can be treated in terms
of information geometry. We argue that the Fisher information is sensitive to
synchronization transition, specifically components of Fisher metric diverge
at the critical point. Our approach is based on the recently proposed relation
between Kuramoto model and geodesics on hyperbolic space.

##  I Introduction

Kuramoto model is the paradigmatic model for synchronization phenomena in non-
linear systems. Despite it deceptively simple form, there are still enough
white spots, concerning different aspects of the model. Whereas some of these
aspects turn around the synchronization in systems with complicated topology,
which were comprehensively described in Rodrigues  _et al._ (2016). Another
aspects are mostly focused on the rigorous description of the Kuramoto model
with all-to-all couplings (complete graph case). It was shown that in case of
identical oscillators the Kuramoto model exhibits low-dimensional dynamics.
The emergent dimensional reduction occurs due to existence of integrals of
motion, initially discovered by trial and error in the papers Watanabe and
Strogatz (1993, 1994). The investigation of such integrals of motion was done
comprehensively in the work Marvel  _et al._ (2009) where it was demonstrated
that they emerge due to the invariance of dynamics under the Möbius
transformation. The Kuramoto model dynamics can be represented as the motion
on the Möbius group orbit. Moreover, recently it was noticed that the
corresponding low-dimensional dynamics can be described in terms of gradient
flows on two dimensional hyperbolic manifold Chen  _et al._ (2017).

However, the origin of hyperbolic manifold was not established. At first
glance, this manifold simply inherits symmetries of Möbius group and there are
no special features. We shall interpret the hyperbolic manifold as induced by
probability measures, which is usually refers as statistical manifold (see
Amari (2016) for the review). In particular the AdS2 metric emerges
immediately for the shifted Gaussian distribution. In a generic situation the
hyperbolic metric emerges as the Fisher metric on the parameter space. The
Fisher metric is the probabilistic counterpart of the quantum metric which is
familiar in the quantum mechanics where one introduces the so-called complex
quantum tensor which has the quantum metric as the real part and the Berry
curvature as the imaginary part Provost and Vallee (1980); Berry (1984). It is
built from the quantum wave function depending on some parameter space however
its classical analogue exists as well.

In this study we focus on the hyperbolic geometry description of the Kuramoto
model and its connection to information geometry. In particular we explain the
hyperbolic geometry involved as the geometry of the statistical manifold. We
shall argue that the equation of motion for Kuramoto model has the form of
gradient flow for the Kullback-Leibler divergence. The properties of the
induced metric or, in other words, the response of the system at the
perturbation of parameters serves as a indicator of the phase transition
Zanardi and Paunković (2006); Zanardi  _et al._ (2007a); You  _et al._ (2007);
Garnerone  _et al._ (2009). In particular the diagonal elements of the quantum
metric provide the good tool for the search of the critical curves which has
been demonstrated in several examples Kolodrubetz  _et al._ (2013, 2017);
Pandey  _et al._ (2020); Buonsante and Vezzani (2007); Dey  _et al._ (2012).
We shall demonstrate in a clear-cut manner that the components of the Fisher
information metric identify in a similar manner the critical curve for the
phase transition in the Kuramoto-Shakaguchi model. Hence the Fisher
information metric yields the proper order parameter for the synchronization
phase transition.

The paper is organized as follows. In section II we recall the Kuramoto model
focusing on its description in terms of the Möbius group. The dimensional
reduction provides the description of the model in terms of motion of a point
at a hyperbolic disc. In section III we shall present the key aspects of the
information geometry. In section IV we apply the information geometry to the
Kuramoto and Kuramoto-Shakaguchi models. We shall treat the coordinates of
point on the disc as the two parameters which the distribution involved depend
on. It turns out that the Cauchy distribution governs the Kuramoto model while
the von Mises distribution is relevant for Kuramoto-Shakaguchi model. In both
cases we shall demonstrate that the singularity of the Fisher metric coincides
with the synchronization transition adding a new example of the relation
between the singularities of the Fisher metric and the phase transitions. In
section V we formulate the results and the open questions. The general
comments concerning the relations between the distributions and the possible
role of the $q$-Gaussian distributions in Kuramoto model can be found in the
Appendices.

##  II Brief review of Kuramoto model

Kuramoto model on a complete graph with identical oscillators has the
following equations of motion,

$\dot{\theta}_{i}=\omega+\frac{\lambda}{N}\sum_{j=1}^{N}\sin\left(\theta_{j}-\theta_{i}\right),$
(1)

where $N$ is the number of oscillators, $\theta_{i}\in[0,2\pi)$, $\lambda>0$
is the coupling constant, and $\omega$ is the oscillator eigenfrequency. The
order parameter (synchronization measure) is defined by

$r(t)=\frac{1}{N}\sum_{j=1}^{N}z_{j}(t),$ (2)

where we introduce complex variable $z_{i}(t)=\exp(i\theta_{i}(t))$. The model
exhibits so called _synchronization transition_ from a given initial state to
a synchronized state with the amplitude of order parameter $|r|\rightarrow 1$,
which does not depend on time. In terms of particles on the unit circle, it
means that in the synchronized state all the particles are concentrated at one
point that moves with a certain average frequency.

The equations of motion become

$\dot{z}_{i}=i\omega
z_{i}+\frac{\lambda}{2}\left(r-\overline{r}z_{i}^{2}\right).$ (3)

The evolution of $z_{i}=z_{i}(t)$ can be represented as the action of Möbius
transformation,

$z_{i}(t)=\mathcal{M}(z_{i}^{0}),\quad z_{i}^{0}\equiv\exp(i\theta_{i}(t=0)),$
(4)

where the parameters of Möbius transformation depend on time. Following the
paper Chen  _et al._ (2017), this transformation can be parametrized as
follows,

$\mathcal{M}(z)=\zeta\frac{z-w}{1-\overline{w}z},$ (5)

where $\zeta\in\mathbb{C}$, $|\zeta|=1$, $w\in\mathbb{C}$, $|w|<1$. The time
derivative of the phase is

$\dot{z}_{i}=-\frac{\dot{w}\zeta}{1-|w|^{2}}+\left(\dot{\zeta}\overline{\zeta}+\frac{\dot{\overline{w}}w-\dot{w}\overline{w}}{1-|w|^{2}}\right)z_{i}+\\\
+\frac{\dot{\overline{w}}\overline{\zeta}}{1-|w|^{2}}z_{i}^{2}.$ (6)

Matching the RHS of eq. (6) and the RHS of eq. (3), we obtain the system of
equations,

$\begin{gathered}\dot{w}=-\frac{\lambda}{2}\left(1-\left|w\right|^{2}\right)\overline{\zeta}r,\\\
\dot{\zeta}=i\omega\zeta-\frac{\lambda}{2}\left(\overline{w}r-w\overline{r}\zeta^{2}\right).\end{gathered}$
(7)

The key feature of these equations is that they are decoupled

$\overline{\zeta}r=\overline{\zeta}\frac{1}{N}\sum_{j=1}^{N}\zeta\frac{z_{j}^{0}-w}{1-\overline{w}z_{j}^{0}}=\\\
=\left|\zeta\right|^{2}\frac{1}{N}\sum_{j=1}^{N}\frac{z_{j}^{0}-w}{1-\overline{w}z_{j}^{0}}=\frac{1}{N}\sum_{j=1}^{N}\frac{z_{j}^{0}-w}{1-\overline{w}z_{j}^{0}}.$
(8)

hence it means that the equation for $w$ does not feel the variable $\zeta$.
This equation is our object of interest and it was shown previously Chen  _et
al._ (2017) that it can be rewritten as the gradient flow on the hyperbolic
disk $\mathbb{D}=\\{z:|z|<1\\}$,

$\dot{w}=\\\
=-\frac{\lambda}{2}\left(1-\left|w\right|^{2}\right)^{2}\frac{\partial}{\partial\overline{w}}\left\\{\frac{1}{N}\sum_{j=1}^{N}\ln
P\left(w,z_{j}^{0}\right)\right\\},$ (9)

where $P=P(w,z_{0})$ is the Poisson kernel,

$P(w,z_{0})=\frac{1-|w|^{2}}{|w-z_{0}|^{2}},\\\
z_{0}\in\mathbb{S}^{1},\,w\in\mathbb{D},\,\mathbb{S}^{1}=\\{z:|z|=1\\}.$ (10)

To determine how the Poisson kernel appears, one should substitute the
expression (8) into the eq. (7) and perform integration over $\overline{w}$.
In the paper Chen  _et al._ (2017) the authors have already noticed that for
$|w|\neq 1$ the fixed point of the eq. (9) coincides with conformal barycenter
of $N$ points on $\mathbb{S}^{1}$, which existence and uniqueness are
guaranteed Cantarella and Schumacher (2022). Also one should note that
$|w|\rightarrow 1$ limit corresponds to the synchronized state.

Using this representation of the Kuramoto model dynamics, we would like to
discuss several features of the model that devoted to the interconnections
between information geometry and synchronization.

##  III Information geometry and Kuramoto model

###  III.1 Basic concepts of information geometry

In this section we discuss the basic concepts of information geometry and
develop an information geometry view on the Kuramoto model. We mainly focus at
the continuum $N\rightarrow\infty$ limit of the Kuramoto model.

To begin with, let us provide some essentials from information geometry. We
follow the book Amari (2016), where more details can be found. Let us consider
the family of probability density functions $p=p(\xi;x)$, where $\xi$ is the
vector of parameters and $x$ is the vector of variables. For all possible
values of $\xi$, this family forms the manifold $\mathbb{M}=\\{p(\xi;x)\\}$,
which is called statistical manifold. To determine how two distributions
$p(\xi_{1};x)$ & $p(\xi_{2};x)$ differ from each other, one can consider the
divergence function $D[\xi_{1};\xi_{2}]$. It is possible to define different
divergence functions but these functions make sense only if they are invariant
and decomposable Amari (2009).

The large class of such divergence functions is called standard
$f$-divergencies, that can be represented as

$D_{f}[p;q]=\int dx\,p(x)f\left(\frac{q(x)}{p(x)}\right),\\\
f(0)=1,\,f(1)=1,\,f^{\prime\prime}(1)=1,$ (11)

where $f$ is the convex function. For $f(v)=-\ln v$ it is easy to see that
corresponding $f$-divergence is nothing more than Kullback-Leibler divergence.
Another important type of divergence is so-called $\alpha$-divergence, which
is defined by (see Amari (2016) and Amari (2009) for details)

$f_{\alpha}(v)=\frac{4}{1-\alpha^{2}}\left[1-v^{(1+\alpha)/2}\right],\quad\alpha\neq
1.$ (12)

The important property of the divergence function is that in case of two close
enough points, $\xi_{1}=\xi+d\xi$ & $\xi_{2}=\xi$, it has the following Taylor
expansion at point $\xi$,

$D[\xi_{1};\xi_{2}]=\frac{1}{2}g_{ij}(\xi)d\xi_{i}d\xi_{j}+\mathcal{O}(|d\xi|^{3}),$
(13)

where $g_{ij}$ is the metric of manifold $\mathbb{M}$. Any standard
$f$-divergence gives the same metric $g_{ij}$, which coincides with the Fisher
metric $g^{F}_{ij}$,

$g_{ij}^{\text{F}}=\int dx\,p(x;\xi)\frac{\partial\ln
p(x;\xi)}{\partial\xi_{i}}\frac{\partial\ln p(x;\xi)}{\partial\xi_{j}}.$ (14)

Strictly speaking, to completely describe the statistical manifold
$\mathbb{M}$ we need one more object, which is called skewness tensor
$T^{\text{F}}$,

$T_{ijk}^{\text{F}}=\\\ =\int dx\,p(x;\xi)\frac{\partial\ln
p(x;\xi)}{\partial\xi_{i}}\frac{\partial\ln
p(x;\xi)}{\partial\xi_{j}}\frac{\partial\ln p(x;\xi)}{\partial\xi_{k}}.$ (15)

Finally, the statistical manifold is the triplet $(\mathbb{M},\,g,\,T)$. For a
given standard $f$-divergence, the metric tensor $g$ (obtained from the Taylor
expansion) coincides with $g^{\text{F}}$, whereas corresponding skewness
tensor is given by $T=\alpha T^{\text{F}}$ with
$\alpha=2f^{\prime\prime\prime}(1)+3$.

The pair of $g^{\text{F}}$ and $T^{\text{F}}$ allows to define so-called
$\alpha$-connections on the statistical manifold, that are given by

$\Gamma^{\pm\alpha}_{ijk}=\Gamma_{ijk}^{0}-\frac{\alpha}{2}T_{ijk}^{\text{F}}.$
(16)

The statistical manifold is called $\alpha$-flat if corresponding
$\alpha$-Riemannian curvature tensor vanishes everywhere with condition
$R_{ijkl}^{\alpha}=R_{ijkl}^{-\alpha}$. Note in contrast with usual Riemannian
manifold, the statistical manifold can have non-zero torsion. It was shown
that such $\alpha$-flat manifolds has dual affine structure, which gives dual
(via Legendre transformation) coordinate system on the statistical manifold.
Such manifold is also called dual flat manifold. For dual flat manifold, the
metric and skewness tensors are given by

$g_{ij}^{\text{F}}=\partial_{i}\partial_{j}\psi(\xi),\,T_{ijk}^{\text{F}}=\partial_{i}\partial_{j}\partial_{k}\psi(\xi),\,\partial_{i}\equiv\frac{\partial}{\partial\xi_{i}}$
(17)

where $\psi(\xi)$ is a convex function. For dual flat manifold, the canonical
divergence is so-called the Bregman divergence.

###  III.2 Gradient flow on statistical manifold

For a dual flat manifold, the existence of $\psi(\xi)$ allows to consider the
gradient flow with respect to Fisher metric Nakamura (1993); Fujiwara and
Amari (1995),

$\frac{d\xi}{dt}=-\left(g^{F}\right)^{-1}\partial_{\xi}\psi(\xi).$ (18)

Some examples of the gradient flows on statistical manifolds are discussed in
Nakamura (1993).

The simplest illustrative example of the mentioned gradient flow corresponds
to the Gaussian distribution statistical manifold. It is straightforward to
check that the statistical manifold formed by Gaussian distributions,
$\mathbb{M}_{\text{G}}=\\{p(\mu,\sigma;x)|\mu\in\mathbb{R},\,\sigma>0\\}$, is
$\alpha$-flat for $\alpha=\pm 1$. One can also generalize this statement to
the exponential distribution family,

$p(\xi;x)=\exp\left[\xi_{i}g_{i}(x)+h(x)-\psi(\xi)\right],$ (19)

which forms dual flat manifold with canonical coordinate system $\xi$ and the
function $\psi=\psi(\xi)$. In case of Gaussian distribution, the coordinate
system is $(\xi_{1},\xi_{2})$ with $\xi_{1}=\mu/\sigma^{2}$,
$\xi_{2}=-1/(2\sigma^{2})$ and
$\psi=\mu^{2}/(2\sigma^{2})+\ln\left(\sqrt{2\pi}\sigma\right)$. In case of
exponential distributions family the Bregman divergence coincides with
Kullback-Leibler divergence. The gradient flow for the Gaussian statistical
manifold looks as

$\dot{\mu}=-\mu;\quad\dot{\sigma}=-\frac{\sigma^{2}+\mu^{2}}{2\sigma}.$ (20)

Note that this system has the conserved integral of motion
$H=\sigma^{2}/\mu-\mu$, that corresponds to the arc radius.

###  III.3 Cauchy distribution

One can show that family of univariate elliptic distributions forms
$\alpha$-flat statistical manifolds Mitchell (1988). However, there is one
orphan distribution in this family: the Cauchy distribution,

$p_{\text{C}}(\gamma,\beta;x)=\frac{1}{\pi}\frac{\gamma}{\gamma^{2}+(x-\beta)^{2}},\,\gamma>0,\,\beta\in\mathbb{R},$
(21)

which can be written in more convenient for our goals McCullagh’s
representation McCullagh (1992),

$p_{\text{C}}(w;x)=\frac{\operatorname{\mathrm{Im}\,}w}{\pi|x-w|^{2}},\quad
w=\beta+i\gamma.$ (22)

The corresponding Fisher metric looks as

$g_{ij}^{\text{F}}=\frac{1}{2\gamma^{2}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$ (23)

It is straightforward to verify that for the Cauchy distribution the skewness
tensor $T^{\text{F}}$ vanishes everywhere, so we can not find any value of
$\alpha$ to obtain dual flat structure at first glance.

However, we can interpret the Cauchy distribution as the member of
$q$-Gaussian family, which probability density function is defined as

$\mathcal{N}_{q}(\beta,x)=\frac{\sqrt{\beta}}{C_{q}}\exp_{q}\left\\{-\beta
x^{2}\right\\},\quad\beta>0,$ (24)

where $C_{q}$ is the normalization constant. The Cauchy distribution can be
considered as $q$-Gaussian with $q=2$, which was done in work Nielsen (2020)
(we discuss some properties of $q$-Gaussians in Appendix),

$p_{\text{C}}(\gamma,\beta;x)=\left.\mathcal{N}_{q}\left(\gamma^{-2},(x-\beta)^{2}\right)\right|_{q=2}.$
(25)

Using the definition of $q$-exponent, it is straightforward to verify that the
distribution function $p_{\text{C}}(\gamma,\beta;x$) can be also represented
as

$p_{\text{C}}(\gamma,\beta;x)=\left.\exp_{q}\left(\xi_{i}g_{i}(x)-\psi(\xi)\right)\right|_{q=2}$
(26)

with $\xi_{1}=2\pi\beta/\gamma$, $\xi_{2}=-\pi/\gamma$, $g_{1}(x)=x$,
$g_{2}(x)=x^{2}$ and the potential function $\psi(\xi)$ is,

$\psi(\xi)=-\frac{\pi^{2}}{\xi_{2}}-\frac{\xi_{1}^{2}}{4\xi_{2}}-1.$ (27)

Next, it was shown that the following divergence

$D_{\text{B-T}}[\xi_{1};\xi_{2}]=\\\ =\left(\int
dx\,p(\xi_{2};x)^{2}\right)^{-1}\left[\int
dx\,\frac{p(\xi_{2};x)^{2}}{p(\xi_{1};x)}-1\right]$ (28)

is the Bregman divergence, i.e. describes the dual flat manifold. Such
divergence is called Bregman-Tsallis divergence. The metric corresponding to
Bregman-Tsallis divergence does not coincide with Fisher metric and can be
computed as

$g_{ij}^{q}=\frac{\partial^{2}\psi(\xi)}{\partial\xi_{i}\partial\xi_{j}},$
(29)

where $\xi$ corresponds to the canonical coordinate system on dual flat
manifold. In case of Cauchy distributions, we have

$g_{ij}^{q}=\frac{2\pi}{\gamma}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$ (30)

and it is clear that $g^{q}$ and $g^{\text{F}}_{\text{C}}$ are related via
conformal transformation. The dual coordinates $\eta_{i}$ are defined by

$\eta_{i}=\frac{\partial\psi(\xi)}{\partial\xi_{i}}=\\\
=\left(-\frac{\xi_{1}}{2\xi_{2}},\,\frac{4\pi^{2}+\xi_{1}^{2}}{4\xi_{2}^{2}}\right)=(\beta,\,\gamma^{2}+\beta^{2})$
(31)

with corresponding dual potential,

$\phi(\eta)=1-2\pi\sqrt{\eta_{2}-\eta_{1}^{2}}.$ (32)

The gradient flow (18) for Cauchy distribution looks like

$\begin{cases}\dot{\xi}_{1}=\displaystyle\frac{\xi_{1}}{8\pi^{2}}\left(4\pi^{2}+\xi_{1}^{2}\right),\\\\[10.0pt]
\dot{\xi}_{2}=\displaystyle\frac{\xi_{2}}{8\pi^{2}}\left(4\pi^{2}-\xi_{1}^{2}\right)\end{cases}\leftrightarrow\begin{cases}\dot{\beta}=-\beta,\\\\[10.0pt]
\dot{\gamma}=\displaystyle\frac{\beta^{2}-\gamma^{2}}{2\gamma}\end{cases}$
(33)

Note that these equations are quite similar to the Gaussian case, eq. (20).
Having introduced all relevant concepts from information geometry, we can turn
to the interpretation of Kuramoto model in terms of the information geometry.

##  IV Fisher metric as the order parameter for synchronization transition

###  IV.1 Fisher metric and phase transitions

Quantum tensor is now the effective tool for analysis of the topological and
critical phenomena in the complicated many-body systems. It is defined as

$Q_{ij}=\langle\partial_{i}\Psi|\partial_{j}\Psi\rangle-\langle\partial_{i}\Psi|\Psi\rangle\langle\Psi|\partial_{j}\Psi\rangle$
(34)

where $i,j$ indices correspond to the coordinates on the parameter space. Its
real part is the quantum metric while imaginary part is the Berry curvature
Provost and Vallee (1980); Berry (1984)

$Q_{ij}=g_{ij}+iF_{ij}$ (35)

The quantum metric can be thought as a two-point correlator or fidelity whose
behavior is expected to quantify the properties of the system Zanardi and
Paunković (2006); Zanardi  _et al._ (2007a); You  _et al._ (2007). This
generic argument can be made more precise if we focus at the geometry of the
metric and it was argued that Zanardi and Paunković (2006); Zanardi  _et al._
(2007a) that the singularity of the metric or Ricci curvature corresponds to
the position of the phase transition. The kind of the classification of
induced geometries attributed to the ground state has been found in
Kolodrubetz  _et al._ (2013). The relation between the singularities of the
metric and the phase transitions is not a completely rigorous statement but
there is convincing list of examples, say Buonsante and Vezzani (2007); Dey
_et al._ (2012) for the Dicke and Hubbard models. The review of this subject
can be found in Kolodrubetz  _et al._ (2017). The parameters of the
Hamiltonian or the momenta can be used to evaluate the components of the
quantum metric. More recently it was recognized that the behavior of the
metric nearby the singular point can distinguish the integrable or chaotic
behavior Pandey  _et al._ (2020). The singularities of the different
components of the metric provide the information concerning the criticality in
the different directions of the parameter space.

The Fisher information metric is parallel to the quantum metric in quantum
mechanics when the probability substitutes the squared modulus of the wave
function. The probability obeys the Fokker-Planck (FP) equation instead of the
Schrödinger one hence we shall focus below at solutions to FP equations. The
notion of the phase transition in the stochastic problems also requires some
care however the relation between the quantum and semiclassical criticalities
are seen in the temperature component of the information metric Zanardi  _et
al._ (2007b). It turns out that the synchronization phase transition provides
another clear-cut example when the behavior of the Fisher metric yields the
identification of the phase transition in the classical system. In this
Section we shall demonstrate that the representation of the Kuramoto model in
terms of the distributions does the job. In the pure Kuramoto model we shall
see a kind of classical version of quantum phase transition at zero
temperature while in Kuramoto-Sakaguchi model we have an effective temperature
due to a noise.

###  IV.2 Fisher metric in Kuramoto model

In the paper Chen  _et al._ (2017) the authors have introduced the hyperbolic
description of the Kuramoto model. We argue that the hyperbolic space arises
as the statistical manifold of wrapped Cauchy distributions. The normalized
version of Poisson kernel is given by

$p_{\text{wC}}(w;z)=\frac{1}{2\pi}\frac{1-|w|^{2}}{|w-z|^{2}},\,|z|=1,\\\
\oint_{|z|=1}dz\,\frac{p_{\text{wC}}(w,z)}{iz}=1,$ (36)

where wC denotes “wrapped Cauchy”. It is convenient to use polar
representation, $w=re^{i\phi}$ and $z=e^{i\theta}$, which gives

$p_{\text{wC}}(r,\phi;\theta)=\frac{1}{2\pi}\frac{1-r^{2}}{1-2r\cos(\theta-\phi)+r^{2}},\\\
0\leq r<1,\,\phi=\phi\,\text{mod}\,2\pi$ (37)

This is PDF with two parameters $r$ and $\phi$. All such PDFs form manifold
$\mathbb{M}_{\text{wC}}$,

$\mathbb{M}_{\text{wC}}=\left\\{p_{\text{wC}}(r,\phi;\theta)\,|r\in[0,1),\phi=\phi\,\text{mod}\,2\pi\right\\}.$

and the Fisher metric $g_{ij}^{\text{F}}$ is given by (14), which can be
represented in more compact form,

$g_{ij}^{\text{F}}=4\int_{-\pi}^{+\pi}d\theta\,\partial_{i}\sqrt{p_{\text{wC}}(r,\phi;\theta)}\partial_{j}\sqrt{p_{\text{wC}}(r,\phi;\theta)}.$

It is straightforward to compute this integral and obtain components of the
metric,

$g_{rr}^{\text{F}}=\frac{2}{(1-r^{2})^{2}};\quad
g_{\phi\phi}^{\text{F}}=\frac{2r^{2}}{(1-r^{2})^{2}};\\\
g_{r\phi}^{\text{F}}=g_{\phi r}^{\text{F}}=0.$ (38)

hence, reintroducing $w=re^{i\phi}$, we can write

$ds^{2}_{\text{F}}=\frac{2\left(dr^{2}+r^{2}d\phi^{2}\right)}{(1-r^{2})^{2}}\equiv\frac{2\,dw\,d\overline{w}}{(1-|w|^{2})^{2}},$
(39)

which coincides with the hyperbolic disk metrics. Also, the direct computation
of (15) shows that $T^{\text{F}}\equiv 0$. This is not surprise, because the
wrapped Cauchy distribution is simply related to the usual one and the
manifold formed by usual Cauchy distributions is nothing more than upper half-
plane model of AdS2 space. The wrapped Cauchy distributions also form AdS2
space but in terms of Poincare disk model. Such two models can be mapped to
each other via conformal transformation.

We also can represent wrapped Cauchy distribution as $q$-exponent with $q=2$,
which can be checked in straightforward way. Computing the canonical
coordinates $(\xi_{1},\xi_{2})$ and then deriving potential function, we can
find the metric $g_{ij}^{q}$. In $(w,\,\overline{w})$-coordinates the metric
obtained from the corresponding potential function is given by

$g_{ij}^{q}=\frac{2\pi}{1-|w|^{2}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\rightarrow
ds_{q}^{2}=\frac{4\pi\,dw\,d\overline{w}}{1-|w|^{2}},$ (40)

so, the $q$-metric again is conformally equivalent to Fisher metric.

The wrapped Cauchy distribution plays a crucial role in the Kuramoto model
dynamics since it is invariant under Möbius group action. Let $z$ is wrapped
random variable with Cauchy distribution, $z\sim p_{\text{wC}}(w;z)$. Let
$z^{\prime}$ is the image of $z$ produced by the Möbius transformation
$\mathcal{M}$, $z^{\prime}=\mathcal{M}(z)$. It was proven that if $z\sim
p_{\text{wC}}(w;z)$, then $z^{\prime}\sim p_{\text{wC}}(w;z^{\prime})$. So,
wrapped Cauchy distributions are closed with respect action of Möbius
transformation. This fact drives us to conclude that the wrapped Cauchy
distribution is a kind of universal distribution for the Kuramoto model
dynamics. Such implication resonates with the role of Ott-Antonsen (O-A)
ansatz Ott and Antonsen (2008).

We would like to emphasize that Fisher information blows up in the
synchronized phase, that corresponds to the limit $|w|\rightarrow 1$. In the
Kuramoto- Sakaguchi the relation between the singularity of the Fisher metric
and the phase transition will be made more transparent.

###  IV.3 Fisher metric in Kuramoto-Sakaguchi model

Consider now the Kuramoto-Sakaguchi model Sakaguchi (1988), i.e. the Kuramoto
model with noise,

$\dot{\theta}_{i}=\omega_{i}+\frac{\lambda}{N}\sum_{j=1}^{N}\sin\left(\theta_{j}-\theta_{i}\right)+\eta_{i}(t),$
(41)

where $\eta_{i}=\eta_{i}(t)$ is stochastic term with properties
$\langle\eta_{i}(t)\rangle=0$,
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2D\delta_{ij}\delta(t-t^{\prime})$,
where $D>0$ is noise amplitude. This model was initially considered by
Sakaguchi and it was shown that there is the continuous phase transition for
the conventional order parameter $r$ with respect to the coupling constant
$\lambda$. The critical coupling could be obtained by the analysis of self-
consistency equation or by the stability analysis of incoherent state and is
given by

$\frac{1}{\lambda_{c}}=\frac{D}{2}\int_{-\infty}^{+\infty}\frac{d\omega\,g(\omega)}{\omega^{2}+D^{2}}.$
(42)

Both methods deal with the Fokker-Planck equation, which arises in the
continuum limit of Kuramoto-Sakaguchi model. In case of identical frequencies,
i.e. $g(\omega)=\delta(\omega)$, the critical coupling becomes
$\lambda_{c}=2D$. Moreover, in such case one could easily find the stationary
solution of the Fokker-Planck equation Goldobin  _et al._ (2018). in rotating
reference frame. The stationary solution $\rho_{0}$ reads as

$\rho_{0}(\theta,\omega)=\\\ =\frac{1}{2\pi
I_{0}(\lambda|r|/D)}\exp\left\\{\frac{\lambda|r|}{D}\cos\left(\arg
r-\theta\right)\right\\}.$ (43)

The self-consistency equation for the Kuramoto-Sakaguchi model is very simple,

$|r|=\frac{I_{1}(\lambda|r|/D)}{I_{0}(\lambda|r|/D)},$ (44)

where it is straightforward to notice that the non-trivial solution exists for
$\lambda>\lambda_{c}=2D$ (see fig. 1). This equation is similar with self-
consistency equation appeared in XY model and with self-consistency equation
appeared in stationary Hamiltonian mean field model. It is not a surprise,
because the noise strength $D$ plays role of temperature. In case of non-
identical frequencies, the stationary distribution could be found explicitly
Alexandrov (2023), but the Fisher metric can be evaluated only numerically.

Figure 1: Continuous phase transition in Sakaguchi-Kuramoto model

The described stationary distribution is nothing more than the von Mises
distribution belonging to the exponential family. In general form, it looks
like

$p_{\text{vM}}(\kappa,\mu;\theta)=\frac{1}{2\pi
I_{0}(\kappa)}\exp\left\\{\kappa\cos(\theta-\mu)\right\\},\\\ \kappa\geq
0,\,\quad\mu=\mu\,\text{mod}\,2\pi.$ (45)

The computation of the Fisher metric for von Mises distribution is
straightforward,

$g_{ij}^{\text{F}}=\begin{pmatrix}1-\displaystyle\frac{I_{1}(\kappa)^{2}}{I_{0}(\kappa)^{2}}-\frac{I_{1}(\kappa)}{\kappa
I_{0}(\kappa)}&0\\\ 0&\displaystyle\frac{\kappa
I_{1}(\kappa)}{I_{0}(\kappa)}\end{pmatrix}.$ (46)

where $(i,j)=(\kappa,\mu)$. Setting $\kappa=\lambda|r|/D$, we obtain the
Fisher metric corresponding to the stationary solution of Kuramoto-Sakaguchi
model. We can interpret in twofold way: we can set $|r|$ to be a fixed value
and then plot the components of metric as the functions of $\lambda$ and $D$
or we can solve the self-consistency equation (44) and find the order
parameter $|r|$ as the function of $\lambda$ and $D$ and then consider the
components of metric tensor as the functions of $\lambda$ and $D$ only. The
second interpretation gives us the following dependencies, fig. 2. The
component $g_{11}^{\text{F}}$ does not exist for $\lambda<\lambda_{c}$,
whereas the component $g_{22}^{\text{F}}$ is identically zero for
$\lambda<\lambda_{c}$. The first interpretation says us to plot the components
of Fisher metric as functions of order parameter $|r|$ and $\lambda/D$, see
fig. 3. The transition point coincides with $\lambda_{c}=2D$ hence the Fisher
metric provides the order parameter for the synchronization transition.

Figure 2: Non-zero components of Fisher metric for Kuramoto-Sakaguchi model as
the function of $\lambda/D$ with $\lambda_{c}=2D$ Figure 3: Non-zero
components of Fisher metric for Kuramoto-Sakaguchi model as the function of
$|r|$ & $\lambda/D$ with $\lambda_{c}=2D$

###  IV.4 Kullback-Leibler divergence in Kuramoto model

After discussion of the Fisher metric, we would like to notice the following
fact: the proposed hyperbolic space description of Kuramoto model dynamics
coincides with gradient flow of Kullback-Leibler divergence on the statistical
manifold.

Indeed, in the continuum limit, $N\rightarrow\infty$, we deal with initial
distribution of phases $f_{0}=f_{0}(z)$, $z=e^{i\theta}$. The gradient flow in
the limit $N\rightarrow\infty$ becomes

$\dot{w}=-\frac{\lambda}{2}\left(1-|w|^{2}\right)^{2}\times\\\
\times\frac{\partial}{\partial\overline{w}}\oint_{|z|=1}\frac{dz\,f_{0}(z)\ln
p_{\text{wC}}(w;z)}{iz},$ (47)

where we rewrite the integral over $\theta$ as the contour integral over the
unit circle $|z|=1$ in the complex plane. Our key observation is that the
integral in the right hand side of eq. (47) is nothing more than so-called
cross entropy of two distributions with PDF $f_{0}(z)$ and
$p_{\text{wC}}(w;z)$,

$S_{\text{cross}}=\oint_{|z|=1}\frac{dz\,f_{0}(z)\ln p_{\text{wC}}(w;z)}{iz}.$
(48)

This quantity is tightly connected to the well-known Kullback-Leibler
divergence,

$S_{\text{K-L}}[f,g]=-\oint_{|z|=1}dz\,\frac{f(z)}{iz}\ln\frac{g(z)}{f(z)}=\\\
=-S_{0}[f(z)]-S_{\text{cross}},$ (49)

where $f=f(z)$ and $g=g(z)$ are two different PDFs of wrapped distributions
and $S_{0}[f(z)]$ denotes the entropy of distribution $f(z)$. We can notice
that gradient with respect to $\overline{w}$ in eq. (47) has the form,

$\frac{\partial
S_{\text{K-L}}}{\partial\overline{w}}=-\frac{\partial}{\partial\overline{w}}\oint_{|z|=1}\frac{dz\,f(z)\ln
g(z)}{iz}.$ (50)

So, we conclude that the gradient flow can be expressed in terms of Kullback-
Leibler divergence,

$\dot{w}=+\frac{\lambda}{2}\left(1-\left|w\right|^{2}\right)^{2}\frac{\partial
S_{\text{K-L}}[f_{0},p_{\text{wC}}]}{\partial\overline{w}},$ (51)

where $f_{0}=f_{0}(z)$ is the initial distribution and
$p_{\text{wC}}=p_{\text{wC}}(w;z)$ is the wrapped Cauchy distribution.

As we have mentioned earlier, the Kullback-Leibler divergence in general does
not coincide with Bregman divergence. Nevertheless, the eq. (51) clearly says
that the Kuramoto model dynamics can be treated as the gradient flow of the
Kullback-Leibler divergence on the statistical manifold formed by wrapped
Cauchy distributions. Therefore, to understand such dynamics we can compute
the Kullback-Leibler divergence between a given initial distribution
$f_{0}(z)$ and wrapped Cauchy distribution $p_{\text{wC}}(w;z)$. Let us notice
that the uniform distribution $p_{\text{U}}(z)=(2\pi)^{-1}$ is nothing more
than wrapped Cauchy distribution with $w=0$ and the Dirac-delta distribution
is also wrapped Cauchy distribution with $w=1$.

If the initial distribution $f_{0}(z)$ is uniform, so $f_{0}(z)=(2\pi)^{-1}$,
the Kullback-Leibler divergence it is easy to compute, which gives

$S_{\text{K-L}}[f_{0},p_{\text{wC}}]=-\ln\left(1-|w|^{2}\right).$ (52)

Substituting this expression into the gradient flow, we find

$\dot{w}=\frac{\lambda}{2}\left(1-|w|^{2}\right)w.$ (53)

The obtained equation is interesting for two reasons. First, it is similar to
the equation appeared in Ott-Antonsen ansatz. Second, it tells us that in
large-$N$ limit the Kuramoto model with all-to-all couplings and identical
frequencies is _dual_ to the _single_ Landau-Stuart oscillator. It is quite
straightforward to obtain the solution $w=w(t)$ and than we can see how the
Kullback-Leilber divergence evolves in time. Until the transition moment, the
divergence is negligibly small (it is quite clear because an instant
distribution on the unit circle is still similar to uniform $f_{0}$). At the
transition moment, the Kullback-Leibler starts to grow significantly and
finally blows up: the final distribution on the unit circle is drastically
different from the uniform, which corresponds to the complete synchronization.

##  V Conclusion

In this work we revised the hyperbolic description of the Kuramoto model
proposed in Chen  _et al._ (2017) relating it to the statistical manifolds.
The key observation of our study is that the singularity of the Fisher
information metric captures the synchronization transition. Namely the metrics
corresponding to the Kuramoto and Kuramoto-Sakaguchi models blow up in the
synchronized state. Therefore we extend the list of examples when the phase
transitions get identified via the singularities of the Fisher metric. We have
focused on the information geometry describing the all-to-all Kuramoto-
Shakaguchi model but it would be interesting to recognize the information
metric for more general graph architecture like the star when the exact
solution is available Alexandrov (2023).

The presented approach can be also useful for case with complicated topology,
i.e. when an analytical treatment is extremely tough. Based on the fact that
Fisher metric has a singularity at the transition point, it seems possible to
extract an explicit expressions for model parameters, which raises
synchronized state. Moreover, there is no restriction for generalizations of
Kuramoto model, which is emphasized by the described example of model with
noise. This fact allows to use information geometry framework for
generalizations, that includes phase lags, external forces and so on.

We have noticed that the gradient flow in Kuramoto model can be represented as
gradient flow on statistical manifold. However, instead of a potential
function, one deals with Kullback-Leibler divergence. As was emphasized,
Kullback-Leibler divergence is not Bregmann divergence for (wrapped) Cauchy
distribution and it is reasonable to represent such flow in terms of Bregmann-
Tsallis divergence. But the metric obtained via the potential function of
Bregmann-Tsallis divergence does not coincide with Fisher metric (it is
conformally equivalent, but different). We address a closer treatment of a
connection between synchronization and Bregmann-Tsallis divergence for further
research. Next, the discussion of Cauchy distribution as member of
$q$-exponential family naturally raises the question concerning wrapped
$q$-exponential distributions. As far as we know, the properties of such
distributions were not examined. It seems that this research can be
interesting in information geometry context.

In our study the coordinates on the $\mathrm{SL}(2,\mathbb{R})$ orbit play the
role of the parameters due to the dimensional reduction of dynamics. Hence to
some extend one could be confused why the effective degrees of freedom play
the role of the effective parameters. However we know the very precise example
of the same nature — the duality between the inhomogeneous spin chains and
integrable many-body system with long-range interaction of Calogero-
Ruijsenaars type which has been formulated in probabilistic terms in Gorsky
_et al._ (2022). It that case the parameters-inhomogeneities in the spin
chains get identified with the coordinates of the particles in Calogero model
which on the other hand are the coordinates on the group orbit. That is the
Fisher metric on the parameters in the spin chain gets mapped into the clear-
cut object at the Calogero-Ruijsenaars side. We shall discuss this analogy and
relation elsewhere.

The hyperbolic geometry and the large number of degrees of freedom at its
boundary invites for the holographic description. However in this case there
are some subtle points since the boundary theory is classical and the
equations are of the first order in the time derivatives. Nevertheless the
description in terms of the conformal barycenter has a lot in common with the
dynamics of a kind of baryonic vertex since the Poisson kernels are related
with the geodesics connecting the points at the boundary and the bulk. We
postpone these issues for the separate study. The general discussion
concerning the relation of the probabilistic manifold with holography can be
found in Erdmenger  _et al._ (2020).

The concluding remark concerns the application of the information geometry to
the neuroscience. The idea is not new Amari (2016) however our findings
provide the new perspectives for this issue. The Kuramoto model is widely used
as the model for the synchronization of the functional connectomes hence one
could expect that the properties of the Fisher metric for the Kuramoto model
at more general graphs shall provide the new approach for the brain rhythms
generation. In particular it would be very interesting to investigate the
synchronization of the several functional connectomes via the properties of
the Fisher metric and possible dependence of the synchronization of the
several functional connectomes on the architecture of the structural
connectome.

## Acknowledgements

We are grateful to A. Polkovnikov for the useful discussions, S. Kato for the
discussion on Kato-Jones distribution properties and to F. Nielsen for the
comments concerning Cauchy distribution statistical manifold. A.G. thanks IHES
and Nordita where the part of the work has been done for the hospitality and
support. A.A. thanks grant by Brain Program of the IDEAS Research Center and
grant №18-12-00434$\Pi$ by Russian Science Foundation. A.G. thanks grant
№075-15-2020-801 by Ministry of Science and Higher Education of Russian
Federation and grant by Basis Foundation.

##  Appendix A Appendix. On Möbius transformation and wrapped distributions

In this Appendix let us describe some facts concerning the wrapped
distributions which can be obtained from non-wrapped one via so-called
“compactification”. For random variable $x\in(-\infty,+\infty)$ with
probability density function $p(x)$, we introduce the complex variable
$z=e^{ix}$ and consider its phase $\theta=\arg z$, $\theta\in(-\pi,+\pi]$,
which has _wrapped_ probability density function $p_{\text{w}}(\theta)$. The
connection between $p(x)$ and its wrapped version is following,

$p_{\text{w}}(\theta)=\sum_{k=-\infty}^{+\infty}p(\theta+2\pi k).$ (54)

All the common concepts of probability theory work for wrapped distributions
as well.

There are four (at least) common wrapped distributions that usually discussed
in the context of Kuramoto model: uniform distribution, wrapped Cauchy
distribution, von Mises distribution and recently appeared Kato-Jones
distribution. The uniform distribution is trivial, the wrapped Cauchy
distribution is well-known. The von Mises distribution arises in the
Sakaguchi-Kuramoto model as the stationary solution of the corresponding
Fokker-Planck equation. The Kato-Jones distribution was introduced in work
Kato and Jones (2010) and it is Möbius transformed von Mises distribution.
This distribution is invariant under the action of Möbius group. The mentioned
list of distribution is easy to extend. For the sake of completeness, we can
also mention cardioid distribution and its Möbius image, wrapped normal
distribution and its Möbius image.

Figure 4: Kuramoto model related statistical manifolds. One can obtain
$p_{\text{K-J}}$ via Möbius transformation $\mathcal{M}$ and then reduce
$M_{\text{K-J}}$ to the wrapped Cauchy submanifold $\mathbb{M}_{\text{wC}}$
and the von Mises submanifold $\mathbb{M}_{\text{vM}}$. Both $p_{\text{K-J}}$
and $p_{\text{wC}}$ belong to the Möbius invariant submanifold
$\mathbb{M}_{\text{M}}$, whereas $p_{\text{vM}}$ does not

The invariance under Möbius group action is crucial in the Kuramoto model. So,
here we focus on the triple of wrapped Cauchy distrubition, von Mises
distribution and Kato-Jones distribution. We would like to establish the
interconnections between members of such triple in the information geometry
context. First of all, let us write down the probability density functions for
each member of triplet,

$p_{\text{wC}}(r,\phi;\theta)=\frac{1}{2\pi}\frac{1-r^{2}}{1-2r\cos(\theta-\phi)+r^{2}},\\\
0\leq r<1,\,\phi=\phi\,\text{mod}\,2\pi,$ (55)
$p_{\text{vM}}(\kappa,\mu;\theta)=\frac{1}{2\pi
I_{0}(\kappa)}\exp\left\\{\kappa\cos(\theta-\mu)\right\\},\\\ \kappa\geq
0,\,\mu=\mu\,\text{mod}\,2\pi,,$ (56)
$p_{\text{K-J}}(\kappa,r,\nu,\mu;\theta)=\\\ =\frac{1}{2\pi
I_{0}(\kappa)}\frac{1-r^{2}}{1-2r\cos(\theta-\mu-\nu)+r^{2}}\times\\\
\times\exp\left\\{\frac{\kappa\cos(\mu+\theta)+\kappa
r^{2}\cos(\mu+2\nu-\theta)-2r\kappa\cos\nu}{1-2r\cos(\theta-\mu-\nu)+r^{2}}\right\\},\\\
\kappa\geq 0,\,0\leq
r<1,\,\mu=\mu\,\text{mod}\,2\pi,\,\nu=\nu\,\text{mod}\,2\pi$ (57)

From these expressions it is straightforward to see the following facts:
wrapped Cauchy and von Mises distributions form two dimensional statistical
manifolds, the von Mises distribution belongs to exponential family, the Kato-
Jones distribution forms four dimensional statistical manifold. The wrapped
Cauchy distribution can be derived from the Kato-Jones by setting
$\kappa\rightarrow 0$, whereas the von Mises distribution can be derived by
setting $r\rightarrow 0$. Therefore, we can treat the statistical manifolds
corresponding to the wrapped Cauchy distribution and to the von Mises
distribution as the submanifolds of the Kato-Jones statistical manifold.
Wrapped distributions $p_{\text{wC}}$ and $p_{\text{K-J}}$ belong to Möbius
invariant statistical manifold $\mathbb{M}_{\mathcal{M}}$, whereas
$p_{\text{vM}}$ does not. However, from $p_{\text{K-J}}$ one can obtain
$p_{\text{vM}}$ and $p_{\text{wC}}$ taking an appropriate limit, $r\rightarrow
0$ and $\kappa\rightarrow 0$, respectively (see fig. 4).

##  Appendix B On $q$-Gaussian distributions in Kuramoto model

The Kuramoto model shares some similarities with the famous Hamiltonian mean
field (HMF) model Miritello  _et al._ (2009a); Pluchino and Rapisarda (2006),
which describes $N$ interacting particles on $\mathbb{S}^{1}$ via cosine
potential with all-to-all couplings,

$H_{\text{HMF}}=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2m}+\frac{\lambda}{2N}\sum_{i<j}\cos\left(\theta_{j}-\theta_{i}\right),$
(58)

where $\lambda$ is the coupling constant. This model is the prototypical for
study of systems with long-range interactions (LRI). In the HMF model LRI is
caused by all-to-all couplings. The effects of LRI in HMF model are quite
well-known: it was shown that LRI causes violent relaxation phenomenon,
existence of quasi-stationary states Yamaguchi  _et al._ (2004). It is worth
mentioning that quantum version of HMF model also exhibits such phenomena. One
quite interesting consequences of LRI is the appearance of so-called Tsallis
$q$-statistics Tsallis and Tirnakli (2010). Due to the presence of LRI, the
entropy of a system becomes non-extensive. Tsallis with co-authors developed
the appropriate machinery to describe this non-extensivity Umarov  _et al._
(2008). Roughly speaking, the parameter $q$ measures the non-extensivity in a
system. The $q$-statisics deals with so-called $q$-Gaussian distributions (see
Umarov  _et al._ (2008) for detailed discussion). Such $q$-Gaussian
distributions differ from the usual Gaussian distribution for $q\neq 1$: they
have heavier tails (see fig. 5) and in the limit $q\rightarrow 1$ one restores
usual Gaussian distribution.

Figure 5: $q$-Gaussians

Next, in the paper Tirnakli  _et al._ (2007) the authors presented the method
to capture correlations between degrees of freedom based on $q$-statistics.
Despite the fact that there is no mathematically rigorous derivations of
$q$-statistics for a system with LRI, such method allows one to verify (at
least numerically) that $q$-statistics takes place. For instance, it was shown
that $q$-statistics appears in the HMF model Pluchino  _et al._ (2009, 2008).

Based on the idea that HMF model and Kuramoto model are related to each other,
in Miritello  _et al._ (2009b) the authors examined fingerprints of
$q$-statistics in the Kuramoto model. The key observation was following. From
the phases $\theta_{i}(t)$ one should construct the so-called CLT-variable
$y_{i}$ (see Tirnakli  _et al._ (2007) for details),

$y_{i}=\frac{1}{\sqrt{M}}\sum_{k=1}^{M}\theta_{i}(k\delta),$ (59)

where $\delta>0$ is the pre-defined time interval In the Kuramoto model it was
demonstrated that for $\lambda<\lambda_{c}$ the variables $y_{i}$ obey
$q$-Gaussian distribution with $q\approx 1.7$ , i.e. $q$-CLT appears. The
authors considered the case of _non-identical_ frequencies and focused on case
of uniform distribution and case of Gaussian distribution. In the case of
$\lambda>\lambda_{c}$ the variables $y_{i}$ were fitted by usual Gaussian
distribution.

Having briefly discussed the $q$-Gaussians, we would like to draw attention to
the fact that $q$-Gaussian distribution have already appeared in the context
of Kuramoto model. Indeed, the continuum limit of Kuramoto model can be
described in Ott-Antonsen framework Ott and Antonsen (2008). In the case of
identical oscillators, the distribution function for the continuum limit can
be explicitly obtained. This distribution function is nothing more then
normalized Poisson kernel, i.e. the wrapped Cauchy distribution. Of course,
parameters of the distribution depend on time and initial conditions. As we
have shown, the usual Cauchy distribution belongs to $q$-exponential family
with $q=2$, so the wrapped Cauchy distribution also inherits properties of
$q$-exponential family (wrapped and non-wrapped versions of Cauchy
distribution are related to each other via conformal transformation).

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# On the Transcendence of Power Towers of Liouville Numbers

Diego Marques  Departamento de Matemática
Universidade de Brasília
Brasília, DF
Brazil<EMAIL_ADDRESS>,  Marcelo Oliveira  Departamento de Matemática
Universidade Estadual do Maranhão
Brazil<EMAIL_ADDRESS>and  Pavel Trojovský  Faculty of Science
University of Hradec Králové
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

In this paper, among other things, we explicit a $G_{\delta}$-dense set of
Liouville numbers, for which the triple power tower of any of its elements is
a transcendental number.

###### Key words and phrases:

Liouville numbers, power tower, transcendental numbers, linear forms in
logarithms

###### 2010 Mathematics Subject Classification:

Primary 11J81; Secondary 11J86

##  1\. Introduction

The beginning of the transcendental number theory happened in 1844 when
Liouville [10] showed that real algebraic numbers are not “well-approximated”
by rational numbers. By using this fact, Liouville explicitly, for the first
time, provided examples of transcendental numbers (known as Liouville numbers
now). We recall that a real number $\xi$ is called a Liouville number if there
exist infinitely many rational numbers $p_{k}/q_{k}$, with $q_{k}>1$, such
that

$\displaystyle\left|\xi-\frac{p_{k}}{q_{k}}\right|<q_{k}^{-\omega_{k}}$,
$\forall k$,

for some sequence of positive real numbers $(\omega_{k})_{k}$ which tends to
$\infty$ as $k\to\infty$. The first example of a Liouville number (and
consequently, of a transcendental number) is the known Liouville constant
defined by $\ell:=\sum_{n\geq 1}10^{-n!}$ (i.e., the decimal with $1$’s in
each factorial position and $0$’s otherwise). The set of Liouville numbers is
denoted by $\mathbb{L}$.

In his pioneering book, Maillet [14, Chapitre III] discussed some arithmetic
properties of Liouville numbers. One of them is that given a non-constant
rational function $f$, with rational coefficients, if $\xi$ is a Liouville
number, then so is $f(\xi)$. Thus, a natural problem concerns the possible
arithmetic behavior of $f(\xi)$ when $\xi\in\mathbb{L}$ and $f$ is a
transcendental function. In particular, it remains unsolved the question
raised by Mahler [13] about the existence of a transcendental entire function
taking $\mathbb{L}$ into itself. However, there are some results on this
subject. For instance, Chalebgwa and Morris [4] proved, that $e^{\xi}$ is
transcendental whenever $\xi\in\mathbb{L}$. Also, Marques and Oliveira [16]
showed that $\alpha^{\xi}$ is transcendental, for all Liouville number $\xi$
and all algebraic number $\alpha\not\in\\{0,1\\}$. By using Baire’s theorem
(see [9]), one has that for any non-constant function $f$, there is a
$G_{\delta}$-dense set $\mathcal{L}$ of Liouville numbers such that
$f(\mathcal{L})\subseteq\mathbb{L}$.

Set, inductively, $\exp[n](x)=\exp(\exp[n-1](x))$ and $\exp[0](x)=x$. In 2015,
Marques and Moreira [15] defined an ultra-Liouville number $\xi$ as a real
number for which, for every positive integer $k$, there exist infinitely many
rational numbers $p/q$, with $q>1$, such that

$0<\left|\alpha-\frac{p}{q}\right|<\frac{1}{\exp[k](q)}.$

They proved the existence of a transcendental entire function $f$ for which
$f(\mathbb{L}_{\text{ultra}})\subseteq\mathbb{L}_{\text{ultra}}$ (where
$\mathbb{L}_{\text{ultra}}$ denotes the set of ultra-Liouville numbers).

Very recently, Marques and Oliveira [16] showed that $\xi^{\xi}$ is a
transcendental number whenever $\xi$ is an ultra-Liouville number (indeed,
they provided a larger class of Liouville numbers with this property).

We now define sub-classes of Liouville numbers $\xi$ (which satisfy (1))
depending on the growth of the map
$\nu\mapsto\lim_{k\to\infty}\omega_{k}/q_{k}^{\nu}$. More specifically, we
have the following definition:

######  Definition 1.1.

A real number $\xi$ is called a $\nu$-Liouville number if $\nu$ is the infimum
(possibly infinite) of the non-negative real numbers $\nu^{*}$ for which there
exists an infinite sequence of rational numbers $(p_{k}/q_{k})_{k}$ (with
$q_{k}>1$) and a sequence of positive real numbers $(\omega_{k})_{k}$ such
that

$\displaystyle\left|\xi-\frac{p_{k}}{q_{k}}\right|<q_{k}^{-\omega_{k}}$,
$\forall k$,

and $\omega_{k}/q_{k}^{\nu^{*}}$ tends to $\infty$ as $k\to\infty$. The set of
all $\nu$-Liouville numbers is denoted by $\mathbb{L}^{(\nu)}$.

We note that $0$-Liouville numbers are precisely the Liouville numbers and
$\mathbb{L}^{(\nu)}\subseteq\mathbb{L}$, for any $\nu\in[0,\infty]$. Also,
$\mathbb{L}^{(\infty)}$ is a $G_{\delta}$-dense set since it contains the set
of ultra-Liouville numbers (indeed, one has that $|\xi-
p_{k}/q_{k}|<\exp(-\exp(q_{k}))=q_{k}^{-\omega_{k}}$, where
$\omega_{k}:=\exp(q_{k})/\log q_{k}$, satisfies
$\lim_{k\to\infty}\omega_{k}/q_{k}^{\nu}=\infty$, $\forall\nu\geq 0$). For
topological reasons, the set $\mathbb{L}^{(\nu)}$ is not $G_{\delta}$-dense
whenever $\nu\in(0,\infty)$. So a natural question arises: are the sets
$\mathbb{L}^{(\nu)}$ non-empty, for all $\nu\in(0,\infty)$? Our first result
answers this question positively. More precisely,

######  Theorem 1.1.

The set $\mathbb{L}^{(\nu)}$ is uncountable, for any $\nu>0$.

The main goal of this paper is to provide some results on the arithmetic
behavior of a triple power tower of numbers related to the previously defined
sets. In order to state some conditions succinctly, we introduce the following
terminology: a triple of polynomials $(P(X),Q(X),R(X))$ is said to be
$\mathbb{K}$-admissible (where $\mathbb{K}\subseteq\mathbb{C}$), when

  * (i)

$P(X),Q(X)\in\mathbb{K}[X]$ and $R(X)\in\mathbb{Z}[X]$;

  * (ii)

$P(X)\not\equiv 0$, $Q(X)\not\equiv 0$ and $P(X)\not\equiv 1$;

  * (iii)

$P^{\prime}(X)\equiv 0$ implies $Q^{\prime}(X)\not\equiv 0$ and
$R(X)\not\equiv 0$;

  * (iv)

$Q^{\prime}(X)\equiv 0$ implies $P^{\prime}(X)\not\equiv 0$;

  * (v)

$P^{\prime}(X)\equiv 0,\ Q^{\prime}(X)\equiv 0$ implies
$R^{\prime}(X)\not\equiv 0$.

where, as usual, $T(X)\equiv c$ (resp., $T(X)\not\equiv c$) means that
$T(X)=0$, for all $X$ (resp., $T(X)\neq c$, for some $X$).

Thus, our main result can be stated as follows:

######  Theorem 1.2.

Let $(P(X),Q(X),R(X))$ be a $\mathbb{Z}$-admissible triple of polynomials. If
$\xi$ is a $\nu$-Liouville number such that $\nu>6\deg R$, then

$P(\xi)^{Q(\xi)^{R(\xi)}}$

is a transcendental number.

######  Remark 1.

As discussed before, by supposing that the function $f:I\to\mathbb{R}$,
defined by $f(X):=P(X)^{Q(X)^{R(X)}}$ is well-defined and non-constant in some
open interval $I\subseteq\mathbb{R}$, then $f(\xi)$ is transcendental, for any
$\xi$ belonging to an uncountable set of Liouville numbers. However, there is
no information on the approximation structure of this set. Thus, our Theorem
1.2 provides such a structure that depends only on the degree of $R(X)$.

As immediate consequences, we have

######  Corollary 1.1.

If $\xi$ is an $\infty$-Liouville number, then $P(\xi)^{Q(\xi)^{R(\xi)}}$ is a
transcendental number, for any non-constant polynomials
$P(X),Q(X),R(X)\in\mathbb{Z}[X]$.

######  Corollary 1.2.

If $\xi\in\mathbb{L}_{\text{ultra}}$, then $\xi^{\xi^{\xi}}$ is a
transcendental number.

Furthermore, by exploring the admissibility conditions, we obtain

######  Corollary 1.3.

Let $\nu>6$ be a real number. Then, for any $\nu$-Liouville number $\xi$ and
for any integer $n\geq 2$, all the numbers

$n^{n^{\xi}},\ n^{\xi^{\xi}},\ n^{\xi^{n}},\ \xi^{n^{\xi}},\xi^{\xi^{n}}$

are transcendental.

The proofs presented here combine some algebraic and analytic tools together
with a bit of Diophantine approximation and lower bounds for linear forms in
logarithms.

##  2\. Proof of Theorem 1.1

For a given $\nu>0$, let $(s_{k})_{k}$ be the sequence defined by $s_{1}=4$
and $s_{k+1}=\lfloor 10^{\nu s_{k}}\rfloor s_{k}+1$, for $k\geq 1$. We claim
that the number $\xi:=\sum_{n=1}^{\infty}10^{-s_{n}}$ is a $\nu$-Liouville
number.

In order to prove that, we start by defining coprime integers
$A_{k}:=\sum_{n=1}^{k}10^{s_{k}-s_{n}}$ and $B_{k}:=10^{s_{k}}$. Since
$A_{k}/B_{k}$ is a truncation of the series which defines $\xi$, we get

(1) $\absolutevalue{\xi-\frac{A_{k}}{B_{k}}}=\sum_{n\geq
k+1}10^{-s_{n}}<\frac{2}{10^{s_{k+1}}}=\frac{2}{B_{k}^{s_{k+1}/s_{k}}}<\frac{1}{B_{k}^{\lfloor
10^{\nu s_{k}}s_{k}\rfloor}}.$

Note that $\lfloor 10^{\nu s_{k}}\rfloor/B_{k}^{\nu}=\lfloor
B_{k}^{\nu}s_{k}\rfloor/B_{k}^{\nu}$ tends to infinite as $k\to\infty$
yielding that $\xi$ belongs to $\mathbb{L}^{(\tau)}$, for some $\tau\geq\nu$.
To finish the proof, it suffices to show that $\xi$ is not a
$(\nu+\epsilon)$-Liouville number, for all $\epsilon>0$. Indeed, by (1), one
has

$\absolutevalue{\xi-\frac{A_{k}}{B_{k}}}<\frac{1}{B_{k}^{\lfloor 10^{\nu
s_{k}}s_{k}\rfloor}}<\frac{1}{2B_{k}^{2}},$

where we used that $\lfloor 10^{\nu s_{k}}s_{k}\rfloor\geq 10^{\nu
s_{k}}s_{k}-1\geq s_{k}-1\geq 3$. Hence, a criterion due to Legendre yields
that $A_{k}/B_{k}$ is a convergent, say $p_{r(k)}/q_{r(k)}$, of the continued
fraction of $\xi=[b_{0};b_{1},b_{2},\ldots]$. Note that
$q_{r(k)}=B_{k}=10^{s_{k}}$. Also, for $r(k-1)<m\leq r(k)$, it holds that

$b_{m}\leq\displaystyle\prod_{i=r(k-1)+1}^{r(k)}b_{i}<\dfrac{q_{r(k)}}{q_{r(k-1)}}=q_{r(k-1)}^{-1+s_{k}/s_{k-1}}\leq
q_{m-1}^{-1+s_{k}/s_{k-1}}.$

Thus, for any convergent $p_{m}/q_{m}$ of the continued fraction of $\xi$, we
have

(2)
$\left|\xi-\dfrac{p_{m}}{q_{m}}\right|>\frac{1}{3b_{m+1}q_{m}^{2}}\geq\frac{1}{q_{m}^{1+s_{k+1}/s_{k}}}.$

Aiming for a contradiction, suppose that $\xi\in\mathbb{L}^{(\nu+\epsilon)}$,
then there exists infinitely many rational numbers $c_{i}/d_{i}$, with
$d_{i}\geq 2$ and a sequence $(\theta_{i})_{i}$, with $\theta_{i}\geq 3$, such
that

(3) $\left|\xi-\dfrac{c_{i}}{d_{i}}\right|<\dfrac{1}{d_{i}^{\theta_{i}}},\
\forall i,$

where $\lim_{i\to\infty}\theta_{i}/d_{i}^{\nu+\epsilon}=\infty$. Since
$1/d_{i}^{\theta_{i}}\leq 1/(2d_{i}^{2})$, then $c_{i}/d_{i}$ must be a
convergent of the continued fraction of $\xi$, say $p_{m(i)}/q_{m(i)}$, where
$r(k)<m(i)\leq r(k+1)$, for some $k$ (note that $d_{i}=q_{m(i)}$). Therefore,
by combining (2) and (3), we arrive at

$\frac{1}{3q_{m(i)}^{s_{k+1}/s_{k}}}<\left|\xi-\dfrac{p_{m(i)}}{q_{m(i)}}\right|<\dfrac{1}{q_{m(i)}^{\theta_{i}}}$

and so $\theta_{i}<2s_{k+1}/s_{k}$. By reordering indexes if necessary, we may
suppose that $i\mapsto m(i)$ and $k\mapsto r(k)$ are increasing functions
(thus $i$ tends to infinity if and only if $k$ does as well). By dividing both
sides of the previous inequality by $d_{i}^{\nu+\epsilon}$, we obtain

$\frac{\theta_{i}}{d_{i}^{\nu+\epsilon}}<\frac{2\cdot 10^{\nu
s_{k}}s_{k}}{10^{s_{k}(\nu+\epsilon)}}=\frac{2s_{k}}{10^{\epsilon s_{k}}},$

where we used that $d_{i}=q_{m(i)}>q_{r(k)}=10^{s_{k}}$. However, the
inequality $\theta_{i}/d_{i}^{\nu+\epsilon}<2s_{k}/10^{\epsilon s_{k}}$ leads
to a contradiction, since its left-hand side tends to infinity while its
right-hand side tends to zero as $i$ and so $k$ tends to $\infty$. Therefore,
$\xi$ is a $\nu$-Liouville number as desired. An easy modification of this
proof shows that $\sum_{n\geq 1}a_{n}10^{-s_{n}}\in\mathbb{L}^{(\nu)}$, for
any choice of $a_{n}\in\\{1,2\\}$. Hence, there exist uncountably many
$\nu$-Liouville numbers, which completes the proof.∎

##  3\. Proof of Theorem 1.2

###  3.1. Auxiliary results

We start by stating two ingredients for the proof (in all that follows, we use
the notation $[a,b]=\\{a,a+1,\ldots,b\\}$, for integers $a<b$).

The first one is due to Íçen [6], which provides an inequality relating the
height of algebraic numbers (the height of $\gamma\in\overline{\mathbb{Q}}$,
denoted by $H(\gamma)$ is the maximum of the absolute values of the
coefficients of the minimal polynomial of $\gamma$ (over $\mathbb{Z}$)).

######  Lemma 3.1.

Let $\alpha_{1},\ldots,\alpha_{n}$ be algebraic numbers belonging to a
$g$-degree number field $K$, and let $F(y,x_{1},\ldots,x_{n})$ be a polynomial
with integer coefficients and with degree $d>1$ in the variable $y$. If $\eta$
is an algebraic number such that $F(\eta,\alpha_{1},\ldots,\alpha_{n})=0$,
then the degree of $\eta$ is at most $dg$, and

(4) $H(\eta)\leq 3^{2dg+(l_{1}+\cdots+l_{k})g}\cdot H^{g}\cdot
H(\alpha_{1})^{l_{1}g}\cdots H(\alpha_{n})^{l_{n}g},$

where $H$ is the maximum of the absolute values of the coefficients of $F$
(the height of $F$), $l_{i}$ is the degree of $F$ in the variable $x_{i}$, for
$i\in[1,n]$.

Also, the lower bounds for linear form in logarithms à la Baker lie at the
heart of our approach. Among the many extensions (and variations) of the
original Baker’s bounds, we choose to use the lower bound due to Mignotte and
Waldschmidt (see Remark after Theorem in [18]), which is the most convenient
for our purpose. For the sake of preciseness, we state it as a lemma:

######  Lemma 3.2.

Let $\beta,\alpha_{1},\alpha_{2}$ be nonzero algebraic numbers and let
$D=[\mathbb{Q}(\beta,\alpha_{1},\alpha_{2}):\mathbb{Q}]$. Set $A_{j}\geq
H(\alpha_{j})$ ($j=1,2$) and $B\geq\max\\{H(\beta),e^{D}\\}$. If
$\Lambda=\beta\log\alpha_{1}-\log\alpha_{2}$ is not zero, then

$|\Lambda|>\exp\left(-5\cdot 10^{10}D^{4}(\log A_{1})(\log
A_{2})T^{2}\right),$

where $T:=\log B+\log\log A_{1}+\log\log A_{2}+\log D$.

Throughout this text, $\log z$ denotes the main value of the complex logarithm
function.

In the next section, each given inequality holds for all sufficiently large
$k$ (unless stated otherwise), which makes valid all the previous ones.

###  3.2. The proof

First, we note that if $R(X)\equiv 0$, then $P(\xi)^{Q(\xi)^{R(\xi)}}=P(\xi)$
is transcendental, for all $\xi\in\mathbb{L}^{(\nu)}$ (since a non-constant
polynomial takes the set of transcendental numbers into itself).

Aiming for a contradiction, suppose that $P(\xi)^{Q(\xi)^{R(\xi)}}$ is an
algebraic number, say $\gamma$. In particular, $Q(\xi)^{R(\xi)}\log
P(\xi)=\log\gamma$ (here we used that $\gamma\neq 0$, since $P(\xi)\neq 0$,
because $\xi\not\in\overline{\mathbb{Q}}$). In what follows, $m,n$ and $r$
denote the degree of the polynomials $P(x),Q(x)$ and $R(x)$, respectively.

Since $\xi$ is a $\nu$-Liouville number, then there exist infinitely many
rational numbers $p_{k}/q_{k}$ (with $q_{k}>1$) and an increasing sequence
$(\omega_{k})_{k}$ belonging to $\mathbb{R}_{>0}$ such that

(5) $0<\absolutevalue{\xi-\frac{p_{k}}{q_{k}}}<\frac{1}{q_{k}^{\omega_{k}}},$

for all $k\geq 1$, where $\lim_{k\to\infty}\omega_{k}/q_{k}^{\nu}=\infty$. We
may suppose (by reordering the indexes if necessary) that
$p_{k}/q_{k}\in\Omega:=\\{z\in\mathbb{C}:P(z)Q(z)R(z)\neq 0\ \mbox{and}\
P(z)\neq 1\\}$, for all $k\geq 1$. Now, we claim

$\Lambda_{k}:=Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}\log P(p_{k}/q_{k})-\log{\gamma}$

is nonzero for all sufficiently large $k$. In fact, on the contrary, i.e., if
$\Lambda_{k}=0$ for all $k$ belonging to an infinite set of positive integers,
say $N^{\prime}$, then the function $h:\Omega\to\mathbb{C}$ given by
$h(z):=Q(z)^{R(z)}\log P(z)-\log\gamma$ vanishes in the set
$\\{p_{k}/q_{k}:k\in N^{\prime}\\}$ which has an accumulation point (namely
$\xi$). By the Identity Theorem for Analytic Functions, one has that $h(z)=0$,
for all $z\in\Omega$, which implies that the function
$\tilde{h}(z):=h(z)+\log\gamma$ is the constant function. On the other hand,
either $\Omega^{+}:=\\{z\in\Omega\cap\mathbb{Z}_{>0}:R(z)\geq 1\\}$ or
$\Omega^{-}:=\\{z\in\Omega\cap\mathbb{Z}_{>0}:R(z)\leq-1\\}$ must be an
infinite set (actually, only one of them has this property). If $\Omega^{+}$
is infinite, we use that

$|\tilde{h}(z)|\geq|Q(z)|^{R(z)}(\log|P(z)|+i\operatorname{Arg}(P(z)))\geq|Q(z)|(\log|P(z)|-2\pi),$

to infer that $|\tilde{h}(z)|$ tends to infinity as $z\to\infty$
($z\in\Omega^{+}$). This contradicts the fact that $\tilde{h}(z)$ is constant.
For the case in which $\Omega^{-}$ is an infinite set, we use that
$|P(z)|<|z|^{m+1}$ and $|Q(z)|>(|c|/2)|z|^{n}$ for all sufficiently large
$z\in\Omega^{-}$ (here $c$ is the leading coefficient of $Q(z)$). Thus

$|\tilde{h}(z)|\leq\frac{1}{|Q(z)|^{-R(z)}}|\log P(z)|\leq\dfrac{(m+1)\log
z+2\pi}{|c/2|z^{n}}$

yielding that $|\tilde{h}(z)|$ tends to $0$ as $z\to\infty$
($z\in\Omega^{-}$). Therefore $\tilde{h}(z)$ must be the zero function which
is absurd, since $\tilde{h}(z)\neq 0$, for all $z\in\Omega$.

In conclusion, $\Lambda_{k}$ is nonzero, for all sufficiently large $k$.
Hence, all hypotheses of Lemma 3.2 are fulfilled for $\Lambda_{k}$ and
therefore

(6) $\absolutevalue{\Lambda_{k}}>\exp(-5\cdot 10^{10}D_{k}^{4}(\log
A_{1,k})(\log A_{2,k})T_{k}^{2}),$

where $A_{1,k}\geq H(P(p_{k}/q_{k})),A_{2,k}:=H(\gamma)$, $T_{k}\geq\log
B_{k}+\log\log A_{1,k}+\log\log A_{2,k}+\log D_{k}$ (and
$B_{k}=H(Q(p_{k}/q_{k})^{R(p_{k}/q_{k})})$). Now, we need to make some
estimates in order to choose $A_{1,k},T_{k}$, and $D_{k}$.

First, in Lemma 3.1, set $F(Y,X):=Y-P(X)\in\mathbb{Z}[X,Y]$, then
$F(P(p_{k}/q_{k}),p_{k}/q_{k})=0$ and so, by that lemma, one deduces that
$H(P(p_{k}/q_{k}))\leq 3^{m+2}(1+|\xi|)^{m}H(P)q_{k}^{m}$. Thus, we can take
$A_{k}:=c_{0}q_{k}^{m}$, where $c_{0}=3^{m+2}(1+|\xi|)^{m}H(P)$.

Next, by setting $\theta_{k}:=q_{k}^{r}R(p_{k}/q_{k})$ (which is an integer),
we note that $Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}$ is a root of the polynomial
$q_{k}^{n\theta_{k}}X^{q_{k}^{r}}-(q_{k}^{n}Q(p_{k}/q_{k}))^{\theta_{k}}\in\mathbb{Q}[X]$
and so it has degree at most $q_{k}^{r}$. Thus, we can choose
$D_{k}:=q_{k}^{r}$ (since $P(p_{k}/q_{k})\in\mathbb{Q}$). Furthermore, one of
the polynomials
$q_{k}^{n\theta_{k}}X^{q_{k}^{r}}-(q_{k}^{n}Q(p_{k}/q_{k}))^{\theta_{k}}$ and
$(q_{k}^{n}Q(p_{k}/q_{k}))^{\theta_{k}}X^{q_{k}^{r}}-q_{k}^{n\theta_{k}}$ has
integers coefficients (depending on the sign of $R(p_{k}/q_{k})$). Thus, this
polynomial is divisible by the minimal polynomial of
$Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}$ (over $\mathbb{Z}$). In particular, after
some manipulations, we arrive at

$H(Q(p_{k}/q_{k})^{R(p_{k}/q_{k})})\leq q_{k}^{c_{1}q_{k}^{r}},$

where $c_{1}:=3n(1+|\xi|)^{mnr}\max\\{H(P),H(Q),H(R)\\}$. Thus, we can choose
$B_{k}:=q_{k}^{c_{1}q_{k}^{r}}$ and $T_{k}:=c_{2}q_{k}^{r}\log q_{k}$ (where
$c_{2}:=c_{1}+\log c_{0}+m+\log H(\gamma)+r$).

By using these estimates in (6), we deduce (after some straightforward
calculations) that

(7) $\absolutevalue{\Lambda_{k}}>\exp(-c_{3}q_{k}^{6r}(\log q_{k})^{3}),$

where $c_{3}:=5\cdot 10^{10}mc_{2}^{2}(\log c_{0})(\log H(\gamma))$.

On the other hand, we have

$\displaystyle\absolutevalue{\Lambda_{k}}$ $\displaystyle=$
$\displaystyle\absolutevalue{Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}\log
P(p_{k}/q_{k})-Q(\xi)^{R(\xi)}\log P(\xi)}$ $\displaystyle\leq$
$\displaystyle\absolutevalue{\log
P(p_{k}/q_{k})}\absolutevalue{Q(\xi)^{R(\xi)}-Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}}$
$\displaystyle+\absolutevalue{Q(\xi)^{R(\xi)}}\absolutevalue{\log P(\xi)-\log
P(p_{k}/q_{k})}$

where we used that $Q(\xi)^{R(\xi)}\log P(\xi)=\log\gamma$ together with the
triangle inequality. It is well-known that for a holomorphic function
$\phi:U\to\mathbb{C}$, if $z_{1},z_{2}\in U$ are such that the the line
segment with endpoints $z_{1},z_{2}$, denoted by, $[z_{1},z_{2}]$, belongs to
$U$, then
$|\phi(z_{1})-\phi(z_{2})|\leq|z_{1}-z_{2}|\cdot\max\\{|\phi^{\prime}(\zeta)|:\zeta\in[z_{1},z_{2}]\\}$.
By using this fact for the holomorphic functions $f:\Omega\to\mathbb{C}$ and
$g:\Omega\to\mathbb{C}$ given by

$f(z):=\log P(z)$ and $g(z):=Q(z)^{R(z)}$,

we obtain

$\absolutevalue{\log P(\xi)-\log P(p_{k}/q_{k})}\leq
c_{4}\absolutevalue{\xi-\frac{p_{k}}{q_{k}}}$ and $\absolutevalue{\log
P(\xi)-\log P(p_{k}/q_{k})}\leq c_{5}\absolutevalue{\xi-\frac{p_{k}}{q_{k}}}$,

where $c_{4}:=\sup_{\zeta\in I_{k}}|f^{\prime}(\zeta)|$ and
$c_{5}:=\sup_{\zeta\in I_{k}}|g^{\prime}(\zeta)|$ (here $I_{k}$ is the closed
interval with endpoints $\xi$ and $p_{k}/q_{k}$). So, we get from (3.2) that

(9)
$\absolutevalue{\Lambda_{k}}<c_{6}\absolutevalue{\xi-\frac{p_{k}}{q_{k}}}<\dfrac{c_{6}}{q_{k}^{\omega_{k}}},$

for all sufficiently large $k$ (say $k\geq\kappa_{1}$), where $c_{6}:=|\log
P(\xi)|+|Q(\xi)^{R(\xi)}|+c_{4}+c_{5}+1$.

Now, by combining (7) and (9), we obtain

$c_{6}q_{k}^{-\omega_{k}}>\exp(-c_{3}q_{k}^{6r}(\log q_{k})^{3})$

yielding

$\omega_{k}<2c_{3}q_{k}^{6r}(\log q_{k})^{2}$

for all sufficiently large $k$. However, there exist $\epsilon>0$ such that
$\nu=6r+\epsilon$ and so the last inequality can be rewritten as

$\dfrac{\omega_{k}}{q_{k}^{6r+\epsilon}}<2c_{3}\dfrac{(\log
q_{k})^{2}}{q_{k}^{\epsilon}}$

This leads to an absurd, since the left-hand side of the previous inequality
tends to infinity, while its right-hand side tends to zero when $k\to\infty$.
In conclusion, the number $P(\xi)^{Q(\xi)^{R(\xi)}}$ is transcendental and the
proof is complete. ∎

##  4\. Further comments, results, and a conjecture

We close this article by addressing a discussion on some more technical
results and possible problems for further research. In what follows, the
implied constants in $\ll$ and $\gg$ do not depend on $k$.

First, we point out that when $P(X)$ is a constant, it can be replaced by some
classes of transcendental numbers. Indeed, let us define the set $\mathcal{T}$
by

$\mathcal{T}:=\\{e^{\beta_{0}}\alpha_{1}^{\beta_{1}}\cdots\alpha_{n}^{\beta_{n}}:n\geq
1\ \mbox{and}\
(\beta_{0},\ldots,\beta_{n},\alpha_{1},\ldots,\alpha_{n})\in\overline{\mathbb{Q}}^{2n+1}\\}$

Note that the preceding set contains all algebraic numbers (case
$(n,\beta_{0},\beta_{1})=(1,0,1)$) as well as the families of transcendental
numbers proved by Hermite and Lindemann (case $(n,\alpha_{1})=(1,1)$), Gelfond
and Schneider (case $(n,\beta_{0})=(1,0)$) and Baker ($n>1$, under some mild
hypotheses).

We have the following result:

######  Proposition 4.1.

Let $(P(X),Q(X),R(X))$ be a $\overline{\mathbb{Q}}$-admissible triple of
polynomials with $R(X)\in\mathbb{Q}[X]$. If $\zeta\in\mathcal{T}$, then the
numbers

$\zeta^{Q(\xi)^{R(\xi)}}$ and $P(\xi)^{Q(\xi)^{R(\xi)}}$

do not belong to $\mathcal{T}$, for all $\xi\in\mathbb{L}^{(\infty)}$. In
particular, $\xi^{\alpha},\xi^{\xi^{r}},e^{\xi^{\xi}}$ and
$\beta^{\alpha\xi^{\xi}}$ are transcendental numbers, whenever
$\alpha\in\overline{\mathbb{Q}}\backslash\\{0\\}$,
$\beta\in\overline{\mathbb{Q}}\backslash\\{0,1\\},r\in\mathbb{Q}$ and
$\xi\in\mathbb{L}_{\text{ultra}}$.

Before the proof, we state the following lower bound for linear forms due to
Baker (see [2]). For the sake of preciseness, we state it as a lemma:

######  Lemma 4.1.

Let $n>1$ be an integer and let
$\alpha_{1},\ldots,\alpha_{n},\beta_{0},\beta_{1},\ldots,\beta_{n}$ be nonzero
algebraic numbers. Set $A=\prod_{i=1}^{n}\log(\max\\{H(\alpha_{i}),4\\})$ and
$B=\max_{i\in[1,n]}\\{H(\beta_{i}),4\\}$. Let
$D=[\mathbb{Q}(\alpha_{1},\ldots,\alpha_{n},\beta_{0},\ldots,\beta_{n}):\mathbb{Q}]$.
Then,

$|\beta_{0}+\beta_{1}\log\alpha_{1}+\cdots+\beta_{n}\log\alpha_{n}|>(AB)^{-(16nD)^{200n}A\log
A},$

provided that the previous linear form does not vanish.

###### Proof of Proposition 4.1.

We shall proceed along the same lines as in the proof of Theorem 1.2. In fact,
striving for a contradiction, suppose the existence of
$\zeta_{1},\zeta_{2}\in\mathcal{T}$ and $\xi\in\mathbb{L}^{(\infty)}$ such
that $\zeta_{1}^{Q(\xi)^{R(\xi)}}=\zeta_{2}$. By proceeding along the same
lines as before, we obtain that $|\Lambda_{k}|\ll q_{k}^{-\omega_{k}}$, where
$\Lambda_{k}:=Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}\log\zeta_{1}-\log\zeta_{2}$ is a
linear form in logarithms of algebraic numbers. By Lemma 4.1, after some
calculations, one has $|\Lambda_{k}|\gg\exp(-q_{k}^{O(1)})$. We then combine
the upper and lower bounds for $|\Lambda|$ to obtain $\omega_{k}\ll
q_{k}^{O(1)}/\log q_{k}$ which contradicts the fact that
$\lim_{k\to\infty}\omega_{k}/q_{k}^{\nu}=\infty$, for all $\nu>0$ (since $\xi$
is an $\infty$-Liouville number).

For the transcendence of $P(\xi)^{Q(\xi)^{R(\xi)}}$, we simply mimic the same
approach. The novelty here lies in the fact that $P(X)$ and $Q(X)$ have
algebraic coefficients. So, it suffices to prove that the degrees and heights
of the numbers $P(p_{k}/q_{k})$ and
$\gamma_{k}:=Q(p_{k}/q_{k})^{R(p_{k}/q_{k})}$ have the reasonable growth order
compared to the case of integer coefficients (by reasonable, we mean that if
the integer case is $\leq f(k)^{g(k)}$, then the algebraic one is
$O(f(k)^{O(g(k))})$). To prove that, let $\mathbb{K}$ be the $d$-degree number
field which contains all the coefficients of $P(X)$ and $Q(X)$. Clearly, the
degree of $P(p_{k}/q_{k})$ is at most $d$. By supposing, without loss of
generality, that the integer $q_{k}^{r}R(p_{k}/q_{k})$ is positive, then
$\gamma_{k}^{q_{k}^{r}}$ has degree at most $d$ and so $\gamma_{k}$ has degree
$\leq dq_{k}^{r}$. For the heights, we shall use the following relations
(which can be found, for instance, in [17, p. 325]): Let
$y_{1},\ldots,y_{\ell}$ be algebraic numbers of degree
$m_{1},\ldots,m_{\ell}$, respectively. Then

  * •

$(2^{m_{1}n}\sqrt{2m_{1}})^{-1}H(y_{1})^{n}\leq H(y_{1}^{n})\leq
e^{O(n)}H(y_{1})^{n}$;

  * •

$H(y_{1}y_{2})\ll(H(y_{1})H(y_{2}))^{m_{1}m_{2}}$;

  * •

$H(y_{1}+\cdots+y_{\ell})\leq e^{O(\ell)}(H(y_{1})\cdots
H(y_{\ell}))^{m_{1}\cdots m_{\ell}}$,

where the implied constants depend only on the degree of the algebraic
numbers. Therefore, by using these estimates, it is straightforward to show
that $H(P(p_{k}/q_{k}))\ll q_{k}^{m^{2}d^{m+2}}$ and $H(\gamma_{k})\ll
q_{k}^{O(q_{k}^{r})}$. Hence, we confirm that these estimates are reasonable
(in our sense) which ensures that the condition of $\xi$ being an
$\infty$-Liouville number is enough to complete the proof (we leave the
details to the reader). ∎

We now continue by recalling the following definitions:

######  Definition 4.1.

Let $k\geq 1$ be an integer. The power tower function of order $k$,
$h_{k}(x)$, is defined for all real number $x>0$ as

$h_{k}(x)=x^{\,x^{\,\cdot^{\cdot^{\cdot^{x}}}}}$ ($k$ times)

For all $x\in[e^{-e},e^{1/e}]$, $h_{k}(x)$ converges (as $k\to\infty$) to a
function $h(x)$ which is named as the infinite power tower function (see
Eisenstein [7]). In other words,

$h(x)=x^{\,x^{\,x^{\,\cdot^{\cdot^{\cdot}}}}}.$

The previous definitions imply in the following functional equations

(10) $h(x)=x^{\,h(x)},\ h_{k+1}(x)=x^{h_{k}(x)}\ \mbox{and}\
h_{k+2}(x)=x^{x^{h_{k}(x)}}.$

######  Remark 2.

We remark that $h_{1}(\xi),h_{2}(\xi)$ and $h_{3}(\xi)$ are transcendental
numbers whenever $\xi\in\mathbb{L}_{\text{ultra}}$ (by [16, Prop. 3.2] and
Corollary 1.2).

Turning back to our problems, we point out that there are at least two
directions in order to generalize the previous results, namely, by increasing
the power tower of Liouville numbers or by replacing Liouville numbers by
$U_{m}$-numbers (we refer to [3] for their definition). However, our argument
is invalid to deal even with the transcendence of $h_{4}(\xi)$ and
$h_{3}(\zeta)$, where $\xi$ is a Liouville number and $\zeta$ is a
$U_{m}$-number (with $m>1$). Indeed, in both cases, at least one of the
coefficients of the linear forms will be $h_{3}(r)$ and $h_{2}(\alpha)$, for
non-integer rational numbers $r$ and non-rational algebraic numbers $\alpha$.
Since $h_{2}(r)\not\in\mathbb{Q}$, for any
$r\in\mathbb{Q}\backslash\mathbb{Z}$, the Gelfond-Schneider theorem implies in
the transcendence of both $h_{3}(r)$ and $h_{2}(\alpha)$. Therefore, the
linear form $\Lambda$ will not have algebraic coefficients, so it is useless
in the sense of Baker’s theory.

However, there are some consequences of Proposition 4.1 concerning the
previous functions:

######  Corollary 4.1.

Let $\xi$ be a positive $\infty$-Liouville number. We have:

  * (i)

If $\xi\in[e^{-e},e^{1/e}]$, then $h(\xi)$ is a transcendental number;

  * (ii)

For any $k\geq 1$, at least one of the numbers $h_{k}(\xi)$ or $h_{k+1}(\xi)$
is transcendental.

  * (iii)

If $h_{k}(\xi)$ is a rational number, then both $h_{k+1}(\xi)$ and
$h_{k+2}(\xi)$ are transcendental numbers.

###### Proof.

The proofs are direct combinations of Proposition 4.1 and (10). In fact, if
$h(\xi)=\gamma\in\overline{\mathbb{Q}}\backslash\\{0\\}$, then $\xi^{\gamma}$
is transcendental contradicting the fact that $\xi^{\gamma}=\gamma$ (this
proves (i)). For (ii), we use that if $h_{k}(\xi)=\gamma$ is algebraic (and so
nonzero), then $h_{k+1}(\xi)=\xi^{\gamma}$ is transcendental. To prove (iii),
if $h_{k}(\xi)=r$ is a rational number, then $h_{k+1}(\xi)$ is transcendental
(by item (ii)) as well as $h_{k+2}(\xi)=\xi^{\xi^{r}}$. ∎

We finish by posing a conjecture, which we do believe to be true.

######  Conjecture 4.1.

If $\xi$ is an ultra-Liouville number, then $h_{k}(\xi)$ is transcendental,
for all $k\geq 1$.

A partial evidence to support this conjecture is that Corollary 4.1 (ii)
implies in the existence of infinitely many transcendental numbers among
$h_{4}(\xi),h_{5}(\xi),\ldots$ whenever $\xi\in\mathbb{L}^{(\infty)}$.

## Acknowledgement

D.M. was supported, in part, by CNPq-Brazil. M.O. was supported by the
Research Support Grant, Departamento de Matemática, Universidade Estadual do
Maranhão. P.T. was supported by the Project of Excellence, Faculty of Science,
University of Hradec Králové, No. 2210/2023-2024. Part of this work was done
during a very enjoyable visit of D.M. to IMPA (Rio de Janeiro) in July 2023.

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