# Classical and Quantum solutions in Scalar field cosmology via the Eisenhart
lift and linearization

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, PO Box 1334, Durban 4000, South Africa
Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos
0610, Casilla 1280 Antofagasta, Chile

###### Abstract

This study introduces a novel approach for solving the cosmological field
equations within scalar field theory by employing the Eisenhart lift. The
field equations are reformulated as a system of geodesic equations for the
Eisenhart metric. In the case of an exponential potential, the Eisenhart
metric is shown to be conformally flat. By applying basic geometric
principles, a new set of dynamical variables is identified, allowing for the
linearization of the field equations and the derivation of classical
cosmological solutions. However, the quantization of the Eisenhart system
reveals a distinct set of solutions for the wavefunction, particularly in the
presence of symmetry breaking at the quantum level.

Quantum Cosmology; Scalar Field; Eisenhart lift; Wheeler-DeWitt equation

###### pacs:

98.80.-k, 95.35.+d, 95.36.+x

##  I Introduction

Gravitational models with scalar fields play a crucial role in describing
cosmological observations ameb10 . Specifically, scalar fields provide new
degrees of freedom in the Einstein field equations, influencing the dynamics
to reproduce the observed cosmological history dataacc1 ; dataacc2 ; data1 ;
data2 ; Hinshaw:2012aka ; Ade:2015xua . Additionally, scalar fields attributes
the new degrees of freedom provided by the modification of the Einstein-
Hilbert Action with the introduction of new geometric invariants soti .

In the literature, numerous scalar field models have been proposed; for
example, refer to f1 ; f2 ; f3 ; f4 ; f5 ; f6 ; f7 ; f8 ; f9 ; f10 ; f11  and
the references therein. Quintessence represents the simplest among these
scalar field models. In this model, the scalar field is minimally coupled to
gravity, featuring a potential function to describe the scalar field mass.
Quintessence provides a straightforward mechanism for cosmic acceleration
Ratra . The universe experiences acceleration when the potential function
dominates over the kinetic term. Due to its simplicity, the quintessence model
has been applied to describe various cosmological phases, including the early-
time acceleration phase, inflation guth ; Aref1 ; de0 , and the late-time
acceleration associated with dark energy de1 ; de2 ; de3 ; de4 . Moreover, the
quintessence model has been utilized as a framework to unify the dark sector
of the universe, encompassing both dark energy and dark matter ur1 ; ur2 ; ur3
.

While the field equations of quintessence cosmology are second-order, they are
nonlinear. Unfortunately, nonlinear differential equations, in general, are
not conducive to analytic treatment. Typically, they can be addressed using
numerical methods; however, such methods may not always yield a comprehensive
understanding of the solution’s behavior across the entire solution space. On
the other hand, the asymptotic behavior of the field equations can be
examined, yet even in this scenario, information about the solution’s behavior
far from the asymptotic points remains elusive.

A systematic method that has been employed to solve nonlinear field equations
is symmetry analysis lie1 ; lie2 ; lie3 . In this approach, the demand for
field equations to be invariant under certain transformations leads to the
identification of conservation laws. These conservation laws have been
utilized to treat cosmological field equations as systems of algebraic
equations, this approach applied before in various cosmological models ns1 ;
ns2 ; ns3 ; ns4

In this study, we focus on the application of the Eisenhart lift el1 ; el2 ;
el3  to analyze cosmological field equations, specifically for the
quintessence model. The field equations of the quintessence model allow for a
minisuperspace description, forming a two-dimensional Hamiltonian system with
the scale factor and the scalar field as dynamical variables under a potential
function. The Eisenhart approach involves increasing the dimension of the
Hamiltonian system, allowing us to express the equations of motion as a set of
geodesic equations. The resulting metric is referred to as the Eisenhart
metric. We study the geometric properties for the Eisenhart metric and we show
how the field equations can be linearized. In this case we are able to solve
the field equations through linearization. Moreover, except from the classical
solution we focus and on the quantization and we derive the extended Wheeler-
DeWitt equation wdw1 ; wdw2  of quantum cosmology kim ; pinto ; manto ; Stu2 ;
sing1 ; qq1 ; qq2 ; qq3 .

The Eisenhart lift for field theories has been explored in pap0 , and the
quantization process of the Eisenhart lift was discussed in pap0a . Recent
investigations of the Eisenhart lift for the quintessence model are presented
in pap1 , covering both the Eisenhart lift and the quantization process. In a
broader context, the Eisenhart lift has been applied to analyze cosmological
field equations in pap2 . While an alternative approach involving the Jacobi
metric has been discussed in pap3 , our focus in this work remains on the
Eisenhart lift. The structure of the paper is as follows.

In Section II, we present the gravitational model under consideration, which
is that of a Friedmann–Lemaître–Robertson–Walker (FLRW) geometry with a
minimally coupled scalar field. The field equations admit a minisuperspace
description, and we derive the Wheeler-DeWitt (WDW) equation for quantum
cosmology. In Section III, we discuss the basic mathematical elements for the
Eisenhart lift. Section IV reviews some previous results on the symmetries and
conservation laws for the geodesic equations. Additionally, for the
exponential potential, the Eisenhart metric for the scalar field model is
three-dimensional, and it possesses the property that the Cotton-York tensor
is always zero. Thus, the field equations, which are invariant under conformal
transformations, can be expressed in the form of three free particles. Indeed,
the requirement for the Eisenhart metric to be conformally flat is equivalent
to the requirement for the field equations to be linearized. The classical and
quantum solutions of the linearized system are presented in Section V.
Finally, in Section VI, we summarize our results and draw conclusions.

##  II Scalar field cosmology

In Einstein’s General Relativity, the contribution of the matter source to the
curvature of the physical space with metric $g_{\mu\nu}$ is given by the field
equations ameb10

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=k\ T_{\mu\nu},$ (1)

in which the left-hand side of expression (1) corresponds to the Einstein
tensor described by the Ricci tensor $R_{\mu\nu}$, which includes the Ricci
tensor and Ricci scalar $R$. The right-hand side of expression (1) gives the
contribution of the matter source in the field equations, where $T_{\mu\nu}$
is the effective energy-momentum tensor, and its equation of motion is
$T_{~{}~{};\nu}^{\mu\nu}=0$. The parameter $k$ is the Einstein constant,
defined as $k=8\pi G$. Without loss of generality, in the following, we
consider that $k=1$.

In very large scales, the universe is described by the homogeneous and
isotropic FLRW spacetime with the line element

$ds^{2}=-dt^{2}+a^{2}(t)\frac{1}{(1+\frac{\mathit{K}}{4}\mathbf{x}^{2})^{2}}(dx^{2}+dy^{2}+dz^{2}),$
(2)

where $a(t)$ is the scale factor of the universe and $\mathit{K}=0,\pm 1$ is
the spatial curvature parameter.

For the line element (2), Einstein’s field equations (1) are reduced from a
system of nonlinear second-order partial differential equations to the
following system of nonlinear second-order ordinary differential equations
ameb10

$H^{2}+\frac{\mathit{K}}{a^{2}}=\frac{1}{3}\rho,$ (3)
$2\dot{H}+3H^{2}+\frac{\mathit{K}}{a^{2}}=-p,$ (4)

where $\rho$ and $p$ are the energy density and pressure components for the
fluid, that is

$T_{\mu\nu}=\left(\rho+p\right)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (5)

the vector field $u_{\mu}$ is the comoving observer, $u^{\mu}u_{\mu}=-1$, and
$H$ is the Hubble function, $H=\frac{\dot{a}}{a}$, related to the expansion
rate $\theta=3H,$ $\theta=u_{;\mu}^{\mu}$.

The equation of motion for the matter source reads

$\dot{\rho}+3H\left(\rho+p\right)=0.$ (6)

Assume now that the energy-momentum tensor $T_{\mu\nu}$ attributes the degrees
of freedom of a scalar field minimally coupled to gravity, which inherits the
symmetries of the background geometry. That is, Ratra

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{\phi})}{\delta
g^{\mu\nu}},$ (7)

where

$L_{\phi}\left(\phi,\phi_{;\nu}\right)=-\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-V(\phi),$
(8)

is the scalar field Lagrangian, $\phi=\phi\left(t\right)$,
$\phi_{,\nu}=\dot{\phi}\delta_{\nu}^{0}$, and $V\left(\phi\right)$ is the
scalar field potential.

The energy density and pressure components are defined as Ratra

$\rho\equiv\frac{1}{2}\dot{\phi}^{2}+V(\phi)~{},$ (9)
$p=\frac{1}{2}\dot{\phi}^{2}-V(\phi)~{},$ (10)

in which Ratra

$w_{\phi}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)},$
(11)

correspond to the equation of state parameter. In the limit where the scalar
field is negligible with respect to the potential energy, that is,
$\frac{\dot{\phi}^{2}}{2}\ll V(\phi)$, then the equation of state parameter
reaches the asymptotic limit $w_{\phi}\simeq-1~{}$Ratra .

Finally, the equation of motion (6) for the matter source is written in the
form of the Klein-Gordon equation

$\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0.$ (12)

According to cosmological observations, the spatial curvature is zero; thus,
in the following, we assume $\mathit{K~{}}=0$.

###  II.1 Minisuperspace description

The cosmological field equations (3), (4), (9), (10) and (12) have the
property to admit a minisuperspace description. Specifically, there exist the
point-like Lagrangian function ns1

$L\left(a,\dot{a},\phi,\dot{\phi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\dot{\phi}^{2}-a^{3}V\left(\phi\right),$
(13)

where variation with respect to dynamical variables $a$ and $\phi$ gives the
second-order field equations. The constraint equation (3) can be seen as the
Hamiltonian of the Lagrangian function (13).

On the other hand, if we introduce the lapse function $N\left(t\right)$ in the
metric tensor (2), that is $dt\rightarrow N\left(t\right)dt$, the point-like
Lagrangian is singular, that is,

$\bar{L}\left(N,a,\dot{a},\phi,\dot{\phi}\right)=\frac{1}{N}\left(-3a\dot{a}^{2}+\frac{1}{2}a^{3}\dot{\phi}^{2}\right)-Na^{3}V\left(\phi\right),$
(14)

in which now the constraint equation (3) follows from the variation of (14)
with respect to the lapse function $N$, that is, the constraint equation reads
$\frac{\partial L}{\partial N}=0$.

From the point-like Lagrangian we define the two-dimensional minisuperspace
$\gamma_{ij}$ with diagonal line element

$ds_{m}^{2}=-6ada^{2}+a^{3}d\phi^{2}.$ (15)

We introduce the Legendre transformation, $p_{a}=\frac{\partial
L}{\partial\dot{a}},~{}p_{\phi}=\frac{\partial L}{\partial\dot{\phi}},$ where
$p_{a}$ and $p_{\phi}$ are the momentum, thenthe field equations read ns1

$-\frac{1}{12a}p_{a}^{2}+\frac{1}{2a^{3}}p_{\phi}^{2}+a^{3}V\left(\phi\right)=0.$
(16)
$\dot{a}=-\frac{1}{6a}\dot{p}_{a}~{},~{}\dot{\phi}=\frac{1}{a^{3}}p_{\phi}~{},~{}\dot{p}_{\phi}=-a^{3}V_{,\phi},$
(17)
$\dot{p}_{a}=-\frac{1}{12}\frac{p_{a}^{2}}{a^{2}}+\frac{3}{2}\frac{p_{\phi}^{2}}{a^{4}}-3a^{2}V\left(\phi\right).$
(18)

or, in the presence of the lapse function

$\frac{1}{N}\left(-\frac{1}{12a}p_{a}^{2}+\frac{1}{2a^{3}}p_{\phi}^{2}\right)+Na^{3}V\left(\phi\right)=0.$
(19)
$\frac{1}{N}\dot{a}=-\frac{1}{6a}p_{a}~{},~{}\frac{1}{N}\dot{\phi}=\frac{1}{a^{3}}p_{\phi}~{},~{}\frac{1}{N}\dot{p}_{\phi}=-a^{3}V_{,\phi},$
(20)
$\frac{1}{N}\dot{p}_{a}=-\frac{1}{12}\frac{p_{a}^{2}}{a^{2}}+\frac{3}{2}\frac{p_{\phi}^{2}}{a^{4}}-3a^{2}V\left(\phi\right).$
(21)

###  II.2 Quantization

In the context of the $3+1$ decomposition notation, the WDW equation is
derived from the Hamiltonian constraint of General Relativity and can be
expressed as follows wdw1 ; wdw2

$\mathcal{H}\Psi=\left[-4\kappa^{2}\mathcal{G}_{ijkl}\frac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}+\frac{\sqrt{h}}{4\kappa^{2}}\left(-\mathcal{R}+2\Lambda+4\kappa^{2}T^{00}\right)\right]\Psi=0,$
(22)

where

$\mathcal{G}_{ijkl}=\frac{1}{2\sqrt{h}}\left(h_{\mu\kappa}h_{\nu\lambda}+h_{\mu\lambda}h_{\nu\kappa}-h_{\mu\nu}h_{\kappa\lambda}\right),$
(23)

is the the metric of superspace, that is, the space of all 3-geometries, with
metric $h_{ij}$ and Ricci scalar $R$, and the matter configuration.

In the case of a scalar field we find

$T^{00}=-\frac{1}{2h}\frac{\delta^{2}}{\delta\phi^{2}}+\frac{1}{2}h^{\mu\nu}\phi_{,\mu}\Phi_{,\nu}+V(\Phi).$
(24)

We note that equation (22) does not represent a single differential equation
but rather a family of equations on the -dimensional hypersurface. However, in
the presence of a minisuperspace, the infinite degrees of freedom of the
superspace are reduced to a finite number, leading to the simplification of
the WDW equation into a single equation.

The Hamiltonian constraint (19) reads wdw1 ; wdw2

$\mathcal{H}\equiv\frac{1}{N^{2}}\left(\gamma_{ij}p^{i}p^{j}\right)+2\tilde{V}=0$
(25)

and the WDW equation follows from the quantization of (25), that is,

$\overset{\char 94\relax}{\mathcal{H}}\Psi=0.$

Yet, since the field equations remain invariant regardless of the chosen lapse
function, which functions as a conformal factor, we ensure the invariance of
the WDW equation under conformal transformations. Instead of employing the
typical Laplace operator, we utilize the conformal Laplace operator

$\hat{L}\equiv-\Delta-\frac{n-2}{4\left(n-1\right)}R_{\gamma},$ (26)

in which
$\Delta=~{}\frac{1}{\sqrt{\left|\gamma\right|}}\frac{\partial}{\partial
x^{i}}\left(\sqrt{\left|\gamma\right|}\frac{\partial}{\partial x^{j}}\right)$
is the Laplacian operator, $n=\dim\gamma_{ij}$, and $R_{{}_{\gamma}}$ is the
Ricci scalar for the minisuperspace. In the case of two-dimensional
minisuperspace it follows $\hat{L}=\Delta.$

Thus, for the minisupespace (15), we write the WDW equation qq1

$\Delta\Psi-2a^{3}V\left(\phi\right)\Psi=0,$ (27)

in which operator $\Delta$ is defined as

$\Delta\equiv-\frac{1}{6a}\left(\frac{\partial^{2}}{\partial
a^{2}}+\frac{\partial}{\partial
a}\right)+\frac{1}{a^{3}}\frac{\partial^{2}}{\partial\phi^{2}}.$ (28)

##  III The Eisenhart lift

The Jacobi metric and the Eisenhart lift delineate distinct methods for
geometrically representing dynamical systems. Notably, autonomous dynamical
systems can be formulated as a set of geodesic equations. In this
investigation, we focus on the Eisenhart lift, a technique that involves
augmenting the dimensionality of the dynamical system. Specifically, this
geometrization process entails introducing additional dimensions through the
inclusion of new dependent variables. A novel kinetic metric is introduced,
characterized by at least one isometry associated with a Noetherian
conservation. When this isometry is applied, the geodesic equations are
reduced back to the original dynamical system.

To demonstrate this, we consider the Hamiltonian

$H_{n}\equiv\frac{1}{2}\gamma^{ij}\left(q\right)p_{i}p_{j}+V\left(q\right)-h_{n}=0,$

with equations of motion

$\dot{q}^{i}=\gamma^{ij}p_{j}~{},~{}\dot{p}_{i}=-\frac{1}{2}\gamma_{~{}~{},i}^{jk}p_{j}p_{k}-V_{,i}.$
(29)

###  III.1 Riemannian lift

We introduce the new Hamiltonian eisn

$H_{n+1}=\frac{1}{2}\gamma^{ij}\left(q\right)p_{i}p_{j}+\frac{1}{2}\alpha
V\left(q\right)p_{z}^{2}-h_{n+1}=0$ (30)

with equations of motion

$\dot{q}^{i}=\gamma^{ij}p_{j}~{},~{}\dot{z}=\alpha V\left(q\right)p_{z},$ (31)

and

$\dot{p}_{i}=-\frac{1}{2}\gamma_{~{}~{},i}^{jk}p_{j}p_{k}-\frac{1}{2}aV_{,i}p_{z}^{2}~{},~{}\dot{p}_{z}=0.$
(32)

From the latter we defin the conservation law $p_{z}=p_{z0}$. The dynamical
system described by the Hamiltonian equation $H_{n}$ is recovered from the
Hamiltonian $H_{n+1}$, when $\frac{1}{2}ap_{z}^{2}=1$ and $h_{n+1}=h_{n}.$

The new $n+1$ metric with line element

$ds_{EL\left(n+1\right)}^{2}=\gamma_{ij}dq^{i}dq^{j}+\frac{1}{\alpha
V\left(q\right)}dz^{2},$ (33)

is the Eisenhart metric. This lift is also known as Riemannian lift.

###  III.2 Lorentzian lift

An alternative apporach for the constuction of the Eisenhart metric is given
by the Lorentzian lift, also known as Eisenhart-Duval lift pap2 . In this
approach we consider the new Hamiltonian system eisn

$H_{n+2}=\frac{1}{2}\gamma^{ij}\left(q\right)p_{i}p_{j}+V\left(q\right)p_{u}^{2}+p_{u}p_{v}-h_{n+2},$
(34)

where the Eisenhart metric is

$ds_{EL\left(n+2\right)}^{2}=\gamma_{ij}dq^{i}dq^{j}+2dudv-2Vdu^{2}\text{.}$

The corresponding equations of motion are

$\dot{q}^{i}=\gamma^{ij}p_{j}~{},~{}\dot{u}=2V\left(q\right)p_{u}+p_{v}~{},~{}\dot{v}=p_{v},$
(35)

and

$\dot{p}_{i}=-\frac{1}{2}\gamma_{~{}~{},i}^{jk}p_{j}p_{k}-V_{,i}p_{u}^{2}~{},~{}\dot{p}_{u}=0,~{}\dot{p}_{v}=0\text{.}$
(36)

Therefore, the dynamical system described by the Hamiltonian function $H_{n}$
is recovered from the latte equtions of motion for $p_{u}^{2}=1$ and
$p_{u}p_{v}-h_{n+2}=h_{n}$.

##  IV Symmetries and conservation laws

We proceed our discussion by presenting some significant findings regarding
the symmetries of the geodesic equations and the conformal Laplace operator.

###  IV.1 Noether symmetries of the geodesic Lagrangian

Consider the geodesic Lagrangian

$L\left(q^{k},\dot{q}^{k}\right)=\frac{1}{2}\gamma_{ij}\left(q^{k}\right)\dot{q}^{i}\dot{q}^{j}~{},~{}\dot{q}^{i}=\frac{dq^{i}}{dt}$
(37)

with “energy”
$\frac{1}{2}\gamma_{ij}\left(q^{k}\right)\dot{q}^{i}\dot{q}^{j}=h$. Then under
the Action of the infinitesimal transformation

$\displaystyle\bar{t}$
$\displaystyle=t+\varepsilon\xi\left(t,q^{k}\right)~{},$ (38)
$\displaystyle\bar{q}^{i}$
$\displaystyle=q^{i}+\varepsilon\eta^{i}\left(t,q^{k}\right)~{},$ (39)

the variation $\delta S$ of the Action $S=\int
L\left(q^{k},\dot{q}^{k}\right)dt$, remain invariant, if and only if, there
exist a function $f\left(t,q^{k}\right)$ such that noe

$X^{\left[1\right]}L+L\dot{\xi}=\dot{f},$ (40)

where $X^{\left[1\right]}$ is the first extension for the generator
$X=\xi\left(t,q^{k}\right)\partial_{t}+\eta^{i}\left(t,q^{k}\right)\partial_{i}$
of the infinitesimal transformation defined as

$X^{\left[1\right]}=X+\left(\dot{\eta}^{i}-\dot{\xi}\dot{q}^{i}\right)\partial_{\dot{q}^{i}}.$
(41)

The condition (40) for the variational symmetry is recognized as Noether’s
first theorem, and the generator associated with the transformation $X$ is
identified as the Noether symmetry.

For a given Noether symmetry $X$, Noether’s second theorem states that the
function

$\Phi\left(X\right)=\xi h-\frac{\partial L}{\partial\dot{q}^{i}}\eta^{i}+f.$
(42)

is a conservation law for the equations of motion.

The Noether symmetries for the geodesic Lagrangian (37) have been widely
studied in the literature. Specifically, it has been found that the Noether
symmetries for the geodesice Lagrangian (37) are constructed by the elements
of the Homothetic algebra for the metric tensor
$\gamma_{ij}\left(q^{k}\right)$, that is, by the Killing symmetries and the
Homothetic symmetry, for the exact mathematical relation we refer the reader
to noegrg .

However, in the special case where the system is constraint, that is, $h=0$,
the geodesic equations are invariant under conformal transformations palgrg .
As a result, the Noether symmetries are constructed by the elements of the
Conformal algebra of the metric tensor $\gamma_{ij}\left(q^{k}\right)$, that
is, by the Killing symmetries, the Homothetic symmetry and the Conformal
symmetries. For this consideration, if $\eta^{i}\left(q^{k}\right)$ is a
Conformal symmetry for the metric tensor $\gamma_{ij}\left(q^{k}\right)$, then
the quantity $\Phi\left(\eta^{i}\right)=\frac{\partial
L}{\partial\dot{q}^{i}}\eta^{i}$ is a conservation law, for more details see
ndim  and references therein.

###  IV.2 Symmetries for the conformal Laplace operator

Consider the Yamabe equation

$\left(\Delta+\frac{n-2}{4\left(n-1\right)}R\right)\Psi\left(q^{i}\right)=0,$
(43)

in which $\Delta$ is the Laplacian for the metric tensor
$\gamma_{ij}\left(q^{k}\right)$, with $n=\dim\gamma_{ij}$.

Let
$Y=\eta^{i}\left(q^{k},\Psi\right)\partial_{i}+\zeta\left(q^{k},\Psi\right)\partial_{\Psi}$
be the generator of the infinitesimal transformation

$\displaystyle\bar{q}^{i}$
$\displaystyle=q^{i}+\varepsilon\eta^{i}\left(q^{k},\Psi\right)~{},$ (44)
$\displaystyle\bar{\Psi}$
$\displaystyle=\Psi+\varepsilon\zeta\left(q^{k},\Psi\right)~{}.$ (45)

Then, equation (43) remain invariant if and only if yamabe

$Y=\eta^{i}\left(q^{k}\right)\partial_{i}+\left(\frac{2-n}{2}\psi\left(q^{k}\right)\Psi+a_{0}\Psi+b\left(q^{k}\right)\right)\partial_{\Psi}$

where $\eta^{i}\left(q^{k}\right)$ is a Conformal symmetry for the metric
tensor $\gamma_{ij}\left(q^{k}\right)$ with conformal factor
$\psi\left(q^{k}\right)$, $a_{0}$ is a constant and $b\left(q^{k}\right)$ is
an arbitrary solution of equation (43). The latter two symmetry vectors
indicate that the Yamabe equation is linear. There exist a unique connection
between the symmetries of the Yamabe equation and the Noetherian conservation
laws for the null geodesic Lagrangian.

The maximum number of admitted conformal symmetries by the metric
$\gamma_{ij}\left(q^{k}\right)~{}$are
$\frac{1}{2}\left(n+1\right)\left(n+2\right)$. In that case the Yamabe
equation will be characterized as maximally symmetric. When the latter is
true, $\gamma_{ij}\left(q^{k}\right)$ is conformally flat, and the Yamabe
equation can be written as the wave equation of the flat space, while the
classical geodesic equations can be linearized.

The criterio in order the tensor $\gamma_{ij}\left(q^{k}\right)$ depends on
the dimension of the space. Indeed, all two-dimensional spaces are conformally
flat ky . A three-dimensional space is conformally flat when the Cotton-York
tensor ky

$C_{\mu\nu\kappa}=R_{\mu\nu;\kappa}-R_{\kappa\nu;\mu}+\frac{1}{4}\left(R_{;\nu}g_{\mu\kappa}-R_{;\kappa}g_{\mu\nu}\right),~{}n=3$
(46)

is zero. Finally, for higher-dimensional spaces, the necessary condition in
order the metric tensor to be conformally flat is that the Weyl tensor be
zero, that is, ky

$C_{\mu\nu\kappa\lambda}=R_{\mu\nu\kappa\lambda}-\frac{2}{n-2}\left(g_{\mu[\kappa}R_{\lambda]\nu}-g_{\nu[\kappa}R_{\lambda]\mu}\right)+\frac{2}{\left(n-1\right)\left(n-2\right)}Rg_{\mu[\kappa}g_{\lambda]\nu}.$
(47)

##  V The exponential potential

We apply the Riemannian lift in order to write the gravitational field
equations of scalar field cosmology (3), (4), (9), (10) and (12) as a set of
geodesic equations. We introduce the Hamiltonian function

$H_{RL}\equiv-\frac{1}{12a}p_{a}^{2}+\frac{1}{2a^{3}}p_{\phi}^{2}+\frac{1}{2}a^{3}V\left(\phi\right)p_{z}^{2}=0.$
(48)

and Lagrangian function

$L_{RL}\left(a,\dot{a},\phi,\dot{\phi},z,\dot{z}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\dot{\phi}^{2}+\frac{1}{2a^{3}V\left(\phi\right)}\dot{z}^{2}.$
(49)

The three-dimensional minisuperspace has the following line element

$ds_{RL}^{2}=-6ada^{2}+a^{3}d\phi^{2}+\frac{1}{a^{3}V\left(\phi\right)}dz^{2}=0.$
(50)

The requirement the latter spacetime to be conformally flat gives the
constraint equations

$\displaystyle V_{,\phi\phi}V-V_{,\phi}^{2}$ $\displaystyle=0,$ (51)
$\displaystyle V^{2}V_{,\phi\phi\phi}-4VV_{,\phi}V_{,\phi\phi}+3V_{,\phi}^{3}$
$\displaystyle=0.$ (52)

Hence, it follows the unique potential

$V\left(\phi\right)=V_{0}e^{-\lambda\phi}.$ (53)

We remark that that for the exponential scalar field potential, the geodesic
equations described by the Hamiltonian (48) can be linearized.

###  V.1 The classical solution through linearization

We introduce the change of variables

$a=x^{\frac{1}{6-\sqrt{6}\lambda}}y^{\frac{1}{6+\sqrt{6}\lambda}}~{},~{}\phi=\frac{1}{\lambda^{2}-6}\left(\left(\sqrt{6}-\lambda\right)\ln
y-\left(\sqrt{6}+\lambda\right)\ln x\right).$ (54)

In the new coordinates the geodesic Lagrangian (49) becomes

$L_{RL}\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=x^{1+\frac{3}{\sqrt{6}\lambda-6}}y^{1-\frac{3}{\sqrt{6}\lambda+6}}\left(\frac{2}{\lambda^{2}-6}\dot{x}\dot{y}+\frac{1}{2V_{0}}\dot{z}^{2}\right).$
(55)

We introduce the lapse function
$N\left(t\right)=x^{1+\frac{3}{\sqrt{6}\lambda-6}}y^{1-\frac{3}{\sqrt{6}\lambda+6}}$,
that is$~{}N\left(t\right)=a^{3}e^{-\lambda\phi}$. Hence, the conformal
Lagrangian reads

$\bar{L}_{RL}\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=\left(\frac{2}{\lambda^{2}-6}\dot{x}\dot{y}+\frac{1}{2V_{0}}\dot{z}^{2}\right).$
(56)

The cosmological field equations are

$\ddot{x}=0~{},~{}\ddot{y}=0~{},~{}\ddot{z}=0,$ (57)

with constraint equation

$\left(\frac{2}{\lambda^{2}-6}\acute{x}\dot{y}+\frac{1}{2V_{0}}\dot{z}^{2}\right)=0~{},~{}\frac{1}{2}~{}\left(\frac{1}{V_{0}}\dot{z}\right)^{2}=1\text{.}$
(58)

The analytic solution for quintessence model with exponential potential
derived before in Russo . However, in this study we solve the field equations
through the linearization.

On the other hand, for $\lambda=\sqrt{6}$, the transformation which linearizes
the field equations is

$a=x^{\frac{1}{12}}e^{\frac{y}{2}}~{},~{}\phi=\sqrt{6}\left(\frac{y}{2}-\frac{1}{12}\ln
x\right),$ (59)

where the geodesic Lagrangian becomes

$\bar{L}_{RL}\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=-\frac{1}{2}\acute{x}\dot{y}+\frac{1}{2V_{0}}\dot{z}^{2}.$
(60)

On the other hand, for $\lambda=-\sqrt{6}$ the corresponding transformation is

$a=6^{\frac{1}{6}}e^{\frac{x}{12}}y^{\frac{1}{12}}~{},~{}\phi=\frac{\sqrt{6}}{12}\left(\ln
y-x\right),$ (61)

where the resulting Lagrangian is again (60).

###  V.2 New solutions for the WDW equation

From the Lagrangian functions (56) and (60) we derive the WDW equation

$\Psi_{,xy}-\frac{1}{2}\bar{V}_{0}\Psi_{,zz}=0,$ (62)

in which $\bar{V}_{0}=\frac{V_{0}}{2}$ for $\lambda^{2}=6$ or
$\bar{V}_{0}=\frac{2}{\left(\lambda^{2}-6\right)}V_{0}$ for $\lambda^{2}\neq
6$.

Equation (62) is the wave equation for the three-dimensional flat space and it
is maximally symmetric. Except from the trivial symmetries
$Y^{0}=\Psi\partial_{\Psi}$ and $Y^{\inf}=b\left(x,y,z\right)\partial_{\Psi}$,
equation (62) admits ten symmetry vectors generated by the ten Conformal
symmetries of the flat space.

The ten symmetry vectors are expressed as follows

$Y^{1}=\partial_{x}~{},~{}Y^{2}=\partial_{y}~{},~{}Y^{3}=\partial_{z}~{},~{}$
$Y^{4}=x\partial_{x}-y\partial_{y}~{},~{}Y^{5}=z\partial_{x}+\bar{V}_{0}y\partial_{z}~{},~{}Y^{6}=z\partial_{y}+\bar{V}_{0}x\partial_{z}~{},$
$Y^{7}=x\partial_{x}+y\partial_{y}+z\partial_{z}~{},$
$Y^{8}=2\left(x^{2}\partial_{x}+z^{2}\partial_{y}+2xz\partial_{z}\right)-\bar{V}_{0}x\Psi\partial_{\Psi}~{},$
$Y^{9}=2\left(y^{2}\partial_{y}+z^{2}\partial_{y}+2yz\partial_{z}\right)-\bar{V}_{0}y\Psi\partial_{\Psi}~{},$
$Y^{10}=2z\left(x\partial_{x}+y\partial_{y}\right)+\left(2xy\bar{V}_{0}+z^{2}\right)-z\Psi\partial_{\Psi}~{}.$

The symmetry vectors keep invariant the differential equation and the solution
$\Psi$. Thus, if $Y$ is a symmetry vector it follows that $Y\Psi=0$.

From the vector field $Y^{3}-\alpha_{0}Y^{0}~{}$
$\left(\partial_{z}-\alpha_{0}\Psi\partial_{\Psi}\right)$, we define the
operator $\partial_{z}\Psi-\alpha_{0}\Psi=0$, which gives that
$\Psi\left(x,y,z\right)=\Psi_{0}\left(x,y\right)e^{\alpha_{0}z}$. By replacing
in (62) we end with the two-dimensional equation

$\Psi_{0,xy}-\frac{\alpha_{0}^{2}}{2}\bar{V}_{0}\Psi_{0}=0$ (63)

which is nothing else that the WDW equation for the scalar field model with
exponential potential. The symmetry vectores of the latter equation are the
fields $Y^{1},~{}Y^{2}\,$and$~{}Y^{4}$. The application of these vector fields
leads to the the solutions for the WDW equation derived in anmpla .
Nevertheless, the application of other symmetry vector fields lead to new
solutions for the WDW equation.

For instance from the vector fields $Y^{1}-\alpha_{0}Y^{0}$ and
$Y^{6}-\beta_{0}Y^{0}$ we derive the closed-form solution

$\Psi\left(x,y,z\right)=\frac{\Psi_{0}}{\sqrt{y}}\exp\left(\alpha_{0}xy-\frac{\left(\beta_{0}-\alpha_{0}z\right)^{2}}{2\alpha_{0}\bar{V}_{0}}\right).$
(64)

On the other hand, from the symmetries $Y^{4}-\alpha_{0}Y^{0}$ and
$Y^{7}-\beta_{0}Y^{0}$ we end with the solution

$\Psi\left(x,y,z\right)=x^{\frac{\alpha_{0}-\beta_{0}}{2}}y^{\frac{\alpha_{0}+\beta_{0}}{2}}\Psi_{1}\left(\sigma\right)~{},~{}\sigma=\frac{z}{\sqrt{xy}},$
(65)

in which

$\left(2\bar{V}_{0}-\sigma^{2}\right)\Psi_{1,\sigma\sigma}-\sigma\left(1+2\beta_{0}\right)\Psi_{1,\sigma}+\left(\alpha_{0}^{2}-\beta_{0}^{2}\right)\Psi_{1}=0.$
(66)

In a similar way we can construct various solutions for the WDW equation (62).
For other solutions of the wave equation we refer the reader to barb .

##  VI Conclusions

In the classical limit, the existence of additional conservation laws for the
field equations in the Eisenhart metric helps us linearize the field
equations, allowing us to express analytical solutions in a simplified form.
However, the situation is different for the WDW equation.

In the Eisenhart metric, additional symmetry vectors exist, leading to the
derivation of new solutions for the universe’s wavefunction. Notably, these
new solutions are derived without applying any initial or boundary conditions
for the partial differential equations. The necessity of the condition
$\left(\partial_{z}-\alpha_{0}\Psi\partial_{\Psi}\right)\Psi=0$ arises to
derive the WDW equation for the original scalar field model.

Nevertheless, the geodesic Lagrangian (49) and the scalar field Lagrangian
(14) describe the same dynamical system, wherein the potential terms are
expressed as dynamical variables. Consequently, the quantization process
differs, as the Eisenhart metric introduces new dynamical degrees of freedom.
However, all the wavefunctions describe the same classical system. The
imposition of the condition
$\left(\partial_{z}-\alpha_{0}\Psi\partial_{\Psi}\right)\Psi=0$ is solely
based on the mathematical requirement to restore the WDW equation for the
original model. Thus, the violation of this condition, which can be viewed as
a symmetry breaking, leads to new wavefunctions and potentially introduces new
behaviors in the very early stages of the universe. The possible new solution
for the quantum equation throught the Eisenhart lift has been discussed before
pap0a , and it was mentioned for the case of the quintessence scalar field in
pap1 . However, no new solutions derived.

In the pressence of an ideal gas with equation of state parameter
$p_{m}=\left(\gamma-1\right)\rho_{m}$ the cosmological Lagrangian becomes

$L\left(a,\dot{a},\phi,\dot{\phi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\dot{\phi}^{2}-a^{3}V\left(\phi\right)-\rho_{m0}a^{-3\left(\gamma-1\right)}.$
(67)

Hence, the Eisenhart lift leads to the three-dimensional metric with line
element

$ds_{EL}^{2}=-6ada^{2}+a^{3}d\phi^{2}+\frac{1}{a^{3}V\left(\phi\right)+\rho_{m0}a^{-3\left(\gamma-1\right)}}dz^{2}.$
(68)

The requirement the latter line element to be conformally flat gives the
scalar field potentials

$V\left(\phi\right)=V_{0}\exp\left(\pm\frac{\sqrt{6}}{2}\gamma\phi\right).$
(69)

Thus, there exists a transformation, as before, where the field equations for
this specific scalar field potential can be linearized.

In this study, we have considered the linearization of the field equations as
an approach to construct solutions through the Eisenhart lift. However, this
is not the only approach that we can apply for the study of the dynamical
system.

Although the Eisenhart lift is a mathematical method for solving nonlinear
dynamical systems, we demonstrate that it can offer new insights into the
physical properties of quantum solutions. In a future study, we plan to
investigate the physical properties of symmetry breaking for the WDW equation.

Data Availability Statements: Data sharing is not applicable to this article
as no datasets were generated or analyzed during the current study.

###### Acknowledgements.

The author thanks the support of Vicerrectoría de Investigación y Desarrollo
Tecnológico (Vridt) at Universidad Católica del Norte through Núcleo de
Investigación Geometría Diferencial y Aplicaciones, Resolución Vridt No -
096/2022 and 098/2022.

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# Matching of observations of dynamical systems, with applications to sequence
matching

Théophile Caby,
CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do
Porto,
Rua do Campo Alegre s/n, 4169007 Porto, Portugal.
<EMAIL_ADDRESS>
###### Abstract

We study the statistical distribution of the closest encounter between generic
smooth observations computed along different trajectories of a rapidly mixing
dynamical system. At the limit of large trajectories, we obtain a distribution
of Gumbel type that depends on both the length of the trajectories and on the
Generalized Dimensions of the image measure. It is also modulated by an
Extremal Index, that informs on the tendency of nearby observations to diverge
along with the evolution of the dynamics. We give a formula for this quantity
for a class of chaotic maps of the interval and regular observations. We
present diverse numerical applications illustrating the theory and discuss the
implications of these results for the study of physical systems. Finally, we
discuss the connection between this problem and the problem of the longest
matching block common to different sequences of symbols. In particular, we
obtain a distributional result for strongly mixing processes.

##  1 Introduction

Certain real-world systems, such as climate, take place in high-dimensional
spaces and exhibit chaotic and multi-scaled properties. To study such complex
dynamics, physicists often have access to only a limited number of observable
quantities through the measurement process. The latter can be modeled by
computing an observation function along a typical trajectory of the system.
Understanding the geometric and statistical properties of such observations,
and their relationship to the properties of the original underlying system is
a problem of great interest in physics, that has been instigated only
recently. The study of the recurrence properties of observations have been
initiated by Rousseau and Saussol in [33, 32], in which asymptotic and
distributional results were obtained for both hitting times and return times
of observations. In a recent paper [12], this problem was studied from the
point of view of Extreme Value Theory (EVT). This approach turned out to
provide information on the local geometry of the image measure, which, for a
good choice of observation, can characterize the local fractal structure of
the original underlying attractor. In this paper, we pursue the statistical
analysis of observed dynamical systems by studying the statistics of the
shortest distance between several observed trajectories. Closely related
problems have gained interest in recent years. The case of real, unobserved
trajectories was considered in [19] and [11], using EVT techniques, while
asymptotic results for the shortest distance between two orbits were obtained
in [7] and then generalized to multiple orbits [8] and finally to observed
orbits [14].

Yet another motivation to study this problem is its deep relationship with a
seemingly distinct one; the length of the longest matching block common to
different sequences of symbols drawn from the same probability distribution.
This old problem has been initiated by Waterman and Arratia, who brought a
plethora of results in the i.i.d. case [5, 3], most of which are presented in
the reference book [35]. Several authors have extended these results, giving
for example distributional results in the i.i.d case [4, 31]. In many
applications, however, the sequences cannot be modeled as i.i.d. sequences.
For example, in biological applications, genes are specific sequences encoding
information, and DNA brands do not constitute independent sequences of
nucleotides. When it comes to written text, a complex dependence structure can
arise from specific sequences of letters, such as words, and higher-order
syntactic and narrative structures. Recently, Barros, Liao and Rousseau
adopted a dynamical system point of view to give the asymptotic behavior of
the length of the longest sub-sequence common to different $\alpha-$mixing
sequences [7, 8]. This problem is different from the present one, because the
sub-sequences may be present at different locations of the different strings
of symbols, but we will also follow a dynamical system approach to derive our
results.

The paper is organized as follows: In the first section, we present the
problem and derive our main result concerning the convergence of the
statistics of observation matching to a Gumbel distribution. In the following
sections, we discuss the parameters of the limit law, since these quantities
provide relevant dynamical information on the system and can be estimated
numerically. We first focus on the generalized dimensions of the image measure
by emphasizing their central role in the statistical properties of
observations. We also study their relations with the generalized dimension
spectrum of the original measure. In the third section, we derive a formula
for the Extremal Index associated with this problem for a class of chaotic
maps of the interval and perform a numerical study of this index for higher
dimensional systems. In the last part, we present some applications of our
results to sequence matching problems. In particular, we obtain distributional
results for the length of the longest sequence of symbols common to
independent strings of symbols drawn from the same strongly mixing probability
measure.

##  2 The general approach

Let us consider the dynamical system $(\mathscr{M},T,\mu)$, where
$\mathscr{M}$ denotes the phase space and $T:\mathscr{M}\to\mathscr{M}$ is a
discrete transformation 111it could be a discretized version of a flow that
leaves the probability measure $\mu$ invariant. In order to model the process
of measurement, we consider a $C^{1}$ function $f:\mathscr{M}\to\mathscr{J}$,
which we refer to as the observation. Both the phase space $\mathscr{M}$ and
the observation space $\mathscr{J}$ are compact metric spaces endowed with two
distances that we will both call $d$ to simplify notations. For physical
applications, we take $\mathscr{J}\subset\mathbb{R}^{m}$, as observational
data usually consists of a collection of real numbers that can be arranged
into vectors. For applications to the problem of sequence matching, we will
take $\mathscr{M}$ to be the space of all infinite sequences of symbols of a
given alphabet $\mathscr{A}$. Because we are interested in the statistical
properties of observations, we need a measure that is supported in the
observational space.

###### Definition 1

We call the push-forward, or image of the measure $\mu$ by the function $f$,
the measure $\mu_{f}$ defined by

$\mu_{f}(A)=\mu(f^{-1}(A)),$

for all $A\subset\mathscr{J}$ such that $f^{-1}(A)$ is $\mu-$measurable.

A more detailed presentation of this object is available in [33] and a
discussion of its properties can be found in [12].

###### Definition 2

We call the generalized dimension of order $q\neq 1$ of the image measure
$\mu_{f}$ the following quantity (if it exists):

$D_{q}^{f}=\underset{r\to
0}{\lim}\frac{\log\int_{\mathscr{J}}\mu_{f}(B(y,r))^{q-1}d\mu_{f}(y)}{(q-1)\log
r}.$ (1)

$B(y,r)$ denotes a ball centered at $y\in\mathscr{J}$ of radius $r$.

The information dimension of $\mu_{f}$ is defined as

$D_{1}^{f}=\underset{q\to 1}{\lim}D_{q}^{f}.$ (2)

We write $D_{q}=D_{q}^{Id}$, the generalized dimension of order $q$ of the
original measure.

We will place ourselves in physical situations where the limits defining the
previous quantities exist. Now that we have introduced the important objects
of the theory, we go forward and consider the following process:

$Y_{i}=-\log\max_{j=2,\dots,q}d(f(T^{i}x_{1}),f(T^{i}x_{j})),$

$(x_{1},...,x_{q})\in\mathscr{M}^{q}$ being a starting point drawn from the
product measure $\mu_{q}$ with support in $\mathscr{M}^{q}$. To follow the
usual procedure of Extreme Value Theory, we consider a sequence of thresholds
$u_{n}(s)$, where $s\in\mathbb{R}$, such that:

$\mu_{q}(Y_{0}>u_{n}(s))\sim\frac{e^{-s}}{n}.$ (3)

Since the $q$ trajectories are independent, we also have:

$\displaystyle\mu_{q}(Y_{0}>u_{n}(s))$
$\displaystyle=\int_{\mathscr{J}}\mu_{f}(B(y,e^{-u_{n}}))^{q-1}d\mu_{f}(y)$
(4) $\displaystyle\sim e^{-u_{n}D^{f}_{q}(q-1)},$

from definition 2. To satisfy both scalings 3 and 4, we take

$u_{n}(s)=\frac{\log n}{D_{q}^{f}(q-1)}+\frac{s}{D_{q}^{f}(q-1)}.$

Now, for a given threshold $u_{n}$, the quantity $\mu_{q}(Y_{0}>u_{n})$ gives
the probability of having all the observations contained in the same small
region of the observational space; a ball of radius $e^{-u_{n}}$ centered at
$f(x_{1})$. Equivalently, it gives the probability that the product dynamics
has entered the following target set:

$S^{q}_{n}=\\{(s_{1},...,s_{q})\in\mathscr{M}^{q},\max_{j=2,\dots,q}d(f(s_{1}),f(s_{j}))<e^{-u_{n}}\\}.$

Following the ideas of [23], studying the behavior of the maximum of the
process $(Y_{i})$ over a trajectory of size n:

$M_{n,q}(x_{1},...,x_{q})=\max\\{Y_{0},\dots,Y_{n-1}\\},$

and in particular its cumulative distribution:

${F_{n}}(u_{n})=\mu_{q}(\\{(x_{1},...,x_{q})\in\mathscr{M}^{q}\mbox{ s.t.
}M_{n,q}(x_{1},...,x_{q})\leq u_{n}\\}),$

is equivalent to studying the Hitting Time Statistics of the product dynamics
in the set $S^{q}_{n}$. Indeed, ${F_{n}}(u_{n})$ gives the probability that
the dynamics has not entered the set $S^{q}_{n}$ after $n$ iterations of the
dynamics. We can now apply results from EVT, in particular, the spectral
theory developed by Keller and Liverani [30, 29], to obtain the convergence of
${F_{n}}(u_{n})$ to its limit law.

###### Proposition 1

For a large class of exponentially-mixing systems and regular observations,
there exists $0<\theta_{q}^{f}\leq 1$ such that:

$|F_{n}(u_{n}(s))-\exp(-\theta^{f}_{q}e^{-s})|\underset{n\to\infty}{\to}0.$
(5)

The term $\theta^{f}_{q}$ is called the Extremal Index (EI) and quantifies the
tendency of the process $(Y_{i})$ to form clusters of high values. To be
applicable, the spectral theory requires that the couple system/observation
satisfies the so-called REPFO property [30, 29], which is verified for rapidly
mixing systems for which the measure of the nested target sets $S^{q}_{n}$
goes to zero in a regular fashion. More detailed presentations of the theory
and its domain of application can be found in various publications [19, 11,
30, 29, 16]. The theory is proven to be particularly adapted to expanding maps
of the interval [18, 16] and certain well-behaved 2-dimensional systems [6].

More classical tools can also be used to prove the convergence to the limit
law, in particular under the following conditions, that are particularly
adapted to processes generated by dynamical systems.

###### Definition 3

We say that the condition Д${}_{1}(u_{n})$ is satisfied for the process
$Y_{0},Y_{1},...$ if there exist a function $\gamma(n,t)$ such that for every
$l,t,n\in\mathbb{N}$,

$|\mu_{q}(A_{n}\cap
B_{t,l,n})-\mu_{q}(A_{n})\mu_{q}(B_{0,l,n})|\leq\gamma(n,t),$ (6)

where $A_{n}=\\{Y_{0}>u_{n},Y_{1}\leq u_{n}\\}$,
$B_{t,l,n}=\bigcap_{i=t}^{t+l-1}T^{-i}(A_{n}^{c})$, and the function
$\gamma(n,t)$ is such that it is decreasing in $t$ for each $n$ and such that
there exists a sequence $(t_{n})_{n}\in\mathbb{N}$ satisfying $t_{n}=o(n)$ and
$n\gamma(n,t_{n})\underset{n\to\infty}{\to}0$.

###### Definition 4

We say that Д${}_{1}^{\prime}(u_{n})$ holds if there exist a sequence
$(k_{n})_{n}$ such that:

  1. 1.

$k_{n}\underset{n\to\infty}{\to}\infty$.

  2. 2.

$k_{n}t_{n}=o(n)$, where $(t_{n})_{n}$ is the sequence in definition 3.

  3. 3.

$\underset{n\to\infty}{\lim}n\sum_{j=2}^{\lfloor\frac{n}{k_{n}}\rfloor-1}\mu_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n}\cap Y_{j}>u_{n})=0.$

Under these two conditions, the result of Proposition 1 holds [18]. We stress
that these conditions depend both on the application $T$ and on the
observation $f$. Д${}_{1}(u_{n})$ is expected to hold for rapidly mixing
systems and regular observations. In particular, we show in the annex that, at
least in the context of symbolic dynamics and if $f=Id$, strong exponential
mixing implies Д${}_{1}(u_{n})$. Condition Д${}_{1}^{\prime}(u_{n})$ concerns
the clustering structure of the process $Y_{i}$. More particularly, it
controls the probabilities of short returns to the target set $S^{q}_{n}$. It
is not our focus to give more appropriate conditions of convergence to the
limit law, since these can be hard to check in dimension more than one, or
sometimes two222for simple systems such as automorphisms of the torus [22] or
systems admitting a product structure [21], and even more so when a non-
trivial observation $f$ is introduced. We will however provide numerical
evidence of the convergence to the extreme value law. Let us now discuss the
values of the different parameters of the limit law, that can acquire a
physical interpretation.

##  3 The Generalized Dimensions of the image measure $D_{q}^{f}$

###  3.1 On the relation between the Generalized dimensions of the image
measure and the one of the original invariant measure

We have seen in the preceding section that the quantity $D_{q}^{f}$ appears as
a parameter of the limit law and therefore modulates the synchronization
properties of observations. In fact, these quantities play a central role in
different aspects of the statistical properties of observations, and in
particular their recurrence times. It is well known that both return and
hitting times of certain chaotic systems in small balls (in fact, re-scaled
versions of these quantities) have large deviations that are governed by the
spectrum of generalized dimensions $D_{q}$ of the invariant measure [11, 15].
These kinds of large deviations relations are known to hold for real
trajectories, but similar results are also expected to apply to the recurrence
times of observations for such systems. This matter will be investigated more
in detail in a future publication. For now, let us focus on the properties of
$D_{q}^{f}$, and in particular on their relation to the generalized dimensions
$D_{q}$ of the original system. In [27], Hunt and Kaloshin give results
concerning the effect of typical projections on the generalized dimensions for
$1\leq q\leq 2$. In this range, they show that if $\mathscr{M}$ is a compact
subset of $\mathbb{R}^{n}$ and $\mathscr{J}=\mathbb{R}^{m}$, and if the
generalized dimension of order $q$, $D_{q}$ of the invariant measure exists,
then:

$D_{q}^{f}=\min(D_{q},m),$ (7)

for a prevalent set of $C^{1}$ observables. See [28] for a review of
prevalence, which is a notion of genericity for infinite-dimensional spaces.
For $q>2$, no such result holds and the behavior of $D_{q}^{f}$ in this range
is not yet completely understood. Under the light of Hunt and Kaloshin’s
result, it is possible to access the correlation dimension $D_{2}$ of a
physical system using a generic observation if the rank is large enough
(larger than the correlation dimension of the original attractor). This
quantity can be obtained by fitting the empirical distribution of $M_{n,q}$
and extracting the desired parameter, as we will do in the following
subsection. Such EVT-based methods of computation of fractal dimensions is in
use in climate studies, in particular for the computation of the local
dimension, which can be used as a tool to characterize certain climatic
patterns [17, 10].

Different kinds of large rank observations can be used by physicists to
recover information on the original system. A first approach is to record
simultaneously the value of a scalar quantity at different locations of a
spatially extended system. These measurements can be arranged into a vector
and constitute a so-called gridded observation in $\mathbb{R}^{m}$. Instead of
recording the same quantity at different positions, one can also record
different independent observables (temperature, position, speed, pressure, …)
at a given time. Yet another possibility is to consider delay coordinates
observables used in embedding techniques [34]. In this context, it is well
known that if one considers enough delay coordinates (larger than $\lceil
2D_{0}\rceil$), the dynamics of the observation settles on an object (the so-
called reconstructed attractor) that is a smooth deformation of the original
attractor, which preserves the dimensions [34]. With our approach, only $m\geq
D_{2}$ delay coordinates are required to access the correlation dimension
$D_{2}$, although the reconstructed attractor is now likely to have a
different fine structure from the original one.

###  3.2 Numerical extraction of $D_{q}^{f}$

Let us now investigate the values of $D_{q}^{f}$ for $q>2$ from a numerical
perspective. This procedure will also allow us to experimentally intuit the
convergence of the distribution of $M_{n,q}$ to its limit law. Let us consider
a system for which the explicit values of $D_{q}$ are available; the motion on
a Sierpinski gasket given by the following Iterated Function System on the
unit square $\mathscr{M}=[0,1]^{2}$:

$\left\\{\begin{array}[]{ll}T_{1}(x,y)=(x/2,(y+1)/2),\;p_{1}=1/4,\\\
T_{2}(x,y)=((x+1)/2,(y+1)/2),\;p_{2}=1/4,\\\
T_{3}(x,y)=(x/2,y/2),\;p_{3}=1/2.\\\ \end{array}\right.$ (8)

At each iteration, the application $T_{i}$ is applied with probability
$p_{i}$. The associated generalized dimensions spectrum is given, for $q\neq
1$, by [11]:

$D_{q}=\frac{\log_{2}(p_{1}^{q}+p_{2}^{q}+p_{3}^{q})}{1-q}.$ (9)

In figure (1), we compare the numerical estimates of $D^{f}_{q}$ for different
observations $f$ and the theoretical values of $D_{q}$ given by equation (9).
These estimates are obtained by evaluating the scale parameter of the
empirical maximum distribution of the process $(Y_{i})$ over blocks of size
$5.10^{4}$, using the maximum likelihood estimator provided by the Matlab
function gevfit [25]. The results are averaged over 10 runs, using each time
different randomly selected trajectories of length $2.10^{8}$. The error bars
represent the standard deviations of the results over these 10 runs.
The functions $f_{1}$, $f_{2}$ are diffeomorphisms, which are known to
preserve the generalized dimensions [27]. Indeed, for these two functions,
good agreement is found, so that the two curves are hardly distinguishable
visually in the picture. These results suggest that this method of computation
of $D_{q}$ can be completed and even improved by introducing a diffeomorphism
computed along the orbits of the system, which may, if well chosen, speed up
the convergence of the method and provide better estimates. Function $f_{3}$
is a very oscillatory function, which gives a point in the observational space
many antecedents, having the effect to alter significantly the fine structure
of the image measure. We do not know whether the disagreement with the $D_{q}$
spectrum is due to the method not being at convergence, or if is a sign that
the spectrum is not preserved under the action of $f_{3}$. However, the small
disagreement for $q=2$ seems to indicate that the method may not be at
convergence, since the correlation dimension is preserved by typical
observations. $f_{4}$ is not a diffeomorphism either, but has a more simple
structure. For this function, the generalized dimensions seem to be preserved.
$f_{5}$ is a degenerate function yielding values close to 1.

Figure 1: Numerical estimates of $D_{q}^{f}$ for different observations:
$f_{1}=Id$, $f_{2}(x,y)=(2x+y,2y)$,
$f_{3}(x,y)=(\sin(\frac{1}{x}),\cos(\frac{1}{y}))$,
$f_{4}(x,y)=((x-0.5)^{2},2y)$ and $f_{5}=(1,y^{2}+x)$. In dashed lines is the
$D_{q}$ spectrum of the underlying system. Estimates are computed as described
in the text.

In [12], we showed that for the two-dimensional baker’s map, which has a non
trivial $D_{q}$ spectrum, a typical linear uni-dimensional projection gives
$D_{q}^{f}=1$ for all $q$. Overall, this result, along with our numerical
computations, suggests that Hunt and Kaloshin’s results may extend to $q>2$
for a certain class of measures and certain smooth observations. We hope to
provide their characterization on future investigations.

##  4 The Extremal Index $\theta^{f}_{q}$

###  4.1 An explicit formula for expanding maps of the interval

When considering real trajectories (i.e. when $f=Id$), the Extremal Index
$\theta_{q}$, and more specifically the quantity

$h_{q}=\frac{\log(1-\theta_{q})}{1-q},$

encodes the hyperbolic properties of the system (see [11] for a detailed
review). In particular, $h_{q}$ as a function of $q$ is constant for maps with
constant Jacobian and is close to the metric entropy of the system (its
Lyapunov exponent in dimension 1). When an observation $f$ is introduced, the
use of the Extremal Index to quantify the rate at which nearby trajectories
diverge becomes less relevant, in particular because two nearby points in
observational space may have antecedents far away in the actual phase space of
the system. Let us investigate this matter with more detail. Keller and
Liverani [30] provide a general formula for the Extremal Index of time series
originated by dynamical systems. Applied to the present situation and if the
limits defining the different quantities exist, we have that

$\theta^{f}_{q}=1-\sum_{k=0}^{\infty}p_{k,q},$ (10)

where

$p_{0,q}=\lim_{n\to\infty}\frac{\mu_{q}(S^{q}_{n}\cap
T^{-1}S^{q}_{n})}{\mu_{q}(S^{q}_{n})}$ (11)

and for $k\geq 1$,

$p_{k,q}=\lim_{n\to\infty}\frac{\mu_{q}(S^{q}_{n}\cap\bigcap_{i=1}^{k}T^{-i}(S^{q}_{n})^{c}\cap
T^{-k-1}S^{q}_{n})}{\mu_{q}(S^{q}_{n})}.$ (12)

In this general set up, obtaining a formula for $\theta^{f}_{q}$ is
challenging, so let us place ourselves in the more simple case of expanding
maps of the unit interval $I=[0,1]$. We define the following sets for a given
$x\in I$ :

$A_{0}(x)=\\{y\in I\text{ such that }f(y)=f(x)\text{ and }f(Ty)=f(Tx)\\}$

and

$A_{k}(x)=\\{y\in I\text{ such that }f(y)=f(x),f(T^{i}y)\neq f(T^{i}x),\text{
for }i=1,..,k\text{ and }f(T^{k+1}(y))=f(T^{k+1}(x))\\}.$

###### Proposition 2

Let $T$ be an expanding map of the unit interval $I=[0,1]$ which is $C^{1}$ by
part and admitting an absolutely continuous invariant measure
$d\mu(x)=h(x)dx$. Let $f:I\to J\subset\mathbb{R}$ be $C^{1}$ by part, finite
to one and such that $f^{\prime}\neq 0$ on $I$. Suppose moreover that the
couple $(T,f)$ satisfies the conditions of Proposition 1, that

$\mu(\\{x\in I,A_{0}(x)=\\{x\\}\\})=1$ (13)

and that, for all $k\geq 1$,

$\mu(\\{x\in I,A_{k}(x)=\emptyset\\})=1.$ (14)

Then:

$\theta^{f}_{q}=1-\frac{\int_{I}\frac{h(x)^{q}}{\max(|f^{\prime}(x)|,|(f\circ
T)^{\prime}(x)|)^{q-1}}dx}{\int_{I}\sum_{(y_{1},...y_{q-1})\in(f^{-1}\\{f(x)\\})^{q-1}}\prod_{i=1}^{q-1}\frac{h(y_{i})}{|f^{\prime}(y_{i})|}h(x)dx}.$
(15)

Proof. We write the proof for $q=2$, the cases $q>2$ can be obtained in a
similar fashion. We start from formula (10) and evaluate both the numerators
and the denominators defining the $p_{k,2}$ terms. Let us start by the
denominator, for the case $k=0$. Following the lines of the proof in [16]
(where the case $f=Id$ is treated), and making use of the mean value theorem,
we get:

$\displaystyle\mu_{2}(S^{2}_{n})$ $\displaystyle\sim\int_{I}\sum_{y\in
f^{-1}\\{f(x)\\}}\mu(B(y,\frac{e^{-u_{n}}}{|f^{\prime}(y)|}))d\mu(x)$ (16)
$\displaystyle\sim 2e^{-u_{n}}\int_{I}\sum_{y\in
f^{-1}\\{f(x)\\}}\frac{h(y)}{|f^{\prime}(y)|}h(x)dx.$

On the other hand, still for the case $k=0$, we get for the numerator:

$\displaystyle\mu_{2}(S^{2}_{n}\cap T^{-1}S^{2}_{n})$
$\displaystyle\sim\int_{I}\sum_{y\in A_{0}(x)}\mu(\\{z\in I,z\in
B(y,\frac{e^{-u_{n}}}{|f^{\prime}(y)|})\cap Tz\in
B(Ty,\frac{e^{-u_{n}}}{|f^{\prime}(Ty)|}\\})d\mu(x)$ (17)
$\displaystyle\sim\int_{I}\sum_{y\in A_{0}(x)}\mu(\\{z\in
I,|z-y|\leq\frac{e^{-u_{n}}}{|f^{\prime}(y)|}\cap
T^{\prime}(y)|y-z|\leq\frac{e^{-u_{n}}}{|f^{\prime}(Ty)|}\\})h(x)dx.$
$\displaystyle=\int_{I}\sum_{y\in A_{0}(x)}\mu(\\{z\in
I,|z-y|\leq\min(\frac{e^{-u_{n}}}{|f^{\prime}(y)|},\frac{e^{-u_{n}}}{|T^{\prime}(y)f^{\prime}(Ty)|})\\})h(x)dx.$
$\displaystyle\sim 2e^{-u_{n}}\int_{I}\sum_{y\in
A_{0}(x)}\frac{h(y)h(x)}{\max(|f^{\prime}(y)|,|(f\circ T)^{\prime}(y)|)}dx.$

By a similar reasoning, we get that for $k\geq 1$,

$\mu_{2}(S^{2}_{n}\cap\bigcap_{i=1}^{k}T^{-i}(S^{2}_{n})^{c}\cap
T^{-k-1}S^{2}_{n})\sim 2e^{-u_{n}}\int_{I}\sum_{y\in
A_{k}(x)}\frac{h(x)h(y)}{\max(|f^{\prime}(x)|,|(f(T^{k+1}(y))^{\prime}|)}dx.\\\
$ (18)

Finally, combining eqs. (10),(2), (17) and (18), we obtain

$\theta^{f}_{2}=1-\sum_{k=0}^{+\infty}\frac{\int_{I}\sum_{y\in
A_{k}(x)}\frac{h(x)h(y)}{\max(|f^{\prime}(x)|,|(f\circ
T^{k+1})^{\prime}(y)|)}dx}{\int_{I}\sum_{y\in
f^{-1}\\{f(x)\\}}\frac{h^{\prime}(y)}{|f^{\prime}(y)|}h(x)dx}.$ (19)

This formula is still difficult to handle, but under condition (14),
$p_{k,2}=0$ for $k>0$, and if condition (13) holds, we obtain

$\displaystyle\theta^{f}_{2}$ $\displaystyle=1-p_{0,2}$ (20)
$\displaystyle=1-\frac{\int_{I}\frac{h(x)^{2}}{\max(|f^{\prime}(x)|,|(f\circ
T)^{\prime}(x)|)}dx}{\int_{I}\sum_{y\in
f^{-1}\\{f(x)\\}}\frac{h(y)h(x)}{|f^{\prime}(y)|}dx}.$

We can generalize this result for $q\geq 2$ to obtain the desired result.

###### Remark 1

For a given map $T$, assumptions (13) and (14) should be satisfied for a
generic observation $f$. The cases where these assumptions are not satisfied
are when $T$ and $f$ share some particular symmetries and similarities in
their structures. For example, $\mu(A_{0}(x)=\\{x\\})\neq 1$ if both the
graphs of $T$ and $f$ are symmetric with respect to the straight line of
equation $x=1/2$.

###### Example 1

Let us take $Tx=2x\mod 1$ and

$f(x)=\left\\{\begin{array}[]{ll}2x&\mbox{ if }0\leq x\leq 1/2\\\ 3/2-x&\mbox{
if }1/2<x\leq 1.\\\ \end{array}\right.$

$T$ is strongly mixing and the couple $(T,f)$ satisfies conditions (13) and
(14), so that $(T,f)$ should satisfy the conditions of existence of the limit
law, Д${}_{1}(u_{n})$ and Д${}_{1}^{\prime}(u_{n})$. It constitutes a good
test for our results, since computations can be worked out quite easily.
Applying formula (25), we get

$\theta^{f}_{q}=1-p_{0,q}=1-\frac{2+2^{2-q}}{1+3^{q}}.$

This result is confirmed by numerical experiments (see figure (2)). We used
the estimator $\hat{\theta}_{5}$ introduced in [13], which consists in
evaluating the 5 first $p_{k,q}$ terms appearing in formula (12). To do so, we
compute Birkhoff sums for both the numerator and the denominator defining the
$p_{k,q}$ terms. It requires fixing a high threshold $u$, that we take here
equal to the 0.99999-quantile of the empirical $Y_{i}$ distribution. As
expected, we find that all the $p_{k,q}$ are 0 or very close to 0 for $k\geq
1$. The results are averaged over 10 runs, with trajectories of length
$2.10^{7}$. The error bars in figure (2) represent the standard deviations of
the results over these 10 runs. In this example, the exact limit distribution
can be computed explicitly; since the image measure is absolutely continuous
with a density that does not vanish and that admits no singularities,
$D_{q}^{f}=1$ for all $q$.

Figure 2: Comparison between theory and computation for the $\theta_{q}^{f}$
spectrum of the system in Example 1. Details of the computation can be found
in the text.

###  4.2 Numerical estimation of $\theta_{q}^{f}$ in higher dimensional
systems

A general formula for higher-dimensional system is out of scope, but we expect
that with conditions of ‘non compatibility’ between the dynamics and the
observation analogue to conditions (13) and (14), all the $p_{k,q}$ terms are
0 for $k\geq 1$. The aim of this section is to show that this hypothesis is
corroborated by numerical experiments.

For the uni-dimensional case, the presence of the derivative of the
observation in formula (25) renders the interpretation of $\theta_{q}^{f}$
less apparent than in the case $f=Id$. However, we point out two facts :

  * •

For a given observation $f$, the larger the values of $|T^{\prime}|$ over
phase space, the larger the values of $\theta_{q}^{f}$, so this index can
still quantify the hyperbolic properties of $T$.

  * •

For a given map $T$, the more the points in the observational space have
antecedents by $f$, the larger is the denominator in equation (25), and the
larger is $\theta_{q}^{f}$. Oscillatory observations yield higher values for
the extremal index.

We expect analogous properties to hold for higher dimensional systems. To test
this statement, we compare in figure (3(b)a) the estimates of $\theta_{q}^{f}$
for the 2-dimensional Hénon system, defined by $T(x,y)=(1-ax^{2}+y,bx)$, with
$a=1.4$ and different values of $b$ such that the system admits a strange
attractor [26]. We consider the observation $f(x,y)=\frac{x+y}{2}$. The
determinant of the Jacobian is given by $b$. We find indeed that for this
fixed choice of observation, the more the original system tends to separate
trajectories (the higher is parameter $b$), the higher are the values of
$\theta_{q}^{f}$, even for uni-dimensional projections. The estimates
$\hat{p}_{k,q}$ of the $p_{k,q}$ terms, for $k>0$ are all null or close to 0
for all the observations that we considered, as conjectured earlier.

In figure (3(b)b), we plot the estimates of the extremal index for
2-dimensional Hénon system (using the usual parameters $a=1.4$, $b=0.3$) and
different observations. We observe that for one-to-one observations, ($f_{1}$,
$f_{2}$ and $f_{3}$), the $\theta_{q}^{f}$ spectrum remains relatively low,
although the form of the Jacobian can impact significantly the values of
$\theta_{q}^{f}$. When the observation ceases to be one-to-one, the whole
spectrum of extremal indices increases significantly (see the curve for
$f_{4}$). This effect is even more important for the very oscillatory function
$f_{5}$. For analogous reasons, we expect that for high dimensional systems,
observations that perform a large drop of dimensionality tend to yield higher
values for the $\theta_{q}^{f}$ spectrum.

(a)

(b)

Figure 3: Left: Estimates for the $\theta^{f}_{q}$ spectrum computed for a
Hénon system with different parameters $b$ and for the observation
$f(x,y)=\frac{x+y}{2}$. Right: Estimates for the $\theta^{f}_{q}$ spectrum
computed for the Hénon system (b=0.3) and different observations : $f_{1}=Id$,
$f_{2}(x,y)=(100x+y,100y)$, $f_{3}(x,y)=(x,100y)$, $f_{4}(x,y)=(x^{2},y^{2})$,
$f_{5}(x,y)=(\sin(1/x),\cos(1/y))$. For both figures, we used the estimate
$\hat{\theta}_{5}$ introduced in [13], with trajectories of length $10^{6}$
and a threshold value equal to the $0.999$ quantile of the empirical $Y_{i}$
distribution. The error bars represent the standard deviation of the results
over 10 runs.

##  5 Application to Sequence Matching

In this section, we discuss the connection between the present problem and
sequence matching problems. Let

$\left\\{\begin{array}[]{ll}X^{1}=X_{1}^{1}X_{2}^{1}\ldots X_{n}^{1},\\\
\vdots\\\ X^{q}=X_{1}^{q}X_{2}^{q}\ldots X_{n}^{q}\\\ \end{array}\right.$ (21)

be $q$ sequences of symbols of length $n$, drawn from the finite alphabet
$\mathscr{A}$ with the same probability distribution $\mathbb{P}$. We will
denote $\bar{X}_{i}=(X_{i}^{1},X_{i}^{2},...,X_{i}^{q})$. We suppose that the
sequences have a good dependence structure that we will describe later. We are
interested in deriving the limit distribution of the length of the longest
matching block for the $q$ sequences; the following random variable:

$\Xi_{n,q}(X^{1},...,X^{q})=\underset{l=0,...,n}{\max}\\{X^{1}_{i+k}=X^{2}_{i+k}=...=X^{q}_{i+k}\text{
for }k=0,...,l\text{ and }1\leq i\leq n-l\\}.$ (22)

To make the connection between the previous sections, let us now consider, as
in [7], the discrete symbolic dynamical system
$(\mathscr{A}^{\mathbb{N}},\sigma,\mathbb{P})$, where $\sigma$ is the right-
sided shift and $\mathbb{P}$ is the probability measure associated to the
process. We consider the symbolic distance in $\mathscr{A}^{\mathbb{N}}$
defined by:

$d(x^{1},x^{2})=\exp(-\inf\\{i\geq 0,\sigma^{i}x^{1}\neq\sigma^{i}x^{2}\\}).$
(23)

For our purpose, we take $f=Id$. In this symbolic dynamics, the quantity
$D_{q}$ (if it exists) identifies with a well-known quantity that we now
define.

###### Definition 5

We call the Rényi entropy of order $q$ of $\mathbb{P}$, the following quantity
(if the limit exists):

$H_{q}=\lim_{k\to\infty}\frac{\log\sum_{C_{k}}\mathbb{P}(C_{k})^{q}}{(1-q)k},$
(24)

where $C_{k}(x)=\\{y\in\mathscr{A}^{\mathbb{N}}:\sigma^{i}x=\sigma^{i}y\text{
for all }0\leq i\leq k\\}$ is the cylinder of length $k$ containing
$x\in\mathscr{A}^{\mathbb{N}}$.

To see that $D_{q}$ identifies with $H_{q}$ in this context, it is enough to
start from definition (2), take $f=Id$ and use the symbolic distance, allowing
to replace balls by cylinders.

The Dynamical Extremal Index $\theta_{q}=\theta_{q}^{Id}$ becomes in this set
up (if it exists, and from equation (10)):

$\displaystyle\theta_{q}$ $\displaystyle=1-p_{0,q}$ (25)
$\displaystyle=\lim_{k\to\infty}\mathbb{P}(\sigma^{k+1}x^{1}=\sigma^{k+1}x^{2}=...=\sigma^{k+1}x^{q}|\sigma^{i}x^{1}=\sigma^{i}x^{2}=...=\sigma^{i}x^{q}\text{
for }0\leq i\leq k).$

Indeed one sees easily that only the $p_{0,q}$ in definition (10) is non-zero
in this situation (we provide a more detailed argument in the annex).

The quantity

$\displaystyle Y_{i}$
$\displaystyle=-\log(\max_{s=2,\dots,q}d(x_{i}^{1},x_{i}^{s}))$ (26)
$\displaystyle=\underset{j\geq
0}{\inf}\\{\sigma^{j}x_{i}^{1}\neq\sigma^{j}x_{i}^{s},\text{for some
}s=2,...,q\\}$

is the length of the longest matching sub-sequence starting from the $i^{th}$
symbol of the different sequences. Now, the quantity

$M_{n,q}(x^{1},...,x^{q})=\underset{i=0,n-1}{\max}Y_{i}$ (27)

is equal to

$\underset{l\in\mathbb{N}}{\max}\\{x^{1}_{i+k}=x^{2}_{i+k}=...=x^{q}_{i+k}\text{
for }k=1,...,l\text{ and }0\leq i\leq n-1\\}.$

This object is closely related to the quantity $\Xi_{n,q}$ we are interested
in. Since we work with different sequences of symbols, and $Y_{i}$ is a
variable defined in the product space, we will state our results with respect
to the product measure $\mathbb{P}_{q}$. We prove our results under the
hypothesis that the process has certain mixing properties, that we now recall.

###### Definition 6

The process $(\mathscr{A}^{\mathbb{N}},\sigma,\mathbb{P})$ is said to be
$\alpha-$mixing if there exists $\alpha(n)\to 0$ such that

$\underset{A,B\subset\mathscr{A}^{\mathbb{N}}}{\sup}|\mathbb{P}(A\cap\sigma^{-n}B)-\mathbb{P}(A)\mathbb{P}(\sigma^{-n}B)|\leq\alpha(n).$
(28)

One could obtain a distributional result analogue to Proposition 1, by proving
that conditions Д${}_{1}(u_{n})$ and Д’${}_{1}(u_{n})$ are satisfied. With
this approach, we get the following result, whose detailed proof can be found
in the annex:

###### Result 1

If the sequences are $\alpha$-mixing with $\alpha(n)<\beta e^{-\kappa n}$ for
some $\beta\in\mathbb{R}^{+}$ and some $\kappa>H_{q}(q-1)$, and the limits
defining $\theta_{q}$ and $H_{q}$ exist and are different from 0, then

$|\mathbb{P}_{q}(\Xi^{q}_{n}\leq
u_{n}(s))-\exp(-\theta_{q}\exp(-s))|\underset{n\to\infty}{\to}0,$

with $u_{n}(s)=\lfloor\frac{\log n+s}{H_{q}(q-1)}\rfloor$.

###### Remark 2

We took $f=id$, to ensure a clustering structure that satisfies the different
conditions of existence of the limit law, in particular condition
Д’${}_{1}(u_{n})$. We could also consider, as in the first section of the
paper, a non-trivial $f$. In the context of sequence matching, $f$ is called
the encoding function (or encoder) and can model different treatments of the
original source of information [14]. The clustering structure is however in
this case too complex to yield such a general result.

It is in fact possible to obtain a more general result than Result 1, under
much weaker conditions. The latter is based on results by Abadi and Saussol
concerning the Hitting Time Statistics of symbolic dynamics in cylinders [2].
This idea originates from a discussion with Jérôme Rousseau to whom the author
is thankful.

###### Theorem 1

If $\mathbb{P}$ is $\alpha-$ mixing, and if the limits defining $\theta_{q}$
and $H_{q}$ exist and are different from 0, then

$|\mathbb{P}_{q}(\Xi_{n,q}\leq
u_{n}(s))-\exp(-\theta_{q}\exp(-s))|\underset{n\to\infty}{\to}0,$

with $u_{n}=u_{n}(s)=\lfloor\frac{\log n+s}{H_{q}(q-1)}\rfloor$.

Proof. Let us consider the process $(Z_{i})$ defined by

$Z_{i}=\left\\{\begin{array}[]{ll}1\text{ if
}X_{i}^{1}=X_{i}^{2}=...=X_{i}^{q},\\\ 0\text{ otherwise.}\\\
\end{array}\right.$ (29)

The problem of finding the largest common substring to $X^{1},...,X^{q}$ is
now equivalent to find the longest succession of ones in the process
$(Z_{i})$. Let us consider the dynamical system
$(\mathscr{B},\tilde{\mathbb{P}},\sigma)$, where
$\mathscr{B}=\\{0,1\\}^{\mathbb{N}}$, $z$ a point in $\mathscr{B}$ and
$\tilde{\mathbb{P}}$ the probability measure defined by

$\displaystyle\tilde{\mathbb{P}}(z_{i}=1)$
$\displaystyle=\mathbb{P}_{q}(x^{1}_{i}=...=x^{q}_{i})$ (30)
$\displaystyle=\sum_{a\in\mathscr{A}}\mathbb{P}(x^{1}_{i}=a)^{q}.$

Let us denote $I_{k}$ the cylinder constituted of all sequences having their
first $k$ symbols equal to 1, and denote

$\tau_{I_{k}}(z)=\inf\\{j\geq 1:\sigma^{j}z\in I_{k}\\},$

the first hitting time of the point $z$ in the set $I_{k}$. We notice that

$\mathbb{P}_{q}(M_{n,q}<u_{n})=\tilde{\mathbb{P}}(\tau_{I_{u_{n}}}>n).$ (31)

Since $\mathbb{P}$ is $\alpha-mixing$, so is $\tilde{\mathbb{P}}$, by theorem
5.1 in [9]. We are then in the set up of Theorem 1 in [2]. In particular,
Hypothesis 1 of this theorem is satisfied, from Example 2 in [2]. Therefore:

$\sup_{t\in\mathbb{R}^{+}}|\tilde{\mathbb{P}}(\lambda(I_{u_{n}})\tilde{\mathbb{P}}(I_{u_{n}})\tau_{I_{u_{n}}}>t)-\exp(-t)|\underset{n\to\infty}{\to}0,$
(32)

where, from [1]:

$\displaystyle\lambda(I_{u_{n}})$
$\displaystyle=1-\lim_{k\to\infty}\frac{\tilde{\mathbb{P}}(I_{k+1})}{\tilde{\mathbb{P}}(I_{k})}$
(33)
$\displaystyle=1-\lim_{k\to\infty}\frac{\sum_{C_{k+1}}\mathbb{P}(C_{k+1})^{q}}{\sum_{C_{k}}\mathbb{P}(C_{k})^{q}}$
$\displaystyle=\theta_{q}.$

Notice now that we have from equation (24):

$\displaystyle H_{q}$
$\displaystyle=\lim_{k\to\infty}\frac{1}{(1-q)k}{\log\underset{C_{k}}{\sum}\mathbb{P}(C_{k})^{q}}$
(34)
$\displaystyle=\lim_{k\to\infty}\frac{\log\tilde{\mathbb{P}}(I_{k})}{(1-q)k},$

so that

$\tilde{\mathbb{P}}(I_{u_{n}})\underset{n\to\infty}{\sim}e^{-(q-1)H_{q}u_{n}}.$
(35)

If we put $t=e^{-s}$, equation (32) writes, after rearranging a bit:

$\sup_{s\in\mathbb{R}}|\tilde{\mathbb{P}}(\tau_{I_{u_{n}}}>e^{-s+(q-1)H_{q}u_{n}})-\exp(-\theta_{q}e^{-s})|\underset{n\to\infty}{\to}0.$
(36)

keeping in mind that $u_{n}=\lfloor\frac{\log n+s}{H_{q}(q-1)}\rfloor$, we get

$\sup_{s\in\mathbb{R}}|\tilde{\mathbb{P}}(\tau_{I_{u_{n}}}>n)-\exp(-\theta_{q}e^{-s})|\underset{n\to\infty}{\to}0.$
(37)

Using now equation (31), we obtain that for all $s\in\mathbb{R}$:

$|\mathbb{P}_{q}(M_{n,q}>u_{n})-\exp(-\theta_{q}e^{-s})|\underset{n\to\infty}{\to}0.$
(38)

Now that we have a distributional result for the variable $M_{n,q}$, we can
get one for $\Xi_{n,q}$, which is a slightly different object. In fact we have
that

$\begin{split}\mathbb{P}_{q}(\Xi_{n,q}\leq
u_{n})&=\mathbb{P}_{q}(\Xi_{n,q}\leq u_{n}\cap
M_{u_{n},q}(\sigma^{n-u_{n}}x^{1},...,\sigma^{n-u_{n}}x^{q})\leq u_{n})\\\
&+\mathbb{P}_{q}(\Xi_{n,q}\leq u_{n}\cap
M_{u_{n},q}(\sigma^{n-u_{n}}x^{1},...,\sigma^{n-u_{n}}x^{q})>u_{n}).\end{split}$
(39)

The second term is bounded above by the term
$\mathbb{P}_{q}(M_{u_{n},q}(\sigma^{n-u_{n}}x^{1},...,\sigma^{n-u_{n}}x^{q})>u_{n})$,
which, by invariance of the measure by $\sigma$ equals
$\mathbb{P}_{q}(M_{u_{n},q}(x^{1},...x^{n})>u_{n})$, which is clearly
vanishing to 0 as $n\to\infty$, from (38).

The first term in (39) is exactly equal to
$\mathbb{P}_{q}(M_{n,q}(\overline{x})\leq u_{n})$. Therefore:

$|\mathbb{P}_{q}(\Xi_{n,q}\leq u_{n})-\mathbb{P}_{q}(M_{n,q}\leq
u_{n})|\underset{n\to\infty}{\to}0.$ (40)

We have that for all $s\in\mathbb{R}$:

$|\mathbb{P}_{q}(\Xi_{n,q}\leq
u_{n})-\exp(-\theta_{q}e^{-s})|\leq|\mathbb{P}_{q}(\Xi_{n,q}\leq
u_{n})-\mathbb{P}_{q}(M_{n,q}\leq u_{n})|+|\mathbb{P}_{q}(M_{n,q}\leq
u_{n})-\exp(-\theta_{q}e^{-s})|,$ (41)

which, by relations (38) and (40) goes to 0.

##  6 Acknowledgement

The author was partially supported by CMUP, which is financed by national
funds through FCT – Fundação para a Ciência e Tecnologia, I.P., under the
project with reference UIDB/00144/2020. The author would like to thank Jorge
M. Freitas, Jérôme Rousseau, Benoît Saussol and Sandro Vaienti for the
fruitful discussions we had concerning this work and the anonymous referee for
its constructive comments.

##  7 Annex (proof of Result 1, via EVT)

We first show that both conditions Д${}_{1}(u_{n})$ and Д’${}_{1}(u_{n})$ are
satisfied, so we have an EVL for the random variable $M_{n,q}$. Then we show
that $\Xi_{n,q}$ and $M_{n,q}$ have the same asymptotic distribution. Let us
first take care of condition Д${}^{\prime}_{1}(u_{n})$.
We observe that if $Y_{0}=k\in\mathbb{N}^{*}$, then $Y_{j}=k-j$ for $1\leq
j\leq k$. Therefore, if $Y_{0}>u_{n}$, then $Y_{1}>Y_{j}>u_{n}-j$ for $2\leq
j<u_{n}$, so that all the probabilities in the sum in point 3 of definition 4
are 0 for $2\leq j<u_{n}$, that is

$\underset{n\to\infty}{\lim}n\sum_{j=2}^{u_{n}-1}\mathbb{P}_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n}\cap Y_{j}>u_{n})=0.$ (42)

Let $0<\varepsilon_{2}<\varepsilon_{1}<1$ and $C_{1}=1-\varepsilon_{1}$. We
define $r_{n}=\lfloor C_{1}u_{n}\rfloor$ and $\lambda_{n}=\lfloor
n^{\varepsilon_{2}}\rfloor$. We take $j$ such that $u_{n}\leq
j\leq\lambda_{n}$. We observe that
$\\{Y_{j}>u_{n}\\}\subset\\{Y_{j+r_{n}}>u_{n}-r_{n}\\}$, so that

$\mathbb{P}_{q}(Y_{0}>u_{n}\cap Y_{1}\leq u_{n}\cap
Y_{j}>u_{n})\leq\mathbb{P}_{q}(Y_{0}>u_{n}\cap Y_{1}\leq u_{n}\cap
Y_{j+r_{n}}>u_{n}-r_{n}).$ (43)

Notice that $\\{Y_{0}>u_{n}\cap Y_{1}\leq u_{n}\\}=\\{Y_{0}=u_{n}+1\\}$, and
this event depends only on the realizations of
$\bar{X}_{1},...,\bar{X}_{u_{n}+2}$, whereas $\\{Y_{j+r_{n}}>u_{n}-r_{n}\\}$
depends only on the realizations of
$\bar{X}_{j+r_{n}},\bar{X}_{j+r_{n}+1},...$, which puts a gap of length
$j+r_{n}-u_{n}-2$. We now use the fact that the sequences are $\alpha-$mixing,
which implies that the $q-$fold Cartesian product of the sequences is
$(\alpha_{q})$-mixing, with $\alpha_{q}(n)\leq q\alpha(n)$ (see theorem 5.1 in
[9]). We have

$\displaystyle\mathbb{P}_{q}(Y_{0}>u_{n}\cap Y_{1}\leq u_{n}\cap Y_{j}>u_{n})$
$\displaystyle\leq\alpha_{q}(j+r_{n}-u_{n}-2)+\mathbb{P}_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n})\mathbb{P}_{q}(Y_{j+r_{n}}>u_{n}-r_{n})$ (44)
$\displaystyle\leq
q\alpha(j+r_{n}-u_{n}-2)+\mathbb{P}_{q}(Y_{0}>u_{n})\mathbb{P}_{q}(Y_{j+r_{n}}>u_{n}-r_{n})$
$\displaystyle\leq q\beta
e^{-\kappa(j+r_{n}-u_{n}-2)}+\mathbb{P}_{q}(Y_{0}>u_{n})\mathbb{P}_{q}(Y_{0}>u_{n}-r_{n}).$

To get the last inequality, we used the invariance of the measure. Notice that
$j\geq u_{n}$, so that $e^{-\kappa(j+r_{n}-u_{n}-2)}\leq
e^{-\kappa(r_{n}-2)}$. We also have from relation (17) that
$\mathbb{P}_{q}(Y_{0}>u_{n})\sim e^{-u_{n}\tau_{q}}$, where
$\tau_{q}=H_{q}(q-1)$, so that there exists $C_{2}>1$ such that

$\mathbb{P}_{q}(Y_{0}>u_{n})<C_{2}e^{-u_{n}\tau_{q}}.$

We then have:

$\mathbb{P}_{q}(Y_{0}>u_{n}\cap Y_{1}\leq u_{n}\cap Y_{j}>u_{n})\leq q\beta
e^{-\kappa(r_{n}-2)}+C_{2}^{2}e^{-(2u_{n}-r_{n})\tau_{q}}.$ (45)

Then we can write

$\displaystyle n\sum_{j=u_{n}}^{\lambda_{n}}\mathbb{P}_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n}\cap Y_{j}>u_{n})$
$\displaystyle\leq\sum_{j=u_{n}}^{\lambda_{n}}[nq\beta
e^{-\kappa(r_{n}-2)}+nC_{2}^{2}e^{-(2u_{n}-r_{n})\tau_{q}}]$ (46)
$\displaystyle\leq(\lambda_{n}-u_{n})nq\beta
e^{-\kappa(r_{n}-2)}+(\lambda_{n}-u_{n})nC_{2}^{2}e^{-(2u_{n}-r_{n})\tau_{q}}$
$\displaystyle\leq\lambda_{n}nq\beta
e^{-\kappa(r_{n}-2)}+\lambda_{n}nC_{2}^{2}e^{-(2u_{n}-r_{n})\tau_{q}}$
$\displaystyle\leq(q\beta e^{2\kappa})n\lambda_{n}e^{-\kappa
r_{n}}+C_{2}^{2}n\lambda_{n}e^{-2(u_{n}-r_{n})\tau_{q}}$
$\displaystyle\leq(q\beta e^{2\kappa})n\lambda_{n}e^{-\kappa\lfloor
C_{1}u_{n}\rfloor}+C_{2}^{2}n\lambda_{n}e^{-2(u_{n}-\lfloor
C_{1}u_{n}\rfloor)\tau_{q}}$ $\displaystyle\leq(q\beta
e^{2\kappa})n\lambda_{n}e^{-\kappa(C_{1}u_{n}-1)}+C_{2}^{2}n\lambda_{n}e^{-(2-C_{1})u_{n}\tau_{q}}$
$\displaystyle\leq(q\beta e^{3\kappa})n\lambda_{n}e^{-\kappa
C_{1}\lfloor\frac{\log
n+s}{\tau_{q}}\rfloor}+C_{2}^{2}n\lambda_{n}e^{-(2-C_{1})\lfloor\frac{\log
n+s}{\tau_{q}}\rfloor\tau_{q}}$ $\displaystyle\leq(q\beta
e^{3\kappa})n\lambda_{n}e^{-\kappa C_{1}(\frac{\log
n+s}{\tau_{q}}-1)}+C_{2}^{2}n\lambda_{n}e^{-(2-C_{1})(\frac{\log
n+s}{\tau_{q}}-1)\tau_{q}}$ $\displaystyle\leq C_{3}n\lambda_{n}e^{-\kappa
C_{1}\frac{\log n}{\tau_{q}}}+C_{4}n\lambda_{n}e^{-(2-C_{1})\log n},$

with

$C_{3}=q\beta e^{3\kappa}e^{-\kappa C_{1}(\frac{s}{\tau_{q}}-1)}$

and

$C_{4}=C_{2}^{2}e^{(C_{1}-2)(s-\tau_{q})}.$

For the first term, we have

$\displaystyle C_{3}n\lambda_{n}e^{-\kappa C_{1}\frac{\log n}{\tau_{q}}}$
$\displaystyle=C_{3}n\lfloor n^{\varepsilon_{2}}\rfloor e^{-\kappa
C_{1}\frac{\log n}{\tau_{q}}}$ (47) $\displaystyle\leq
C_{3}nn^{\varepsilon_{2}}e^{-\kappa C_{1}\frac{\log n}{\tau_{q}}}$
$\displaystyle\leq C_{3}n^{1+\varepsilon_{2}-\frac{\kappa C_{1}}{\tau_{q}}}$
$\displaystyle\leq
C_{3}n^{1+\varepsilon_{2}-\frac{\kappa(1-\varepsilon_{1})}{\tau_{q}}}.$

Since $\kappa>\tau_{q}$, we can always chose $\varepsilon_{1},\varepsilon_{2}$
and $\varepsilon_{3}>0$ such that

$\epsilon_{3}>\frac{(\varepsilon_{1}+\varepsilon_{2})\tau_{q}}{1-\varepsilon_{1}}$
(48)

and

$\kappa>\tau_{q}+\varepsilon_{3}.$ (49)

We then have

$\displaystyle 1+\varepsilon_{2}-\frac{\kappa(1-\varepsilon_{1})}{\tau_{q}}$
$\displaystyle<\varepsilon_{1}+\varepsilon_{2}-\frac{\varepsilon_{3}(1-\varepsilon_{1})}{\tau_{q}}<0.$
(50)

and so by relation (47):

$C_{3}n\lambda_{n}e^{-\kappa C_{1}\frac{\log
n}{\tau_{q}}}\underset{n\to\infty}{\to}0.$ (51)

Let us now come to the second term in relation (46):

$\displaystyle C_{4}n\lambda_{n}e^{-(2-C_{1})\log n}$
$\displaystyle=C_{4}n\lfloor n^{\varepsilon_{2}}\rfloor
e^{-(1+\varepsilon_{1})\log n}$ (52) $\displaystyle\leq
C_{4}n^{\varepsilon_{2}-\varepsilon_{1}}.$

And since $\varepsilon_{1}>\epsilon_{2}$:

$C_{4}n\lambda_{n}e^{-(2-C_{1})\log n}\underset{n\to\infty}{\to}0.$ (53)

Combining relations (51), (53) and (46), we have that:

$\underset{n\to\infty}{\lim}n\sum_{j=u_{n}}^{\lambda_{n}}\mathbb{P}_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n}\cap Y_{j}>u_{n})=0.$ (54)

Combining equation (42) and (54), we get that

$\underset{n\to\infty}{\lim}n\sum_{j=2}^{\lambda_{n}}\mathbb{P}_{q}(Y_{0}>u_{n}\cap
Y_{1}\leq u_{n}\cap Y_{j}>u_{n})=0.$ (55)

Taking $k_{n}=\frac{n}{\lambda_{n}}=n^{1-\varepsilon_{2}}$, we have that
points 1 and 3 of condition Д${}_{1}^{\prime}(u_{n})$ are satisfied. To
satisfy point 2, we take $t_{n}=\lfloor n^{\varepsilon_{4}}\rfloor$, with
$\varepsilon_{4}<\varepsilon_{2}$. Д${}_{1}^{\prime}(u_{n})$ is then
satisfied.

Let us now come to condition Д${}_{1}(u_{n})$. Define the event
$\Omega_{n}=\\{Y_{0}<\lfloor t_{n}/2\rfloor\\}$. We have that

$\displaystyle|\mathbb{P}_{q}(A_{n}\cap
B_{t_{n},l,n})-\mathbb{P}_{q}(A_{n})\mathbb{P}_{q}(B_{0,l,n})|$
$\displaystyle\leq|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}\cap
B_{t_{n},l,n})-\mathbb{P}_{q}(A_{n}\cap\Omega_{n})\mathbb{P}_{q}(B_{0,l,n})|$
(56) $\displaystyle+|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c}\cap
B_{t_{n},l,n})-\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c})\mathbb{P}_{q}(B_{0,l,n})|.$

We have just introduced a gap of size $t_{n}-\lfloor t_{n}/2\rfloor-1$ in the
first term of the right hand side of the previous inequation. Indeed, for $n$
large enough, the event $A_{n}\cap\Omega_{n}$ depends only on the realizations
of $\bar{X}_{1},\bar{X}_{2},...,\bar{X}_{\lfloor t_{n}/2\rfloor}$, while
$B_{t_{n},l,n}$ depends on the realizations of
$\bar{X}_{t_{n}},...,\bar{X}_{t_{n}+l}$. We can then bound this term, using
again theorem 5.1 in [9] :

$\displaystyle|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}\cap
B_{t_{n},l,n})-\mathbb{P}_{q}(A_{n}\cap\Omega_{n})\mathbb{P}(B_{0,l,n})|$
$\displaystyle\leq q\alpha(t_{n}-\lfloor t_{n}/2\rfloor-1)$ (57)
$\displaystyle\leq q\beta e^{-\kappa(t_{n}-\lfloor t_{n}/2\rfloor-1)}$
$\displaystyle\leq q\beta e^{-\kappa(t_{n}/2-1)}.$

For the second term, we can write

$\displaystyle|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c}\cap
B_{t,l,n})-\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c})\mathbb{P}_{q}(B_{0,l,n})|$
$\displaystyle\leq|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c}\cap
B_{t,l,n})|+|\mathbb{P}_{q}(A_{n}\cap\Omega_{n}^{c})\mathbb{P}_{q}(B_{0,l,n})|$
(58) $\displaystyle\leq 2\mathbb{P}_{q}(\Omega_{n}^{c})\sim 2e^{-\lfloor
t_{n}/2\rfloor\tau_{q}}$ $\displaystyle\leq C_{5}e^{-\lfloor
t_{n}/2\rfloor\tau_{q}}$ $\displaystyle\leq C_{5}e^{-(t_{n}/2-1)\tau_{q}},$

for some $C_{5}>2$.

Let us now take

$\gamma(n,t_{n})=q\beta e^{-\kappa(t_{n}/2-1)}+C_{5}e^{-(t_{n}/2-1)\tau_{q}}.$

Combining expressions (56), (57), (58), we get

$|\mathbb{P}_{q}(A_{n}\cap
B_{t,l,n})-\mathbb{P}_{q}(A_{n})\mathbb{P}_{q}(B_{0,l,n})|\leq\gamma(n,t_{n}).$
(59)

Let us recall that from condition Д${}_{1}^{\prime}(u_{n})$, $t_{n}=\lfloor
n^{\varepsilon_{4}}\rfloor=o(n).$ $\gamma$ is clearly decreasing and we check
easily that

$n\gamma(n,t_{n})\underset{n\to\infty}{\to}0.$

Condition Д${}_{1}(u_{n})$ is then satisfied. We can now apply corollary 4.1.7
in [18] to get that

$\mathbb{P}_{q}(M_{n,q}\leq
u_{n})-\exp(-\theta_{q}\exp(-s))\underset{n\to\infty}{\to}0.$ (60)

We conclude by using the same arguments as in the proof of Theorem 1, showing
that $M_{n,q}$ and $\Xi_{n,q}$ have the same limit distribution.

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