# A non-symmetric Kesten criterion and ratio limit theorem for random walks on
amenable groups

Rhiannon Dougall and Richard Sharp Department of Mathematical Sciences, Durham
University, Upper Mountjoy, Durham DH1 3LE<EMAIL_ADDRESS>Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
<EMAIL_ADDRESS>
###### Abstract.

We consider random walks on countable groups. A celebrated result of Kesten
says that the spectral radius of a symmetric walk (whose support generates the
group as a semigroup) is equal to one if and only if the group is amenable. We
give an analogue of this result for finitely supported walks which are not
symmetric. We also conclude a ratio limit theorem for amenable groups.

©2023. This work is licensed by a CC BY license.

##  1\. Introduction

Let $G$ be a finitely generated countable group. Let $\mu$ be a probability
measure on $G$, i.e. a function $\mu:G\to\mathbb{R}^{+}$ such that $\sum_{g\in
G}\mu(g)=1$. Let $S_{\mu}:=\\{g\in G\hbox{ : }\mu(g)>0\\}$, the support of
$\mu$. We say that $\mu$ is _non-degenerate_ if $S_{\mu}$ generates $G$ as a
semigroup. (We do not require $S_{\mu}$ to be finite.)

Let $|\cdot|$ be a word metric on $G$ associated to some finite generating
set. (We do not assume any connection between this set and $S_{\mu}$.) We say
that $\mu$ has finite first moment if

$\sum_{g\in G}|g|\mu(g)<\infty$

and that $\mu$ has finite exponential moment of order $c>0$ if

$\sum_{g\in G}e^{c|g|}\mu(g)<\infty.$

The measure $\mu$ defines a random walk on $G$ with transition probabilities
$p(s,t)=\mu(s^{-1}t)$. The convolution $\mu\ast\nu$ of two functions
$\mu,\nu:G\to\mathbb{R}^{+}$ is defined by

$\mu\ast\nu(g)=\sum_{s\in G}\mu(s)\nu(s^{-1}g).$

We will be interested in the spectral radius of this random walk, defined by

$\lambda(G,\mu):=\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n}.$

Clearly, $0\leq\lambda(G,\mu)\leq 1$. A celebrated theorem of Kesten (which
does not even require $G$ to be finitely generated) says that if $\mu$ is
symmetric then $\lambda(G,\mu)=1$ if and only if $G$ is amenable [8]. (We
recall that $G$ is amenable if and only if it admits a Banach mean, i.e. a
linear functional $M:\ell^{\infty}(G)\to\mathbb{R}$ such that
$M(\mathds{1})=1$, $\inf_{g\in G}f(g)\leq M(f)\leq\sup_{g\in G}f(g)$, and
$M(f_{g})=M(f)$, where $f_{g}(x)=f(gx)$. See the papers of Følner [5] and Day
[2] for further discussion.) The aim of this note is to generalise Kesten’s
criterion to the non-symmetric case.

To state our generalisation, we need to consider the abelianisation of $G$.
Since $G$ is finitely generated, this has a finite rank $k\geq 0$. Let
$G^{\mathrm{ab}}=G/[G,G]$ denote the abelianisation of $G$ and let
$G^{\mathrm{ab}}_{\mathrm{T}}$ denote the torsion subgroup of
$G^{\mathrm{ab}}$. Now set
$\overline{G}=G^{\mathrm{ab}}/G^{\mathrm{ab}}_{\mathrm{T}}\cong\mathbb{Z}^{k}$,
for some $k\geq 0$, (the torsion-free part of the abelianisation) and let
$\pi:G\to\overline{G}$ be the natural projection homomorphism. Write
$\bar{\mu}=\pi_{*}(\mu)$, i.e.

$\bar{\mu}(m)=\sum_{\begin{subarray}{c}g\in G\\\
\pi(g)=m\end{subarray}}\mu(g).$

######  Theorem 1.1 (Non-symmetric Kesten criterion).

Let $G$ be a finitely generated group and let $\mu$ be a non-degenerate
probability measure on $G$. Then

$G\;\mathrm{amenable}\;\iff\lambda(G,\mu)=\lambda(\overline{G},\bar{\mu}).$

The special case where $\mu$ has finite support originally appeared in
Dougall–Sharp [4], where it is written in the language of subshifts of finite
type and Gibbs measures.

The value of $\lambda(\overline{G},\bar{\mu})$ may be characterised in the
following way. Define
$\phi_{\mu}:\mathbb{R}^{k}\to\mathbb{R}^{+}\cup\\{+\infty\\}$ by

$\phi_{\mu}(v)=\sum_{g\in G}e^{\langle
v,\pi(g)\rangle}\mu(g)=\sum_{m\in\mathbb{Z}^{k}}e^{\langle
v,m\rangle}\bar{\mu}(m),$

where $\langle v,m\rangle=v_{1}m_{1}+\cdots+v_{k}m_{k}$. By a result of Stone
[15],[16], there is a unique $\xi\in\mathbb{R}^{k}$ at which $\phi_{\mu}(v)$
attains its minimum. Then $\lambda(\overline{G},\bar{\mu})=\phi_{\mu}(\xi)$.
This is discussed in more detail in Section 2.

A probability measure $\mu$ (with finite first moment) is said to be _centred_
if for each homomorphism $\chi:G\to\mathbb{R}$, we have

$\sum_{g\in G}\chi(g)\mu(g)=0.$

Any such homomorphism factors through $\overline{G}$ so it is easy to see that
$\mu$ is centred if and only if either $k=0$ or

$\sum_{g\in G}\pi(g)\mu(x)=\sum_{m\in\bar{G}}m\bar{\mu}(m)=0.$

In particular, $\mu$ is centred if and only if $\bar{\mu}$ is centred.

If, in addition, $\mu$ has a finite exponential moment of some order then we
have the following result.

######  Corollary 1.2.

Let $G$ be a finitely group and let $\mu$ be a non-degenerate probability
measure on $G$. Provided $\mu$ has a finite exponential moment of some order,
we have $\lambda(G,\mu)=1$ if and only if $G$ is amenable and $\mu$ is
centred.

Theorem 1.1 allows us to prove a ratio limit theorem for amenable groups with
an explicit limit. To avoid any parity issues, it is convenient to restrict to
aperiodic walks. We say that $(G,\mu)$ is aperiodic if there exists $n_{0}\geq
1$ such that $\mu^{\ast n}(e)>0$ for all $n\geq n_{0}$.

######  Theorem 1.3 (Ratio limit theorem).

Suppose that $G$ is a finitely generated amenable and that $\mu$ is a non-
degenerate probability measure on $G$. Assume in addition that $(G,\mu)$ is
aperiodic. Then, for each $g\in G$,

$\lim_{n\to\infty}\frac{\mu^{\ast n}(g)}{\mu^{\ast
n}(e)}=e^{-\langle\xi,\pi(g)\rangle},$

where $\xi\in\mathbb{R}^{k}$ is the unique value for which
$\lambda(G,\mu)=\phi_{\mu}(\xi)$.

######  Remark 1.4.

One should compare this with a theorem of Avez [1] that says that if $G$ is
amenable and $\mu$ is _symmetric_ , non-degenerate and aperiodic then
$\lim_{n\to\infty}\mu^{\ast n}(g)/\mu^{\ast n}(e)=1$, for all $g\in G$.

######  Remark 1.5.

It should be noted that there is no a priori mechanism to guarantee that the
ratios do indeed have a limit. However, notice that if one has the ratio
limits

$\lim_{n\to\infty}\frac{\mu^{\ast n}(g)}{\mu^{\ast
n}(e)}=e^{-\langle\xi,\pi(g)\rangle},$

for all $g\in G$, for _some_ $\xi$ then $G$ is necessarily amenable. We give
the short demonstration. From the hypothesis we have for any $s\in G$,

$\frac{\mu^{\ast n}(s^{-1})}{\mu^{\ast n}(e)}=e^{\langle\xi,\pi(s)\rangle}.$

Now, since

$\mu^{\ast(n+1)}(g)=\sum_{s\in G}\mu(s)\mu^{\ast n}(s^{-1}g),$

we then have

(1.1) $\lim_{n\to\infty}\frac{\mu^{\ast(n+1)}(e)}{\mu^{\ast
n}(e)}=\lim_{n\to\infty}\sum_{s\in S_{\mu}}\mu(s)\frac{\mu^{\ast
n}(s^{-1})}{\mu^{\ast n}(e)}\geq\sum_{s\in
S_{\mu}}\mu(s)e^{\langle\xi,\pi(g)\rangle}=\phi_{\mu}(\xi).$

In particular $\phi_{\mu}(\xi)<\infty$. We proceed with the proof assuming
that $\xi=0$ and deduce the general case after.

Now using that

$\frac{\mu^{\ast n}(e)}{\mu^{\ast 1}(e)}=\prod_{m=2}^{n}\frac{\mu^{\ast
m}(e)}{\mu^{\ast(m-1)}(e)}$

we see that (1.1) with $\phi_{\mu}(0)=1$ implies that
$\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n}=1$. This contradicts Day’s [2]
generalisation of Kesten’s criterion to the random walk operator spectral
radius — a consequence of which is that, for any non-degenerate probability,
we have that if $G$ is non-amenable then $\limsup_{n\to\infty}(\mu^{\ast
n}(e))^{1/n}<1$.

For the general case $\xi\neq 0$, we have already shown that
$\phi_{\mu}(\xi)<\infty$, and so

$\hat{\mu}(g)=\frac{e^{\langle\xi,\pi(g)\rangle}}{\phi_{\mu}(\xi)}\mu(g)$

is a well-defined probability measure on $G$ with ratio limits equal to one,
and we again conclude that $G$ is amenable.

Let us now outline the contents of the rest of the paper. In Section 2, we
recall results of Stone on random walks on $\mathbb{Z}^{k}$ that are essential
to the formulation of our results, and the rather general results of Gerl. In
Section 3, we give a proof of Corollary 1.2 assuming Theorem 1.1. We prove
Theorem 1.1 in Sections 4 and 5. Theorem 1.3 is proved in Section 6, as a
consequence of equidistribution results for countable state shifts.

### Acknowledgements

We are grateful to Wolfgang Woess for bringing to our attention the work of
Gerl. We are grateful to Rémi Coulon for asking whether Theorem 1.3 had a
converse.

##  2\. Results of Stone and Gerl

In this section, we recall classic results of Stone concerning random walks on
$\mathbb{Z}^{k}$. Let $\omega$ be a non-degenerate aperiodic probability
measure on $\mathbb{Z}^{k}$ and define
$\phi_{\omega}:\mathbb{R}^{k}\to\mathbb{R}\cup\\{+\infty\\}$ by

$\phi_{\omega}(v)=\sum_{m\in\mathbb{Z}^{k}}e^{\langle v,m\rangle}\omega(m).$

######  Lemma 2.1 (Stone [15], [16]).

If $\omega$ is non-degenerate then there is a unique $\xi\in\mathbb{R}^{k}$
such that $\phi_{\omega}(\xi)=\inf_{v\in\mathbb{R}^{k}}\phi_{\omega}(v)$.
Furthermore, $\lambda(\mathbb{Z}^{k},\omega)=1$ if and only if
$\phi_{\omega}(\xi)=1$, i.e. if and only if $\xi=0$.

We note that $\phi_{\omega}(\xi)=1$ if and only if $\phi_{\omega}(v)\geq 1$
for all $v\in\mathbb{R}^{k}$.

######  Corollary 2.2.

$\lambda(\mathbb{Z}^{k},\omega)=\phi_{\omega}(\xi)$.

###### Proof.

Suppose that $\phi_{\omega}(\xi)<1$. Then we can define a new probability
measure $\omega_{\xi}$ on $\mathbb{Z}^{k}$ by

$\omega_{\xi}(m)=(\phi_{\omega}(\xi))^{-1}e^{\langle\xi,m\rangle}\omega(m).$

Then $\omega_{\xi}$ has the same support as $\omega$ and

$\omega_{\xi}^{\ast
n}(m)=(\phi_{\omega}(\xi))^{-n}e^{\langle\xi,m\rangle}\omega^{\ast n}(m).$

We have

$\sum_{m\in\mathbb{Z}^{k}}e^{\langle
v,m\rangle}\omega_{\xi}(m)=\frac{1}{\phi_{\omega}(\xi)}\sum_{m\in\mathbb{Z}^{k}}e^{\langle
v+\xi,m\rangle}\omega(m)=\frac{\phi_{\omega}(v+\xi)}{\phi_{\omega}(\xi)}\geq
1.$

Hence, $\lambda(\mathbb{Z}^{k},\omega_{\xi})=1$ and

$\lambda(\mathbb{Z}^{k},\omega)=\phi_{\omega}(\xi)\lambda(\mathbb{Z}^{k},\omega_{\xi})=\phi_{\omega}(\xi).$

∎

We now state a ratio limit theorem due to Stone.

######  Proposition 2.3 (Stone [15]).

Suppose that $\omega$ is non-degenerate and aperiodic. Then, for each
$m\in\mathbb{Z}^{k}$,

(2.1)
$\lim_{n\to\infty}\frac{\omega^{*n}(m)}{\omega^{*n}(0)}=e^{-\langle\xi,m\rangle}.$

Ratio limit theorems are intimately related to the existence of harmonic
functions. Given a random walk $(G,\mu)$, we define the random walk operator
$P_{\mu}:\ell_{\mu}^{1}(G)\to\ell_{\mu}^{1}(G)$ by

$P_{\mu}f(g)=\sum_{s\in G}p(g,s)f(s)=\sum_{s\in G}\mu(g^{-1}s)f(s).$

This may be written as a convolution $P_{\mu}f=f\ast\check{\mu}$, where
$\check{\mu}(s)=\mu(s^{-1})$. A function $f:G\to\mathbb{R}^{+}$ is called
$\lambda$-_harmonic_ if $P_{\mu}f=\lambda f$, i.e. if

$\sum_{s\in G}\mu(g^{-1}s)f(s)=\lambda f(g).$

(Some authors define $f$ to be $\lambda$-harmonic if $\mu\ast f=\lambda f$.)
If we write $h_{\xi}(m)=e^{-\langle\xi,m\rangle}$ for the limit in Proposition
2.3 then we see that the function
$\check{h}_{\xi}(m):=h_{\xi}(-m)=e^{\langle\xi,m\rangle}$ is
$\lambda(\mathbb{Z}^{k},\omega)$-harmonic (for $\omega$). Furthermore
$\lambda=\lambda(\mathbb{Z}^{k},\omega)$ is the smallest value for which there
is a $\lambda$-harmonic function.

One may ask about ratio limit theorems on more general groups than
$\mathbb{Z}^{k}$. Following earlier work by Avez [1] and Gerl [6], a rather
general ratio limit theorem was proved by Gerl [7], where it is obtained as a
corollary of the following limit theorem. A detailed account and discussion
may be found in the recent note by Woess [18].

######  Proposition 2.4 (Gerl’s fundamental theorem [7]).

Suppose that $\mu$ is a non-degenerate probability measure on $G$ such that
$(G,\mu)$ is aperiodic. Then we have

$\lim_{n\to\infty}\frac{\mu^{\ast(n+1)}(e)}{\mu^{\ast n}(e)}=\lambda(G,\mu).$

Gerl used this to prove the following conditional ratio limit theorem.

######  Proposition 2.5 (Gerl’s ratio limit theorem [7]).

Suppose that $\mu$ is a non-degenerate probability measure on $G$ such that
$(G,\mu)$ is aperiodic. Suppose there is a set
$\mathfrak{F}\subset\\{f:G\to\mathbb{R}^{+}\hbox{ : }f(e)=1\\}$ such that

  1. (1)

if $f:G\to\mathbb{R}^{+}$ is defined by

$f(g)=\lim_{j\to\infty}\frac{\mu^{\ast n_{j}}(g)}{\mu^{\ast n_{j}}(e)},$

for some subsequence $(n_{j})_{j=1}^{\infty}$, then $f\in\mathfrak{F}$;

  2. (2)

there exists a unique $h\in\mathfrak{F}$ satisfying the equation $\mu\ast
h=\lambda(G,\mu)h$.

Then

$\lim_{n\to\infty}\frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)}=h(g),$

for all $g\in G$.

In particular, if we have uniqueness of a $\lambda(G,\mu)$-harmonic function
for $(G,\mu)$ then we know the ratio limit theorem holds. The advantage of our
Theorem 1.3, for amenable groups, is that we don’t consider arbitrary harmonic
functions instead we directly work with functions coming from the
abelianisation.

##  3\. Proof of Corollary 1.2

In this section we prove Corollary 1.2, assuming Theorem 1.1. We note that
$\phi_{\mu}=\phi_{\bar{\mu}}$, so we can use Lemma 2.1.

###### Proof of Corollary 1.2.

If $G$ is not amenable then Theorem 1.1 tells us that
$\lambda(G,\mu)<\lambda(\overline{G},\bar{\mu})\leq 1$; so it suffices to show
that, if $G$ is amenable, then $\lambda(\overline{G},\bar{\mu})=1$ if and only
if $\mu$ is centred.

Since $\bar{\mu}$ is non-degenerate, $\phi_{\mu}(v)$ is strictly convex on the
set where it is finite. The hypothesis of a finite exponential moment implies
that $\phi_{\mu}(v)$ is finite and differentiable in some neighbourhood of
$0\in\mathbb{R}^{k}$. Therefore, $\phi_{\mu}(v)$ has its unique minimum at
$v=0$ if and only if $\nabla\phi_{\mu}(0)=0$. Suppose that
$\lambda(\overline{G},\bar{\mu})=1$; then, by Lemma 2.1, $\phi_{\mu}$ has its
minimum at $0$ and so

$0=\nabla\phi_{\mu}(0)=\sum_{m\in\mathbb{Z}^{k}}m\bar{\mu}(m),$

i.e. $\mu$ is centred. On the other hand, if
$\lambda(\overline{G},\bar{\mu})<1$ then, again by Lemma 2.1, the unique
minimum of $\phi_{\mu}$ is not at $0$ and so

$0\neq\nabla\phi_{\mu}(0)=\sum_{m\in\mathbb{Z}^{k}}m\bar{\mu}(m),$

i.e. $\mu$ is not centred. ∎

##  4\. Proof of Theorem 1.1 ($\impliedby$)

In this section we show that if
$\lambda(G,\mu)=\lambda(\overline{G},\bar{\mu})$ then $G$ is amenable. (In
Kesten’s original theorem, this was the harder direction but in our case it is
the easier implication.) Noting that $\phi_{\mu}=\phi_{\bar{\mu}}$, recall
from Section 2 that there is a unique $\xi\in\mathbb{R}^{k}$ such that

$\phi_{\mu}(\xi)=\inf_{v\in\mathbb{R}^{k}}\phi_{\mu}(v)$

and

$\phi_{\mu}(\xi)=\sum_{g\in
G}\mu(g)e^{\langle\xi,\pi(g)\rangle}=\lambda(\overline{G},\bar{\mu}).$

We define a new probability measure $\mu_{\xi}$ on $G$ (analogous to the
measure $\omega_{\xi}$ on $\mathbb{Z}^{k}$ in the proof of Corollary 2.2) by

$\mu_{\xi}(g)=\phi_{\mu}(\xi)^{-1}e^{\langle\xi,\pi(g)\rangle}\mu(g).$

###### Proof of Theorem 1.1 ($\impliedby$).

Assume that $G$ is non-amenable. By Theorem 1 of Day [2] (see also Theorem 5.4
of Stadlbauer [13]), we see that the probability measure $\mu_{\xi}$ satisfies

$\limsup_{n\to\infty}(\mu_{\xi}^{\ast n}(e))^{1/n}<1.$

Unpicking the definitions, $\mu_{\xi}^{\ast
n}(e)=\phi_{\mu}(\xi)^{-n}\mu^{\ast n}(e)$. Hence
$\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n}<\phi_{\mu}(\xi).$ ∎

##  5\. Proof of Theorem 1.1 ($\implies$)

In this section we will show the harder implication that if $G$ is amenable
then $\lambda(\overline{G},\bar{\mu})\leq\lambda(G,\mu)$, and hence that
$\lambda(\overline{G},\bar{\mu})=\lambda(G,\mu)$. We remark that the proof
given here is significantly easier and more direct than the one we gave in
[4]. Following that proof would introduce a family of measures, indexed by
$g\in G$, on the space $S_{\mu}^{\mathbb{N}}\times G$, each describing the
paths that visit $S_{\mu}^{\mathbb{N}}\times\\{g\\}$. These measures (which
are also analysed in [14]) are not required here.

Let us begin by emphasising the following: though we know that $\bar{\mu}$ has
a $\lambda(\overline{G},\bar{\mu})$-harmonic function _it plays no role in
this part of proof_! The first element of the proof is the following
proposition. Subsequently, the rest of the section will be devoted to showing
that show its hypothesis is satisfied with $\lambda=\lambda(G,\mu)$.

######  Proposition 5.1.

If there is a homomorphism $h:\overline{G}\to\mathbb{R}^{>0}$, the
multiplicative group of positive real numbers, such that, for all
$n\in\mathbb{N}$,

$\sum_{g\in G}\mu^{\ast n}(g)h(-\pi(g))\leq\lambda^{n}$

then $\lambda(\overline{G},\bar{\mu})\leq\lambda$.

###### Proof.

Suppose such a homomorphism $h$ exists. Since $h$ is positive so we can throw
away the terms where $-\pi(g)\neq 0$ and obtain

$\sum_{\begin{subarray}{c}g\in G\\\ \pi(g)=0\end{subarray}}\mu^{\ast
n}(g)\leq\lambda^{n}.$

Hence, for any $\delta<1$,

$\sum_{n\in\mathbb{N}}\bar{\mu}^{\ast
n}(0)(\delta^{-1}\lambda)^{-n}=\sum_{n\in\mathbb{N}}\sum_{\begin{subarray}{c}g\in
G\\\ \pi(g)=0\end{subarray}}\mu^{\ast n}(g)\delta^{n}\lambda^{-n}<\infty.$

This says that $\lambda(\overline{G},\bar{\mu})\leq\lambda\delta^{-1}$. Since
we can take $\delta$ arbitrarily close to $1$ we are done. ∎

We view $\lambda(G,\mu)$ as a convergence parameter for the series

$\zeta(t)=\sum_{n\in\mathbb{N}}\mu^{\ast n}(e)t^{-n},$

where $t>0$, i.e.

$\inf\left\\{t\in\mathbb{R}^{+}:\sum_{n\in\mathbb{N}}\mu^{\ast
n}(e)t^{-n}<\infty\right\\}=\limsup_{n\to\infty}(\mu^{\ast
n}(e))^{1/n}=\lambda(G,\mu).$

This formulation is reminiscent of the Poincaré series used in the
construction of Patterson–Sullivan measures on the limit sets of Kleinian
groups and more general limit sets and, indeed, we employ ideas from this
theory The most relevant reference here is Roblin [9], which covers the basic
tools of Patterson-Sullivan theory and the amenability “trick” we will use in
the proof of Proposition 5.6.

The series $\zeta(t)$ does not necessarily diverge at $t=\lambda(G,\mu)$ but
one can modify the series, in a controlled way, to guarantee divergence at
this critical parameter. The following appears as Lemma 3.2 in Denker and
Urbanski [3] (see also [14]).

######  Lemma 5.2.

Let $(a_{n})_{n=1}^{\infty}$ be a sequence in $\mathbb{R}^{+}$ and let
$\rho=\limsup_{n\to\infty}a_{n}^{1/n}$. Then there is a sequence
$(b_{n})_{n=1}^{\infty}$ of positive real numbers such that
$\lim_{n\to\infty}b_{n+1}/b_{n}=1$ for which
$\sum_{n\in\mathbb{N}}a_{n}b_{n}t^{-n}<\infty$ for $t>\rho$ but

$\sum_{n\in\mathbb{N}}a_{n}b_{n}\rho^{-n}=\infty.$

Let $(b_{n})_{n=1}^{\infty}$ be the sequence given by Lemma 5.2 for the series
with $a_{n}=\mu^{\ast n}(e)$. We prefer to use $c_{n}=1/b_{n}$, a decreasing
sequence. Note that we have $\lim_{n\to\infty}c_{n-r}/c_{n}=1$ for all
$r\in\mathbb{N}$. We will work with a modified series $\zeta_{c}^{e}(t)$
defined by

$\zeta^{e}_{c}(t)=\sum_{n\in\mathbb{N}}\frac{\mu^{\ast n}(e)}{c_{n}}t^{-n},$

and also, for each $g\in G$, the series

$\zeta^{g}_{c}(t)=\sum_{n\in\mathbb{N}}\frac{\mu^{\ast n}(g)}{c_{n}}t^{-n}.$

######  Lemma 5.3.

For each $g\in G$,

$0<\inf_{\lambda(G,\mu)<t\leq
2}\frac{\zeta_{c}^{g}(t)}{\zeta_{c}^{e}(t)}\leq\sup_{\lambda(G,\mu)<t\leq
2}\frac{\zeta_{c}^{g}(t)}{\zeta_{c}^{e}(t)}<\infty.$

###### Proof.

We begin by observing that, for every $g,h\in G$, we have

$\mu^{\ast(n+k)}(g)\geq\mu^{\ast k}(gh^{-1})\mu^{\ast n}(h)$

and we may choose $k\geq 1$ for which $\mu^{\ast k}(gh^{-1})>0$. This gives us
the inequality

$\displaystyle\zeta_{c}^{g}(t)=\sum_{m\leq k}\frac{\mu^{\ast
m}(g)}{c_{m}}+\sum_{n\in\mathbb{N}}\frac{\mu^{\ast(n+k)}(g)}{c_{n+k}}\geq\sum_{m\leq
k}\frac{\mu^{\ast m}(g)}{c_{m}}+\sum_{n\in\mathbb{N}}\mu^{\ast
k}(gh^{-1})\frac{\mu^{\ast n}(h)}{c_{n}}\frac{c_{n}}{c_{n+k}}.$

Since the numbers $c_{n}$ are positive and, for a fixed $k$,
$\lim_{n\to\infty}c_{n}/c_{n+k}=1$, we have $\inf_{n}c_{n}/c_{n+k}>0$. Hence

$\zeta_{c}^{g}(t)\geq
C_{1}(g,k)+\frac{C_{2}(g,h,k)}{C_{3}(k)}\sum_{n\in\mathbb{N}}\frac{\mu^{\ast
n}(h)}{c_{n}},$

for positive $C_{1}$, $C_{2}$ and $C_{3}$. We conclude

$0<\inf_{\lambda(G,\mu)<t\leq 2}\frac{\zeta_{c}^{g}(t)}{\zeta_{c}^{h}(t)}.$

Since $g,h$ are arbitrary the lemma follows. ∎

By the previous lemma and a standard diagonal argument, there exists a
sequence $t_{n}\to\lambda(G,\mu)$ for which the following limits are well-
defined

$H(g)=\lim_{n\to\infty}\frac{\zeta_{c}^{g}(t_{n})}{\zeta_{c}^{e}(t_{n})}\in(0,\infty),$

for all $g\in G$. A crucial observation is the following.

######  Lemma 5.4.

For any $r\in\mathbb{N}$, we have

$\sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g)=\lambda^{r}H(g)$

with $\lambda=\lambda(G,\mu)$.

###### Proof.

Fix $r\in\mathbb{N}$ and let $\epsilon>0$. Since
$\lim_{n\to\infty}c_{n-r}/c_{n}=1$, we can choose $n_{0}$ such that
$1-\epsilon\leq c_{n-r}/c_{n}\leq 1+\epsilon$, for all $n>n_{0}$. We will also
use that

$\mu^{\ast n}(g)=\sum_{s\in G}\mu^{\ast r}(s)\mu^{\ast(n-r)}(s^{-1}g).$

Then, for $n>n_{0}$,

$\frac{1-\epsilon}{c_{n-r}}\sum_{s\in G}\mu^{\ast
r}(s)\mu^{\ast(n-r)}(s^{-1}g)\leq\frac{\mu^{\ast
n}(g)}{c_{n}}\leq\frac{(1+\epsilon)}{c_{n-r}}\sum_{s\in G}\mu^{\ast
r}(s)\mu^{\ast(n-r)}(s^{-1}g).$

Setting

$C_{1}(g,t,n_{0})=\sum_{n\leq n_{0}}\frac{\mu^{\ast n}(g)}{c_{n}}t^{-n},$

we have

$\displaystyle t^{r}\sum_{n\in\mathbb{N}}\frac{\mu^{\ast n}(g)}{c_{n}}t^{-n}$
$\displaystyle\leq C_{1}(g,t,n_{0})+t^{r}(1+\epsilon)\sum_{s\in G}\mu^{\ast
r}(s)\sum_{n>n_{0}}t^{-n}\frac{\mu^{\ast(n-r)}(s^{-1}g)}{c_{n-r}}$
$\displaystyle\leq C_{1}(g,t,n_{0})+(1+\epsilon)\sum_{s\in G}\mu^{\ast
r}(s)H(s^{-1}g)\zeta^{e}_{c}(t),$

using that the terms in the series are non-negative. This gives

$\lambda^{r}H(g)\leq\lim_{m\to\infty}\frac{C_{1}(g,t_{m},n_{0})}{\zeta^{e}_{c}(t_{m})}+(1+\epsilon)\sum_{s\in
G}\mu^{\ast r}(s)H(s^{-1}g)=(1+\epsilon)\sum_{s\in G}\mu^{\ast
r}(s)H(s^{-1}g)$

and, since $\epsilon$ is arbitrary,

$\lambda^{r}H(g)\leq\sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g).$

For the lower bound, we have

$\displaystyle t^{r}\sum_{n\in\mathbb{N}}\frac{\mu^{\ast n}(g)}{c_{n}}t^{-n}$
$\displaystyle\geq(1-\epsilon)\sum_{s\in G}\mu^{\ast
r}(s)\sum_{n>n_{0}}t^{-(n-r)}\frac{\mu^{\ast(n-r)}(s^{-1}g)}{c_{n-r}}$
$\displaystyle=(1-\epsilon)\sum_{s\in G}\mu^{\ast
r}(s)\sum_{n\in\mathbb{N}}t^{-n}\frac{\mu^{\ast n}(s^{-1}g)}{c_{n}}.$

This gives

$\displaystyle\lambda^{r}H(g)$ $\displaystyle\geq\sum_{s\in G}\mu^{\ast
r}(s)H(s^{-1}g).$

∎

Lemma 5.4 gives us the following corollary.

######  Corollary 5.5.

For any fixed $\gamma\in G$, we have

$0<\inf_{g\in G}\frac{H(\gamma g)}{H(g)}\leq\sup_{g\in G}\frac{H(\gamma
g)}{H(g)}<\infty.$

###### Proof.

Given $\gamma\in G$, we can find $x_{1},\ldots,x_{k}\in S_{\mu}$, for some
$k\geq 1$, such that $x_{1}\cdots x_{k}=\gamma^{-1}$. This gives us

$\mu(x_{1})\cdots\mu(x_{k})H(\gamma g)\leq\sum_{(s_{1},\ldots,s_{k})\in
G^{k}}\mu(s_{1})\cdots\mu(s_{k})H((s_{1}\cdots s_{k})^{-1}g)=\lambda^{k}H(g),$

and so $\sup_{g\in G}H(\gamma g)/H(g)$ is finite.

Now we put $\gamma g$ on the right hand side and choose
$y_{1},\ldots,y_{\ell}\in S_{\mu}$ such that $y_{1}\cdots y_{\ell}=\gamma$ to
get

$\mu(y_{1})\cdots\mu(y_{\ell})H(g)\leq\sum_{(s_{1},\ldots,s_{\ell})\in
G^{\ell}}\mu(s_{1})\cdots\mu(s_{\ell})H((s_{1}\cdots s_{\ell})^{-1}\gamma
g)=\lambda^{\ell}H(\gamma g),$

and so $\inf_{g\in G}H(\gamma g)/H(g)$ is positive. ∎

We are now ready to use the amenability of $G$. We use the existence of a
Banach mean on $\ell^{\infty}(G)$ to “average over $g$” in the last lemma.

######  Proposition 5.6.

There is a homomorphism $h:\overline{G}\to\mathbb{R}^{>0}$ such that, for all
$n\in\mathbb{N}$,

$\sum_{s\in G}\mu^{\ast n}(s)h(-\pi(s))\leq\lambda^{n},$

with $\lambda=\lambda(G,\mu)$.

###### Proof.

Since any homomorphism $h:G\to\mathbb{R}^{>0}$ factors through $\overline{G}$,
it suffices to show that there is a homomorphism $h:G\to\mathbb{R}^{>0}$ such
that, for all $n\in\mathbb{N}$, we have

(5.1) $\sum_{s\in G}\mu^{\ast n}(s)h(s^{-1})\leq\lambda^{n}.$

Let $M$ be a Banach mean on $\ell^{\infty}(G)$. Jensen’s inequality says that
if $\varphi$ is convex then

$M(\varphi(f))\geq\varphi(M(f)).$

(This is more familiar when the linear functional is integration with respect
to a probability measure. One can check that we only need monotonicity, finite
additivity, and the unit normalisation $M(1)=1$.) We apply this to the
function $g\mapsto(H(\gamma g)/H(g))$ in $\ell^{\infty}(G)$. (Note we have
used Corollary 5.5 to know that $g\mapsto\log(H(\gamma g)/H(g))$ is in
$\ell^{\infty}(G)$.) Thus we obtain

$\displaystyle M\left(g\mapsto\frac{H(\gamma g)}{H(g)}\right)$
$\displaystyle=M\left(g\mapsto\exp\log\frac{H(\gamma g)}{H(g)}\right)$
$\displaystyle\geq\exp M\left(g\mapsto\log\frac{H(\gamma g)}{H(g)}\right).$

Now set

$h(\gamma)=\exp M\left(g\mapsto\log\frac{H(\gamma g)}{H(g)}\right).$

We will show that $h$ satisfies (5.1), recalling that $M$ is only finitely
additive. Let $\\{g_{k}\\}_{k=1}^{\infty}$ be an enumeration of $G$ and, for
$N\geq 1$, let $G_{N}=\\{g_{1},\ldots,g_{N}\\}$. Lemma 5.4 gives us that, for
any $n\geq 1$ and any $N\geq 1$, we have

$\displaystyle\lambda^{n}$ $\displaystyle=M\left(g\mapsto\sum_{s\in
G}\mu^{\ast n}(s)\frac{H(s^{-1}g)}{H(g)}\right)\geq M\left(g\mapsto\sum_{s\in
G_{N}}\mu^{\ast n}(s)\frac{H(s^{-1}g)}{H(g)}\right)$ $\displaystyle=\sum_{s\in
G_{N}}\mu^{\ast
n}(s)M\left(g\mapsto\frac{H(s^{-1}g)}{H(g)}\right)\geq\sum_{s\in
G_{N}}\mu^{\ast n}(s)\exp M\left(g\mapsto\log\frac{H(s^{-1}g)}{H(g)}\right)$
$\displaystyle=\sum_{s\in G_{N}}\mu^{\ast n}(s)h(s^{-1}).$

Taking the supremum over $N$ gives (5.1).

It remains to show that $h$ is a homomorphism. Notice that

$\displaystyle\log h(ab)$
$\displaystyle=M\left(g\mapsto\log\frac{H(abg)}{H(g)}\right)$
$\displaystyle=M\left(g\mapsto\log\frac{H(abg)}{H(bg)}\right)+M\left(g\mapsto\log\frac{H(bg)}{H(g)}\right)$
$\displaystyle=\log h(a)+\log h(b)$

and

$\displaystyle\log h(\gamma^{-1})$
$\displaystyle=M\left(g\mapsto\log\frac{H(\gamma^{-1}g)}{H(g)}\right)=M\left(g\mapsto\log\frac{H(g)}{H(\gamma
g)}\right)$ $\displaystyle=M\left(g\mapsto-\log\frac{H(\gamma
g)}{H(g)}\right)=-\log h(\gamma).$

using translation invariance of $M$. The conclusion follows. ∎

Combining Proposition 5.1 and Proposition 5.6 shows that if $G$ is amenable
then $\lambda(\overline{G},\bar{\mu})\leq\lambda(G,\mu)$.

##  6\. Equidistribution and proof of the ratio limit theorem

In this section we use Theorem 1.1 to prove a ratio limit theorem for amenable
groups, Theorem 1.3. Our arguments will also give a new proof of Proposition
2.4 in this setting. Our approach is based on weighted equidistribution
results for countable state shift spaces. Suppose that $G$ is amenable, that
$\mu$ is non-degenerate and that $(G,\mu)$ is aperiodic. We let $\lambda$
denote the common value

$\lambda=\phi_{\mu}(\xi)=\lambda(\overline{G},\bar{\mu})=\lambda(G,\mu),$

given by Theorem 1.1. As above, $\mu_{\xi}(g)=\lambda^{-1}h(g)\mu(g)$, where
$h(g)=e^{\langle\xi,\pi(g)\rangle}$.

We consider the sequence space $\Sigma=S_{\mu}^{\mathbb{N}}$ and let
$\sigma:\Sigma\to\Sigma$ be the shift map:
$\sigma((s_{i})_{i=1}^{\infty})=(s_{i+1})_{i=1}^{\infty}$. If
$s=(s_{1},\ldots,s_{n})\in S_{\mu}^{n}$, we write $[s]=[s_{1},\dots,s_{n}]$
for the cylinder set

$[s_{1},\dots,s_{n}]:=\\{(x_{i})_{i=1}^{\infty}\in\Sigma\hbox{ : }x_{i}=s_{i}\
\forall i=1,\ldots,n\\}.$

We give $\Sigma$ the topology generated by cylinder sets (which are both open
and closed).

We denote by $\nu_{\xi}$ the Bernoulli measure on $\Sigma$ given by

$\nu_{\xi}([s_{1},\ldots,s_{n}])=\mu_{\xi}(s_{1})\cdots\mu_{\xi}(s_{n}).$

Let

$\Lambda_{n}=\\{s=(s_{1},\ldots,s_{n})\in S_{\mu}^{n}\hbox{ : }s_{1}\cdots
s_{n}=e\\}$

and define a sequence of probability measures $m_{n}$ on $\Sigma$ by

$\displaystyle m_{n}$ $\displaystyle:=\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{s=(s_{1},\ldots,s_{n})\in\Lambda_{n}}\mu_{\xi}(s_{1})\cdots\mu_{\xi}(s_{n})\delta_{s_{\infty}}=\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{s=(s_{1},\ldots,s_{n})\in\Lambda_{n}}\nu_{\xi}([s])\delta_{s_{\infty}},$

where we use the notation $s_{\infty}\in\Sigma$ to mean the one-sided infinite
concatenation of $s=(s_{1},\ldots,s_{n})$ and $\delta_{s_{\infty}}$ denotes
the Dirac measure at this point. We remark that we also have

$m_{n}=\frac{1}{\mu^{\ast
n}(e)}\sum_{s=(s_{1},\ldots,s_{n})\in\Lambda_{n}}\mu(s_{1})\cdots\mu(s_{n})\delta_{s_{\infty}}$

but we do not use this formula. We will need to explicitly evaluate the
measures $m_{n}$ on cylinder sets.

######  Lemma 6.1.

For a cylinder set $[u_{1},\ldots,u_{k}]$ we have that, for $n>k$,

$m_{n}([u_{1},\ldots,u_{k}])=\frac{\mu_{\xi}(u_{1})\cdots\mu_{\xi}(u_{k})}{\mu_{\xi}^{\ast
n}(e)}\mu_{\xi}^{\ast(n-k)}(u^{-1}),$

where $u=u_{1}\cdots u_{k}$.

###### Proof.

This is a straightforward calculation. For $n>k$,

$\displaystyle m_{n}([u_{1},\ldots,u_{k}])$
$\displaystyle=\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{(s_{1},\ldots,s_{n})\in\Lambda_{n}}\mu_{\xi}(s_{1})\cdots\mu_{\xi}(s_{n})\,\delta_{(s_{1},\ldots,s_{n})_{\infty}}([u_{1},\ldots,u_{k}])$
$\displaystyle=\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}(s_{1},\ldots,s_{n})\in\Lambda_{n}\\\
s_{1}=u_{1},\cdots,s_{k}=u_{k}\end{subarray}}\mu_{\xi}(s_{1})\cdots\mu_{\xi}(s_{n})$
$\displaystyle=\frac{\mu_{\xi}(u_{1})\cdots\mu_{\xi}(u_{k})}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}(s_{k+1},\ldots,s_{n})\in S_{\mu}^{n-k}\\\
s_{k+1}\cdots
s_{n}=u^{-1}\end{subarray}}\mu_{\xi}(s_{k+1})\cdots\mu_{\xi}(s_{n})$
$\displaystyle=\frac{\mu_{\xi}(u_{1})\cdots\mu_{\xi}(u_{k})}{\mu_{\xi}^{\ast
n}(e)}\mu_{\xi}^{\ast(n-k)}(u^{-1}).$

∎

We will show that, for each cylinder set $[u_{1},\ldots,u_{k}]$.
$m_{n}([u_{1},\ldots,u_{k}])$ converges to $\nu_{\xi}([u_{1},\ldots,u_{k}])$,
as $n\to\infty$. In order to have _convergence_ (as opposed to an accumulation
point) we need $\mu_{\xi}^{\ast n}(e)^{1/n}$ to have a limit. This is a
consequence of aperiodicity, as the next lemma shows.

######  Lemma 6.2.

We have

$\lim_{n\to\infty}(\mu_{\xi}^{\ast n}(e)^{1/n})=1.$

###### Proof.

We know that $\limsup_{n\to\infty}(\mu_{\xi}^{\ast n}(e))^{1/n}=1$. Since
$\mu$ is aperiodic, we have $\mu^{\ast n}(e)>0$ for all sufficiently large
$n$. Recall also that $\mu_{\xi}^{\ast(n+m)}(e)\geq\mu_{\xi}^{\ast
n}(e)\mu_{\xi}^{\ast m}(e)$. This tells us that $-\log\mu_{\xi}^{\ast n}(e)$
is sub-additive. Hence by Fekete’s lemma

$\lim_{n\to\infty}\frac{-\log\mu^{\ast n}(e)}{n}=\inf_{n\geq
1}\frac{-\log\mu^{\ast n}(e)}{n},$

in particular the limit exists and is equal to the limsup. ∎

In order to show the required convergence for the $m_{n}$, we introduce some
ideas and terminology from thermodynamic formalism and large deviation theory.
A function $\varphi:\Sigma\to\mathbb{R}$ is called locally Hölder continuous
if

(6.1) $\sup_{s\in S_{\mu}^{n}}\sup_{x,y\in[s]}|\varphi(x)-\varphi(y)|\leq
C\theta^{n},$

for some $C>0$ and $0<\theta<1$, for all $n\geq 1$. For a locally Hölder
continuous function $\varphi:\Sigma\to\mathbb{R}$, we define the _Gurevič
pressure_ $P_{\mathrm{G}}(\varphi)$ by

$P_{\mathrm{G}}(\varphi)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{s\in
S_{\mu}^{n}}\exp\sum_{j=0}^{n-1}\varphi(s_{\infty})\in\mathbb{R}\cup\\{+\infty\\}.$

(The original definition given by Sarig in [10] is somewhat different, and
only requires, (6.1) to hold for $n\geq 2$, but, by Corollary 1 of [11], the
above formula gives the Gurevič pressure in our setting.) We now fix

$\varphi((s_{i})_{i=1}^{\infty}):=\log\mu_{\xi}(s_{1})=\log\nu([s_{1}]),$

so that, in particular, $P_{\mathrm{G}}(\varphi)=0$. Let $\chi$ be the
indicator function of some cylinder. We can easily calculate from the
definition that, for $t\in\mathbb{R}$,
$P_{\mathrm{G}}(\varphi+t\chi)\leq\max\\{0,|t|\\}$ for all $t\in\mathbb{R}$.
Hence, by Corollary 4 of [11], $t\mapsto P_{\mathrm{G}}(\varphi+t\chi)$ is
real analytic for $t\in\mathbb{R}$ and, by Theorems 6.12 and 6.5 of [12],

(6.2) $\frac{dP(\varphi+t\chi)}{dt}\Bigg{|}_{t=0}=\int\chi\,d\nu_{\xi}.$

(The same discussion remains true if $\chi$ is replaced with any _bounded_
locally Hölder function but indicator functions of cylinders are sufficient
for our purposes.)

For $s\in S_{\nu}^{n}$, let $\tau_{s,n}$ denote the orbital measure

$\tau_{s,n}:=\frac{1}{n}\sum_{j=0}^{n-1}\delta_{\sigma^{j}(s_{\infty})}.$

Following, Theorem 7.4 of [12], we have the following large deviations bound.

######  Proposition 6.3.

Given $\epsilon>0$, there exists $C>0$ and $\eta(\epsilon)>0$ such that

$\sum_{\begin{subarray}{c}s\in S_{\mu}^{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|>\epsilon\end{subarray}}\nu_{\xi}([s])\leq
Ce^{-\eta(\epsilon)n}.$

###### Proof.

The proof is standard but we include it for completeness. We consider $s\in
S_{\mu}^{n}$ such that $\int\chi\,d\tau_{s,n}>\int\chi\,d\nu_{\xi}+\epsilon$
and $\int\chi\,d\tau_{s,n}<\int\chi\,d\nu_{\xi}-\epsilon$ separately. For
$t>0$, we have

$\displaystyle\sum_{\begin{subarray}{c}s\in S_{\mu}^{n}\\\
\int\chi\,d\tau_{s,n}>\int\chi
d\nu_{\xi}+\epsilon\end{subarray}}\nu_{\xi}([s])$ $\displaystyle\leq\sum_{s\in
S_{\mu}^{n}}e^{\varphi^{n}(s_{\infty})+t\chi^{n}(s_{\infty})-nt\int\chi\,d\nu_{\xi}-nt\epsilon},$

so that,

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log\sum_{\begin{subarray}{c}s\in
S_{\mu}^{n}\\\ \int\chi\,d\tau_{s,n}>\int\chi
d\nu_{\xi}+\epsilon\end{subarray}}\nu_{\xi}([s])\leq
P_{\mathrm{G}}(\varphi+t\chi)-t\int\chi\,d\nu_{\xi}-t\epsilon.$

Using $P_{\mathrm{G}}(\varphi)=0$ and (6.2), we see that, for sufficiently
small $t>0$, we have

$P_{\mathrm{G}}(\varphi+t\chi)-t\int\chi\,d\nu_{\xi}-t\epsilon<0.$

Similarly, for $t<0$,

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log\sum_{\begin{subarray}{c}s\in
S_{\mu}^{n}\\\ \int\chi\,d\tau_{s,n}<\int\chi
d\nu_{\xi}-\epsilon\end{subarray}}\nu_{\xi}([s])\leq
P_{\mathrm{G}}(\varphi+t\chi)-t\int\chi\,d\nu_{\xi}+t\epsilon.$

and, for sufficiently small $t<0$, this upper bound is negative. Combing these
estimates gives the result. ∎

Since $\Lambda_{n}\subset S_{\mu}^{n}$, we have the following immediate
corollary.

######  Corollary 6.4.

For $\epsilon>0$, we have

$\sum_{\begin{subarray}{c}s\in\Lambda_{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|>\epsilon\end{subarray}}\nu_{\xi}([s])\leq
Ce^{-\eta(\epsilon)n}.$

The following equidistribution result is now an easy consequence.

######  Proposition 6.5.

For any cylinder set $[u_{1},\ldots,u_{k}]$, we have that

$\lim_{n\to\infty}m_{n}([u_{1},\ldots,u_{k}])=\mu_{\xi}(u_{1})\cdots\mu_{\xi}(u_{k}).$

###### Proof.

Let $\chi:\Sigma\to\mathbb{R}$ be the indicator function of
$[u_{1},\ldots,u_{k}]$, then we need to show

$\lim_{n\to\infty}\int\chi\,dm_{n}=\int\chi\,d\nu_{\xi}.$

We have

$\displaystyle\int\chi\,dm_{n}-\int\chi\,d\nu_{\xi}$
$\displaystyle=\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}s\in\Lambda_{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|>\epsilon\end{subarray}}\nu_{\xi}([s])\int\chi\,d\tau_{s,n}$
$\displaystyle+\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}s\in\Lambda_{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|\leq\epsilon\end{subarray}}\nu_{\xi}([s])\int\chi\,d\tau_{s,n}-\int\chi\,d\nu_{\xi}.$

By Lemma 6.2 and Corollary 6.4, the first term on the right hand side tends to
zero exponentially fast. Also,

$\displaystyle\left|\frac{1}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}s\in\Lambda_{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|\leq\epsilon\end{subarray}}\nu_{\xi}([s])\int\chi\,d\tau_{s,n}-\int\chi\,d\nu_{\xi}\right|$
$\displaystyle\leq\epsilon+\frac{\left|\int\chi\,d\nu_{\xi}\right|}{\mu_{\xi}^{\ast
n}(e)}\sum_{\begin{subarray}{c}s\in\Lambda_{n}\\\
\left|\int\chi\,d\tau_{s,n}-\int\chi
d\nu_{\xi}\right|>\epsilon\end{subarray}}\nu_{\xi}([s])\leq\epsilon+Ce^{-\eta(\epsilon)n},$

which, since $\epsilon$ is arbitrary, gives the result. ∎

Combining Proposition 6.5 with Lemma 6.1, we see that for $u=u_{1}\cdots
u_{k}$ we have

(6.3) $\lim_{n\to\infty}\frac{\mu_{\xi}^{\ast(n-k)}(u^{-1})}{\mu_{\xi}^{\ast
n}(e)}=1.$

Most of the work is done. We prove Proposition 2.4 for amenable groups.

###### Proof of Proposition 2.4.

It suffices to show that

$\lim_{n\to\infty}\frac{\mu_{\xi}^{\ast(n+1)}(e)}{\mu_{\xi}^{\ast n}(e)}=1.$

If we choose $(u_{1},\ldots,u_{k})\in S_{\mu}^{k}$ with $u_{1}\cdots u_{k}=e$,
then equation (6.3) gives that
$\lim_{n\to\infty}\mu_{\xi}^{\ast(n-k)}(e)/\mu_{\xi}^{\ast n}(e)=1$, so that
$\lim_{n\to\infty}\mu_{\xi}^{\ast(n+k)}(e)/\mu_{\xi}^{\ast n}(e)=1$. We then
write

$\frac{\mu_{\xi}^{\ast(n+k)}(e)}{\mu^{\ast
n}(e)}=\frac{\mu_{\xi}^{\ast(n+k)}(e)}{\mu_{\xi}^{\ast(n+k-1)}(e)}\cdots\frac{\mu_{\xi}^{\ast(n+1)}(e)}{\mu_{\xi}^{\ast
n}(e)}$

and deduce that $\lim_{n\to\infty}\mu_{\xi}^{\ast(n+1)}(e)/\mu_{\xi}^{\ast
n}(e)=1$, as required. ∎

We can now establish the ratio limit theorem for amenable groups.

###### Proof of Theorem 1.3.

Let $g\in G$ be arbitrary. Choosing $(u_{1},\ldots,u_{k})$ with $u_{1}\cdots
u_{k}=g^{-1}$ and applying (6.3) gives that
$\lim_{n\to\infty}\mu_{\xi}^{\ast(n-k)}(g)/\mu_{\xi}^{\ast n}(e)=1$ and hence
that

$\lim_{n\to\infty}\frac{\mu^{\ast(n-k)}(g)}{\mu^{\ast
n}(e)}=\frac{\lambda^{-k}}{h(g)}.$

Now, applying Proposition 2.4,

$\frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)}=\frac{\mu^{\ast
n}(g)}{\mu^{\ast(n+k)}(e)}\frac{\mu^{\ast(n+k)}(e)}{\mu^{\ast n}(e)}\to
e^{-\langle\xi,\pi(g)\rangle},$

as $n\to\infty$. ∎

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# Existence of exponentially spatially localised breather solutions for
lattices of nonlinearly coupled particles: Schauder’s fixed point theorem
approach

Dirk Hennig  Nikos I. Karachalios  Department of Mathematics, University of
Thessaly, Lamia GR 35100, Greece

###### Abstract

Abstract: The problem of showing the existence of localised modes in nonlinear
lattices has attracted considerable efforts from the physical but also from
the mathematical viewpoint where a rich variety of methods has been employed.
In this paper we prove that a fixed point theory approach based on the
celebrated Schauder’s Fixed Point Theorem may provide a general method to
establish concisely not only the existence of localised structures but also a
required rate of spatial localisation. As a case study we consider lattices of
coupled particles with nonlinear nearest neighbour interaction and prove the
existence of exponentially spatially localised breathers exhibiting either
even-parity or odd-parity symmetry under necessary non-resonant conditions
accompanied with the proof of energy bounds of the solutions.

##  I Introduction

The study of nonlinear lattice dynamical systems is an active area of research
of continuously growing interest. For the motivation, efforts, and research
achievements in this major field of nonlinear science we refer to BK -TD2 . A
turning point can be traced back to the end of 30’s of the previous century
when the striking studies of Frenkel and Kontorova FK1  appeared. These
studies on crystal dislocations brought about the derivation of the Discrete
Klein-Gordon equation (DKG). In the 50’s came a revolution with the famous
Fermi-Pasta-Ulam-Tsingou (FPUT) experiment FPUT1 -TD2 , investigating the
equipartition of energy in a weakly nonlinear system of coupled oscillators.
The unexpected and surprising results of this experiment are identified with
the birth of the mathematics and physics of solitons. The FPUT system, the DKG
equation, the Discrete Nonlinear Schrödinger equation (DNLS) and their
variants are the prototype systems for the study of localised phenomena (in
crystals, nonlinear-waveguide arrays, Bose-Einstein condensates, DNA
macromolecules-Davydov mechanism).

Due to their relevance to the above mentioned nonlinear phenomena of
restrained energy redistribution, the problem of the existence of discrete
localised modes in the form of solitons and breathers in the above lattice
systems has attracted tremendous interest, not only from the physical but also
from the rigorous mathematical viewpoint. Results regarding proofs of
existence and stability of discrete solitons and breathers are based on a
variety of mathematical methods: on the one hand, integrability theory applies
to only very few systems such as the Toda and the Ablowitz-Ladik lattices
Toda1 -AL2 , while the wide class of non-integrable systems has been studied
with the continuation of (trivially localised) solutions from the anti-
integrable limit for sufficiently weak coupling strengths, Lyapunov-Schmidt
decomposition, normal form and center manifold reduction methods, Floquet
monodromy theory, as well as, direct and minimax variational approaches. We
refer to MacKay -APB  and cited works therein.

In this work, pursuing further our studies initiated in JFPT  for DKG-chains,
we claim that a fixed point theory approach, based on the celebrated
Schauder’s fixed point theorem Schauder , can offer a comparatively concise
and to our knowledge, novel approach compared to the aforementioned methods,
for the proof of existence of localised waveforms for nonlinear lattices. In
the present paper, we facilitate this fixed point approach to a general class
of lattices with nonlinear interaction between its constituents and with
general potentials whose second derivative is bounded by a power-law growth.
Prominent examples are the $\alpha$-FPUT and $\beta$-FPUT systems or
combinations of both. Other important examples of interaction potentials
include periodic and bounded soft potentials. We think that one of the main
assets of the Schauder’s fixed point method is that it establishes, apart from
existence, directly a requested degree of spatial localisation Pankov2 ,
Zhang3 , Pankov3 . That is, the breathers are proved to be exponentially
localised as it is the case in physically significant applications (see PRL1 ,
PhysRep1  and references therein). Another potential advantage, is that the
method is not restricted to the existence of small amplitude oscillations
James , and is promising for its extension to higher dimensional set-ups
(where localisation effects are more intricate FG ), as its major
characteristics (particularly the derivation of suitable estimates), seem not
to be affected by higher-dimensionality.

The presentation of the paper is as follows. In section II we discuss the
model systems and main assumptions on the interaction potential. Section III
contains the description and properties of the functional setting essentially
involving weighted sequence spaces and the proof of the main result. We claim
that the whole approach can be of general interest: Firstly, it implements a
Fourier series representation of the solutions to establish compactness
embedding properties of the considered functional spaces; these can be useful
generically for the analysis of lattice dynamical systems (existence and
dynamics, see NT2005 ,DNat  and references therein). Secondly, facilitating
the Schauder’s fixed point approach and investigating the contraction regime
for the nonlinear map in this set-up, gives rise to physically relevant and
consistent restrictions on the characteristics of the breather solutions,
namely, their frequency (natural non-resonant conditions) and their magnitude
measured in the suitable weighted norm (norm upper and lower bounds);
importantly, the latter exhibits dependence on the localisation degree (as
dictated by the relevant weighted norm) and the growth of the nonlinearity for
the general class of potentials considered herein (see Theorem III.2 and
subsection III.1). Finally, section IV summarises the main findings and gives
an outlook to extensions for future studies.

##  II Description of the system

The system of interacting particles on infinite lattices is given by the
following set of coupled equations

$\frac{d^{2}q_{n}}{dt^{2}}=-V^{\prime}(q_{n}-q_{n-1})+V^{\prime}(q_{n+1}-q_{n}),\,\,\,n\in{\mathbb{Z}},$
(2.1)

where $q_{n}$ is the displacement of a particle from its equilibrium position.
The function $V(x)$ is the interaction potential between nearest neighbours
and the prime ′ stands for the derivative with respect to the argument $x$.
This system has a Hamiltonian structure related to the energy

$H=\sum_{n\in{\mathbb{Z}}}\left(\frac{1}{2}p_{n}^{2}+V(q_{n}-q_{n-1})\right),$

and it is time-reversible with respect to the involution $p\mapsto-p$. In
addition to the energy $H=E$, the total momentum $P=\sum_{n}p_{n}$ is
conserved too.

Discrete breathers can be characterised as follows:

$\displaystyle q_{n}(t+T)$ $\displaystyle=$ $\displaystyle
q_{n}(t),\,\,\,p_{n}(t+T)=p_{n}(t),\,\,n\in{\mathbb{Z}},$ (2.2)
$\displaystyle\lim_{n\rightarrow\pm\infty}q_{n}$ $\displaystyle=$
$\displaystyle c_{\pm},\,\,\,\lim_{n\rightarrow\pm\infty}p_{n}=0,$ (2.3)

and the difference $c_{+}-c_{-}$ is the lattice deformation induced by the
breather. Furthermore, breather solutions are characterised by zero time
average, i.e.

$\int_{0}^{T}q_{n}(t)dt=0,\,\,\,n\in{\mathbb{Z}}.$ (2.4)

We consider periodic solutions exhibiting even-parity and odd-parity symmetry.
To this end we introduce the following linear mappings:

$\displaystyle
R_{odd}\,:\,(b)_{n\in\mathbb{Z}}\mapsto(b)_{n\in\mathbb{Z}},\,\,\,\left(R_{odd}\cdot
b\right)_{n}$ $\displaystyle=$ $\displaystyle-
b_{-n},\,\,\,b_{n\in\mathbb{Z}}\in\mathbb{R},$ $\displaystyle
R_{even}\,:\,(b)_{n\in\mathbb{Z}}\mapsto(b)_{n\in\mathbb{Z}},\,\,\,\left(R_{even}\cdot
b\right)_{n}$ $\displaystyle=$ $\displaystyle-
b_{-(n+1)},\,\,\,b_{n\in\mathbb{Z}}\in\mathbb{R}.$

A periodic solution $q(t)=q(t+T)$ has even-parity symmetry if

$R_{even}\cdot q(t)=q(t),\,\,\,\forall t\in\mathbb{R},$ (2.5)

and possesses odd-parity symmetry if

$R_{odd}\cdot q(t+T/2)=q(t),\,\,\,\forall t\in\mathbb{R}.$ (2.6)

The even-parity symmetric solution is characterised by amplitude profiles
centered between sites $n=-1$ and $n=0$ while the odd-parity symmetry solution
belongs to amplitude profiles centered at site $n=0$.

The existence theorem will be proved under the following assumption:

  * •

A: The interaction potential $V(x)$ has a global minimum at $x=0$ and is of
such form that

$V(x)=\frac{1}{2}x^{2}+W(x),$ (2.7)

with $W:\,\,{\mathbb{R}}\,\rightarrow\,{\mathbb{R}}$ at least twice
continuously differentiable and $W(0)=W^{\prime}(0)=0$. $W(x)$ is of the form
that $V^{\prime\prime}(0)=a>0$ holds. Further, we assume that $W$ satisfies
for some constant $\overline{K}>0$ and $\alpha\geq 0$, such that

$|W^{\prime\prime}(x)|\leq\overline{K}|x|^{\alpha},\;\;\forall
x\in\mathbb{R}.$ (2.8)

Examples of interaction potentials which satisfy condition $\mathbf{A}$ are
given by the $\alpha$-FPUT and $\beta$-FPUT system with $W(x)=\alpha x^{3}/3$
and $W(x)=\beta x^{4}/4$ as well as combinations of both Wattis ,Arioli .

The solutions of the system obtained when linearising equations (2.1) around
the equilibrium $q_{n}=0$, are superpositions of plane wave solutions
(phonons)

$q_{n}(t)=\exp(i(kn-\omega t)),$

with frequencies

$\omega^{2}(k)=4\sin^{2}\left(\frac{k}{2}\right),\,\,\,k\in[-\pi,\pi].$

These (extended) states disperse. Therefore, the frequency $\Omega$ of a
localised time-periodic solution must satisfy the non-resonance condition
$\Omega\neq|\omega(k)|/m$ for any integer $m\geq 1$. This requires
$\Omega^{2}>4$ as a necessary condition for the existence of localised time-
periodic solutions of system (2.1).

For the forthcoming analysis, it is convenient to pass to relative variables
defined by

$\displaystyle x_{n}$ $\displaystyle=$ $\displaystyle q_{n}-q_{n-1},$ (2.9)
$\displaystyle p_{n}$ $\displaystyle=$ $\displaystyle y_{n}-y_{n+1},$ (2.10)

where the original variables satisfy

$\sum_{n\in\mathbb{Z}}|p_{n}|<\infty,\,\,\,\sum_{n\in\mathbb{Z}}|q_{n}-q_{n-1}|<\infty.$
(2.11)

which imply that

$\sum_{n\in\mathbb{Z}}|y_{n}-y_{n+1}|<\infty,\,\,\,\sum_{n\in\mathbb{Z}}|x_{n}|<\infty.$
(2.12)

From (2.3), it follows that $\lim_{n\rightarrow-\infty}y_{n}=0$ so that the
change of variables (2.9)-(2.10) is invertible. That is, for given
$\\{(x_{n},y_{n})\\}_{n\in\mathbb{Z}}$, one gets the relations

$p_{n}=y_{n}-y_{n+1},\,\,\,q_{n}=\sum_{l=-\infty}^{n}x_{l}.$ (2.13)

The transformation (2.9)-(2.10) is symplectic, because

$\sum_{n}dp_{n}\wedge dq_{n}=\sum_{n}\left(dy_{n}\wedge dq_{n-1}+dy_{n}\wedge
dx_{n}-dy_{n+1}\wedge dq_{n}\right)=\sum_{n}dy_{n}\wedge dx_{n},$ (2.14)

and from the Hamiltonian

$H=\sum_{n}\left(\frac{1}{2}(y_{n}-y_{n+1})^{2}+V(x_{n})\right),$ (2.15)

one derives the canonical equations

$\dot{x}_{n}=2y_{n}-y_{n+1}-y_{n-1},\,\,\,\dot{y}_{n}=-V^{\prime}(x_{n}),$
(2.16)

or equivalently

$\ddot{x}_{n}=V^{\prime}(x_{n+1})+V^{\prime}(x_{n-1})-2V^{\prime}(x_{n}).$
(2.17)

In (2.14) the symbol $\wedge$ denotes the standard wedge product. The
relations (2.2)-(2.4), satisfied by breather solutions of the original system
(2.1), when passing to the new variables, read as

$\displaystyle x_{n}(t+T)$ $\displaystyle=$ $\displaystyle
x_{n}(t),\,\,\,y_{n}(t+T)=y_{n}(t),\,\,n\in{\mathbb{Z}}$
$\displaystyle\lim_{n\rightarrow\pm\infty}x_{n}$ $\displaystyle=$
$\displaystyle 0,\,\,\,\lim_{n\rightarrow\pm\infty}y_{n}=0,$
$\displaystyle\int_{0}^{T}x_{n}(t)dt$ $\displaystyle=$ $\displaystyle
0,\,\,\,n\in{\mathbb{Z}}.$

##  III Existence of exponentially localised breathers

In the following we prove the existence of localised periodic solutions of
system (2.1) on the infinite one-dimensional lattice. To this end, some
appropriate function spaces are introduced on which the original problem is
presented as a fixed point problem for a corresponding operator. Utilising
Schauder’s Fixed Point Theorem, we establish the existence of exponentially
localised solutions, extending the approach followed in JFPT , to a suitable
functional spaces- setup.

To establish the required spatial localisation of the solutions, we introduce
suitable weighted function spaces. First we consider the exponentially
weighted Hilbert space of square-summable sequences,
${l}^{2}_{w}(\mathbb{Z})$, defined as

$l^{2}_{w}=\bigg{\\{}u_{n}\in\mathbb{R}\,:\,||u||_{l_{w}^{2}}^{2}:=\sum_{n}\,w_{n}|u_{n}|^{2}\,\bigg{\\}}\,.$
(3.1)

and we use $w_{n\in\mathbb{Z}}=\exp(\lambda|n|)$ and $\lambda\geq 0$ complying
according to (2.5) and (2.9) with the even-parity amplitude profile. Denoting
by $u^{even}$ and $u^{odd}$ the even and odd parity amplitude pattern,
respectively, it can be readily shown that

$||u^{odd}||_{l^{2}_{w}}<||u^{even}||_{l^{2}_{w}}.$ (3.2)

Then, we denote by

${X}_{0}=\bigg{\\{}u\in
L^{2}_{per}([0,T];{l}_{w}^{2})\,:\,\int_{0}^{T}u_{n}(t)dt=0,\,\,\,n\in{\mathbb{Z}}\bigg{\\}},$

the space of $T-$periodic square integrable functions in time with zero time
average, with values in $l^{2}_{w}$. Evidently, $X_{0}$ is a closed convex
subspace of $L^{2}_{per}([0,T];{l}_{w}^{2})$. We also consider the Sobolev
space

$X_{2}=\bigg{\\{}u\in H^{2}_{per}([0,T];{l}_{w}^{2})\bigg{\\}},$

containing $T-$periodic functions of time assuming values in $l_{w}^{2}$
which, together with their weak derivatives up to second order are in $X_{0}$.
The above spaces are endowed with the following norms:

$\displaystyle||u||_{X_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\,||u(t)||^{2}_{{l}_{w}^{2}}dt,$
$\displaystyle||u||_{X_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\,\left(||u(t)||^{2}_{{l}_{w}^{2}}+||D{u}(t)||^{2}_{{l_{w}}^{2}}+||D^{2}{u}(t)||^{2}_{{l^{2}_{w}}}\right)dt.$

For an element $u\in X_{2}$, we may consider the Fourier-series expansion of
$u_{n}(t)$ with respect to time $t$ and space variable $n$, determined by

$\displaystyle u_{n}(t)$ $\displaystyle=$
$\displaystyle\sum_{m\in{\mathbb{Z}}\setminus\\{0\\}}\hat{u}_{n,m}\exp\left(i\Omega
mt\right),\,t\in[0,T],\,\,\,\,\,\,{\hat{u}}_{n,m}=\frac{1}{T}\int_{0}^{T}u_{n}(t)\exp(-i\Omega
mt)dt,$ (3.3) $\displaystyle\hat{u}_{n,m}$ $\displaystyle=$
$\displaystyle\overline{\hat{u}}_{n,-m},$ (3.4)

and

$\displaystyle u_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{u}_{k}(t)\exp(ikn)dk,\,\,\,\,\,\tilde{u}_{k}(t)=\sum_{n}u_{n}(t)\exp(-i\,k\,n),$
(3.5) $\displaystyle\tilde{u}_{k}$ $\displaystyle=$
$\displaystyle\overline{\tilde{u}}_{-k}.$ (3.6)

Then, using (3.3)-(3.6), we may have the representation of $u$, given by

$u_{n}(t)=\sum_{m\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{2\pi}\int_{0}^{2\pi}{\hat{\tilde{u}}}_{k,m}\exp(ikn)dk\exp(i\Omega
mt).$ (3.7)

The following compactness lemma will be particularly useful for our approach.

###### Lemma III.1

$X_{2}$ is compactly embedded in $X_{0}$ (denoted as $X_{2}\Subset X_{0}$).

Proof: The continuous embedding $X_{2}\subseteq X_{0}$ is clear due to the
inequality

$||u||_{X_{0}}\leq||u||_{X_{2}}.$

Now, in order to show the compactness of the embedding, let
$(u_{n})_{n\in\mathbb{N}}\in X_{2}$ be a bounded sequence, i.e.
$||u_{n}||_{X_{2}}\leq M<\infty$ for all $n\in\mathbb{N}$. Then, there exists
a weakly convergent subsequence (not relabeled) $u_{n}\rightharpoonup u\in
X_{2}$. The continuous embedding $X_{2}\subseteq X_{0}$ implies the weak
convergence $u_{n}\rightharpoonup u$ in $X_{0}$. For the strong convergence
$u_{n}\rightarrow u$ in $X_{0}$, considering the difference
$||u_{n}-u||_{X_{0}}^{2}$, we have the estimate:

$\displaystyle||u_{n}-u||_{X_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}||u_{n}(t)-u(t)||_{l^{2}_{w}}^{2}dt$
(3.8) $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{k\in\mathbb{Z}}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt$
$\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt+\frac{1}{T}\int_{0}^{T}\sum_{|k|>K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt$
$\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt+\sum_{|k|>K}w_{k}\sum_{m\neq
0}|\hat{u}_{n,m}^{k}-\hat{u}_{m}^{k}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt+\sum_{|k|>K}w_{k}\sum_{m\neq
0}|\hat{u}_{n,m}^{k}|^{2}+\sum_{|k|>K}w_{k}\sum_{m\neq
0}|\hat{u}_{m}^{k}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt+\sum_{|k|>K}w_{k}|{u}_{n}^{k}|^{2}\sum_{m\neq
0}\frac{1}{m^{4}}+\sum_{|k|>K}w_{k}|{u}^{k}|^{2}\sum_{m\neq 0}\frac{1}{m^{4}}$
$\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt+2\zeta(4)\left(\sum_{|k|>K}w_{k}|{u}_{n}^{k}|^{2}+\sum_{|k|>K}w_{k}|{u}^{k}|^{2}\right),$

where $\zeta(s)=\sum_{m=1}^{\infty}m^{-s}$ is the Riemann zeta function,
$({u}_{n}^{k})_{k\in I}\in l^{2}_{w}$, $({u}^{k})_{k\in I}\in l^{2}_{w}$,
$I=(-K,...,0,...K)$. We also used that since $u\in
H^{2}_{per}[0,T]\Rightarrow|\hat{u}_{m}|\leq C/{|m|^{2}}$. Then for
sufficiently large $K$ we get

$\displaystyle
2\zeta(4)\left(\sum_{|k|>K}w_{k}|{v}_{n}^{k}|^{2}+\sum_{|k|>K}w_{k}|{v}^{k}|^{2}\right)$
$\displaystyle\leq$ $\displaystyle\frac{2\epsilon}{3}.$ (3.9)

Exploiting that weak and strong convergence coincide in finite-dimensional
spaces, by choosing $n$ sufficiently large, we get for the first term on the
right-hand side of (3.8), that

$\frac{1}{T}\int_{0}^{T}\sum_{|k|\leq
K}w_{k}|u_{n}^{k}(t)-u^{k}(t)|^{2}dt\leq\frac{\epsilon}{3}.$ (3.10)

Combining (3.9) and (3.10) we get

$||u_{n}-u||_{X_{0}}^{2}\leq\epsilon,$ (3.11)

and the proof is finished.

$\Box$

Let us also recall the two versions of the Schauder’s fixed point Theorem
Schauder .

###### Theorem III.1

  1. 1.

(First version): Let $G$ be a closed bounded convex subset of a Banach space
$X$. Assume that $f:\,G\mapsto G$ is compact. Then, $f$ has at least one fixed
point in $G$.

  2. 2.

(Second version): Let $G$ be a closed convex subset of a Banach space $X$ and
$f$ a continuous map of $G$ into a compact subset of $G$. Then, $f$ has at
least one fixed point.

With the preparations above, we may proceed to the statement and proof of the
main result.

###### Theorem III.2

(i). Let condition A hold and suppose

$\Omega^{2}>4.$ (3.12)

Then there exist nonzero sequences
$x^{even}\equiv\\{x_{n}^{even}\\}_{n\in{\mathbb{Z}}}\in H^{2}([0,T];l^{2})$
and $x^{odd}\equiv\\{x_{n}^{odd}\\}_{n\in{\mathbb{Z}}}\in H^{2}([0,T];l^{2})$
with zero time average $\int_{0}^{T}x_{n}(t)dt=0$ and $||x||_{X_{0}}\leq R$,
where

$R\leq\left(\frac{\Omega^{2}-4}{\overline{K}\,\sqrt{2(1+\cosh(\lambda))}}\right)^{1/\alpha}:=R_{\mathrm{max}}.$
(3.13)

These solutions are exponentially localised time-periodic solutions of system
(2.17) with period $T=\frac{2\pi}{\Omega}$ possessing even-parity and odd-
parity symmetry, respectively.
(ii). Assume further that

$\Omega^{2}>4+\overline{K}\,\sqrt{2(1+\cosh(\lambda))}.$ (3.14)

Then, the corresponding solutions satisfy the lower and upper bounds in their
$X_{0}$-norm

$R_{\mathrm{crit}}:=\left(\frac{\Omega^{2}-4}{\overline{K}\,\sqrt{2(1+\cosh(\lambda))}}\right)^{\frac{1}{1+\alpha}}\leq
R\leq R_{\mathrm{max}}.$ (3.15)

Proof: For part (i), we prove the existence of solutions when $\Omega^{2}>4$,
which is a natural condition for the existence of localised modes. We will
show that the solutions are satisfying the upper bound (3.13), by implementing
the two versions of the Schauder’s fixed point Theorem III.1 to the suitably
defined map. For part (ii), assuming (3.14), we examine the contraction
mapping regime for this map, identifying a ring in $X_{0}$ of nontrivial
solutions satisfying the lower bound of (3.15).
(i): Existence by the Schauder’s fixed point Theorem of even-parity and odd-
parity solutions when (3.12) holds. First we show the existence of even-parity
symmetry breathing solutions. Afterwards we argue that by virtue of relation
(3.2) the existence proof applies to the odd-parity breathing mode too. For
the ease of notation we denote the solution by $x$ instead of $x^{even}$. We
shall provide two alternatives of the proof by implementing the two versions
of the Schauder’s fixed point Theorem III.1. For this purpose, it is
convenient to rewrite Eq. (2.17) as:

$\ddot{x}_{n}-[x_{n+1}-2x_{n}+x_{n-1}]=W^{\prime}(x_{n+1})+W^{\prime}(x_{n-1})-2W^{\prime}(x_{n}).$
(3.16)

Thus, only the right-hand side of (3.16) features terms nonlinear in $x$. We
shall express (3.16) as a fixed point equation in $x$.
I. $1^{st}$ version of the proof: We relate the left-hand side of (3.16) to
the linear mapping: $M\,:\,X_{2}\,\rightarrow\,X_{0}$:

$M(u_{n})=\ddot{u}_{n}-[u_{n+1}-2u_{n}+u_{n-1}].$

Then, applying the operator $M$ to the Fourier elements
$\exp(ikn)\exp(i\,\Omega mt)$ in the representation (3.7), we get that

$M\exp(ikn)\exp(i\,\Omega mt)=\nu_{m}(k)\exp(ikn)\exp(i\,\Omega mt),$

where

$\nu_{m}(k)=-\Omega^{2}\,m^{2}+4\sin^{2}\left(\frac{k}{2}\right),\,\,m\in\mathbb{Z}\setminus\\{0\\},\,\,k\in[0,2\pi].$

As by assumption (3.14) $\Omega^{2}>4$, it is guaranteed that $\nu_{m}(k)\neq
0$, for all $m\in{\mathbb{Z}}\setminus\\{0\\}$ and for all $k\in[0,2\pi]$, the
mapping $M$ possesses an inverse $M^{-1}$, obeying $M^{-1}\exp(i\,(\Omega
mt+kn))=(1/\nu_{m}(k))\exp(i\,(\Omega mt+kn))$. For the norm of the linear
operator $M^{-1}:\,\,X_{0}\rightarrow X_{2}$ one derives the upper bound:

$\displaystyle||M^{-1}||_{X_{0},X_{2}}^{2}$ $\displaystyle=$
$\displaystyle\sup_{{}_{||u||_{X_{0}}=1}}||M^{-1}\,u||_{X_{2}}^{2}$ (3.17)
$\displaystyle=$
$\displaystyle\sup_{{}_{||u||_{X_{0}}=1}}\frac{1}{T}\int_{0}^{T}\,\left[||M^{-1}u(t)||^{2}_{{l}_{w}^{2}}+||D{M^{-1}u(t)}||^{2}_{{l}_{w}^{2}}+||D^{2}{M^{-1}u(t)}||^{2}_{{l}_{w}^{2}}\right]dt$
$\displaystyle=$
$\displaystyle\sup_{||u||_{X_{0}}=1}\frac{1}{T}\int_{0}^{T}\,\sum_{n\in\mathbb{Z}}w_{n}\left(\left|\sum_{m^{\prime}}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\hat{\tilde{u}}_{k,m}}{\nu_{m}(k)}\exp(ikn)\exp(i\Omega
mt)dk\right|^{2}\right.$ $\displaystyle+$
$\displaystyle\left|\sum_{m^{\prime}}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{i\Omega
m\hat{\tilde{u}}_{k,m}}{\nu_{m}(k)}\exp(ikn)\exp(i\Omega mt)dk\right|^{2}$
$\displaystyle+$
$\displaystyle\left.\left|\sum_{m^{\prime}}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(i\Omega
m)^{2}\hat{\tilde{u}}_{k,m}}{\nu_{m}(k)}\exp(ikn)\exp(i\Omega
mt)dk\right|^{2}\right)dt$ $\displaystyle\leq$
$\displaystyle\sup_{m\in{\mathbb{Z}}\setminus\\{0\\}}\sup_{k\in[0,2\pi]}\frac{1+(\Omega
m)^{2}+(\Omega m)^{4}}{|\nu_{m}(k)|^{2}}$ $\displaystyle\cdot$
$\displaystyle\sup_{||u||_{X_{0}}=1}\frac{1}{T}\int_{0}^{T}\,\sum_{n\in\mathbb{Z}}w_{n}\left|\sum_{m^{\prime}}\frac{1}{2\pi}\int_{0}^{2\pi}\hat{\tilde{u}}_{k,m}\exp(ikn)\exp(i\Omega
mt)dk\right|^{2}dt$ $\displaystyle\leq$
$\displaystyle\frac{1+\Omega^{2}+\Omega^{4}}{(\Omega^{2}-4)^{2}}\sup_{||u||_{X_{0}}=1}||u||_{X_{0}}^{2}=\frac{1+\Omega^{2}+\Omega^{4}}{(\Omega^{2}-4)^{2}}\leq\frac{(1+\Omega^{2})^{2}}{(\Omega^{2}-4)^{2}}<\infty,$

verifying the boundedness of $M^{-1}$ and we used the notation
$\sum_{l^{\prime}}=\sum_{l\in{\mathbb{Z}}\setminus\\{0\\}}$. When we consider
$M^{-1}$ as a map $M^{-1}:\,\,X_{0}\rightarrow X_{0}$, we note that

$||M^{-1}||_{{X}_{0},{X}_{0}}\leq\frac{1}{\Omega^{2}-4}.$ (3.18)

In order to deal with the nonlinear terms, we attribute to the right-hand side
of (3.16) the nonlinear operator $N\,:\,X_{0}\,\rightarrow\,X_{0}$, as

$(N(u))_{n}=W^{\prime}(u_{n+1})+W^{\prime}(u_{n-1})-2W^{\prime}(u_{n}).$

To establish that the operator $N$ is continuous on $X_{0}$, we prove that $N$
is Frechet differentiable at any $u$, with bounded derivative. We observe that

$N^{\prime}(u):\,h\in X_{0}\,\mapsto N^{\prime}(u)[h]=W^{\prime\prime}(u)h\in
X_{0},$

where
$(W^{\prime\prime}(u))_{n}=W^{\prime\prime}(u_{n+1})+W^{\prime\prime}(u_{n-1})-2W^{\prime\prime}(u_{n})$.
Therefore, we derive the estimate

$\displaystyle||N^{\prime}(u)[h]||_{X_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{n\in\mathbb{Z}}{w_{n}}\left|\,\left(W^{\prime\prime}(u_{n+1}(t))+W^{\prime\prime}(u_{n-1}(t))-2W^{\prime\prime}(u_{n}(t))\right)\,h_{n}(t)\right|^{2}\,dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{T}\int_{0}^{T}\sum_{n\in\mathbb{Z}}w_{n}\overline{K}^{2}(|u_{n+1}(t)|^{2\alpha}+|u_{n-1}(t)|^{2\alpha}+2|u_{n}(t)|^{2\alpha})\,|h_{n}(t)|^{2}\,dt$
$\displaystyle\leq$ $\displaystyle
4\overline{K}^{2}\sup_{n\in{\mathbb{Z}}}\max_{t\in[0,T]}|u_{n}(t)|^{2\alpha}\,||h||^{2}_{X_{0}}=A^{2}||h||^{2}_{X_{0}},$

where
$A^{2}=4\overline{K}^{2}\sup_{n\in{\mathbb{Z}}}\max_{t\in[0,T]}|u_{n}(t)|^{2\alpha}$.
That is

$||N^{\prime}(u)||_{{\cal{L}}(X_{0},X_{0})}\leq A,$ (3.19)

implying the (uniform) boundedness of the differential. Let us now use as the
closed convex subset $Y_{0}$ of $X_{0}$, its closed ball centered at $0$ of
radius $R$,

${Y}_{0}=\left\\{u\in X_{0}\;\;:\;\;||u||_{X_{0}}\leq R\right\\}.$

For even-parity solutions we get for the range of $N$ on $Y_{0}$ the bound

$\displaystyle||N(u)||_{X_{0}}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\sum_{n\in\mathbb{Z}}{w_{n}}|W^{\prime}(u_{n+1}(t))+W^{\prime}(u_{n-1}(t))-2W^{\prime}(u_{n}(t))|^{2}\,dt$
(3.20) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\sum_{n\in\mathbb{Z}}{w_{n}}\left(|W^{\prime}(u_{n+1}(t))|^{2}+|W^{\prime}(u_{n-1}(t))|^{2}+2|W^{\prime}(u_{n}(t))|^{2}\right)\,dt$
$\displaystyle\leq$
$\displaystyle\overline{K}^{2}\int_{0}^{T}\sum_{n\in\mathbb{Z}}\left(\frac{w_{n}}{w_{n+1}}w_{n+1}|u_{n+1}(t)|^{2(\alpha+1)}+\frac{w_{n}}{w_{n-1}}w_{n-1}|u_{n-1}(t)|^{2(\alpha+1)}+2w_{n}|u_{n}(t)|^{2(\alpha+1)}\right)\,dt$
$\displaystyle\leq$ $\displaystyle
2\overline{K}^{2}(1+\cosh(\lambda))||u||_{\tilde{X}_{0}}^{2(\alpha+1)}\leq
2\overline{K}^{2}(1+\cosh(\lambda))||u||_{{X}_{0}}^{2(\alpha+1)}$
$\displaystyle\leq$ $\displaystyle
2\overline{K}^{2}(1+\cosh(\lambda))R^{2(\alpha+1)},\qquad\forall u\in Y_{0},$

where $\tilde{X}_{0}=\\{u\in
L_{per}^{2}([0,T];l_{w}^{2(\alpha+1)})\,:\,\int_{0}^{T}u_{n}(t)dt=0,\,\,n\in{\mathbb{Z}}\\}$
and we used $X_{0}\subset\tilde{X}_{0}$. (A similar expression is obtained for
odd-parity solutions.)

In order to apply Theorem III.1, the final step is to express the problem
(3.16) as a fixed point equation in terms of a mapping
$Y_{0}\,\rightarrow\,Y_{0}$:

$x=M^{-1}\circ N(x)\equiv S(x).$ (3.21)

Clearly, $S$ is continuous on $X_{0}$, as the relations (3.17) and (3.19)
establish that its constituents $M^{-1}$ and $N$ are continuous. Next, we show
that $S({Y}_{0})\subseteq{Y}_{0}$. Using (3.17) and (3.20), we have

$\displaystyle||S(x)||_{{X}_{0}}$ $\displaystyle=$
$\displaystyle||M^{-1}(\,N(x))||_{{X^{0}}}\leq||M^{-1}||_{{X}_{0},X_{0}}\cdot||\,N(x)||_{{X}_{0}}$
(3.22) $\displaystyle\leq$
$\displaystyle\frac{\sqrt{2(1+\cosh(\lambda)}\,\,\overline{K}}{\Omega^{2}-4}R^{\alpha+1}\leq
R,\,\,\,\forall x\in Y^{0},$

assuring by assumption (3.15), that indeed

$S({Y}_{0})\subseteq{Y}_{0}.$

As $M^{-1}$ maps $X_{0}$ to $X_{2}\Subset X_{0}$, one has $S(Y_{0})\subseteq
Y_{0}\cap X_{2}$. Therefore the map $S=M^{-1}\circ N$, viewed as a map
$S:Y_{0}\subseteq X_{0}\mapsto Y_{0}\subseteq X_{0}$, is compact. The first
version of Schauder’s fixed point theorem implies then that the fixed point
equation $x=S(x)$ has at least one solution of even-parity symmetry. In
addition, due to the relation (3.2), the above existence proof is valid for
the odd-parity symmetry solution when $\Omega$ is considered as its associated
breathing frequency.
II. $2^{nd}$ version of the proof: For the application of the second version
of Schauder’s fixed point theorem, we verify that the range of $S$ is
contained in a compact subset of $Y_{0}$. Representing $N(u)\in Y_{0}$ in
terms of its spatial and temporal Fourier-transforms as

$(N(u))_{n}(t)=\sum_{m\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{2\pi}\int_{0}^{2\pi}\hat{\tilde{N}}_{k,m}\exp(ikn)dk\exp\left(i\Omega
mt\right),$ (3.23)

the Fourier coefficients of $M^{-1}(N(u))$ fulfill for all $u\in Y_{0}$, for
all $k\in[0,2\pi]$, and for all $m\in{\mathbb{Z}}\setminus\\{0\\}$

$\displaystyle\left|\frac{\hat{{\tilde{N}}}_{k,m}}{-\Omega^{2}\,m^{2}+4\sin^{2}\left(\frac{k}{2}\right)}\right|^{2}\leq\frac{\sum_{m\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{2\pi}\int_{0}^{2\pi}\left|\hat{{\tilde{N}}}_{k,m}\right|^{2}dk}{m^{4}(\Omega^{2}-4)^{2}}$
(3.24) $\displaystyle=$ $\displaystyle\frac{\parallel
N(u)\parallel_{{X}_{0,0}}^{2}}{m^{4}(\Omega^{2}-4)^{2}},$

where $X_{0,0}$ is determined by

${X}_{0,0}=\left\\{u\in
L^{2}_{per}([0,T];{l}^{2})\,|\,\int_{0}^{T}u_{n}(t)dt=0,\,\,\,n\in{\mathbb{Z}}\right\\}.$

Since $l_{w}^{2}\subset l^{2}$ and the inequality
$||u||_{l^{2}}\leq||u||_{l_{w}^{2}}$ holds, we get

$\displaystyle\left|\frac{\hat{{\tilde{N}}}_{k,m}}{-\Omega^{2}\,m^{2}+4\sin^{2}\left(\frac{k}{2}\right)}\right|^{2}\leq\frac{2{\overline{K}}^{2}(1+\cosh(\lambda))R^{2(\alpha+1)}}{m^{4}(\Omega^{2}-4)^{2}}\leq\left(\frac{R}{m^{2}}\right)^{2}.$
(3.25)

Hence, we conclude that $S$ maps $Y_{0}$ into the subset

$Y_{0,c}=\left\\{u=\left\\{u\right\\}_{n\in{\mathbb{Z}}}\in
Y_{0},\,u_{n}(t)=\sum_{m\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{2\pi}\int_{0}^{2\pi}\hat{{\tilde{u}}}_{k,m}\,\exp(ikn)dk\exp\left(i\Omega
mt\right)\,\left|\,|\hat{{\tilde{u}}}_{k,m}|\leq\frac{R}{m^{2}}\right.\right\\},$

which is compact in $Y_{0}$. That is, the operator $S$ maps closed convex
subsets $Y_{0}\subset X_{0}\subset L_{per}^{2}((0,T);l^{2})$ into compact
subsets $Y_{0,c}$ of $Y_{0}$. The second version of Schauder’s fixed point
theorem implies then that the fixed point equation $u=S(u)$ has at least one
solution.

We conclude the proof of part (i), with the following detail: According to the
(invertible) transformation (2.9)-(2.10), breather solutions to system (2.17)
represent breather solutions to the original system (2.1), satisfying the
relations (2.2)-(2.3).

Moreover, by the hypothesis (3.14) the values of the frequency of oscillations
$\Omega$ lie above the upper edge of the continuous (phonon) spectrum
determined by $2$. Thus the corresponding solutions have to be anharmonic
which necessitates nonzero nonlinearity. Only the latter facilitates
amplitude-depending tuning of the frequency of oscillations so that the
breather frequency $\Omega$ lies outside of the phonon spectrum. Thus, it must
hold that $||N(u)||_{X_{0}}\not\equiv 0$, which, by condition A, is satisfied
if and only if $u\not\equiv 0$. That is, the fixed point equation (3.21)
possesses a nonzero solution and the proof of part (i), is finished.

(ii): Identification of a ring in $X_{0}$ for non trivial solutions when
(3.14) holds. Strengthening the condition $\Omega^{2}>4$ to (3.14), we are
able to identify a ring in $X_{0}$ for nontrivial solutions. In particular,
returning to the estimates (3.18), (3.19) and the definition of the map $S$ in
(3.21), we see that from the upper bound (3.22) we may derive a contraction
regime for the map $S$. Actually, it follows that if we require

$\frac{\sqrt{2(1+\cosh(\lambda)}\,\,\overline{K}}{\Omega^{2}-4}R^{\alpha+1}<1,$

we get that when

$\displaystyle
R\leq\left(\frac{\Omega^{2}-4}{\overline{K}\,\sqrt{2(1+\cosh(\lambda))}}\right)^{\frac{1}{1+\alpha}}:=R_{\mathrm{crit}},$
(3.26)

there exists a unique solution, namely the trivial one $u=0$. To be consistent
with the bound (3.15), and in order to avoid the trivial solution, we need to
assume

$\displaystyle\Omega^{2}>4+\overline{K}\,\sqrt{2(1+\cosh(\lambda))},$

that is, the condition (3.14) on the frequencies holds instead of
$\Omega^{2}>4$. Thus, under condition (3.14), it holds that
$R_{\mathrm{crit}}<R_{\mathrm{max}}$ and non-trivial solutions satisfying
(3.14) are located in the ring $\mathbf{R}_{E}$ of $X_{0}$ determined by

$\displaystyle\mathbf{R}_{E}=\left\\{u\in
X_{0}\;\;:\;\;R_{\mathrm{crit}}\leq||u||_{X_{0}}\leq
R_{\mathrm{max}}\right\\},$ (3.27)

that is the estimates (3.15) are satisfied. This concludes the proof of part
(ii), and of the whole theorem. $\Box$

###  III.1 Comments on the estimates in $X_{0}$ and further properties of the
solutions

The lower and upper bounds (3.15) derived via the fixed-point approach might
not be optimal for the estimation of the actual norm of the solutions, as
relevant studies concerning DNLS systems (which in some cases may approximate
lattice systems described by equation (2.1)), showed b3 ; b4 ; b5 . For
example, in the case of the DNLS-type systems, the studies in b3 ; b4 ; b5
demonstrated that the actual minimum of the power of the solution (excitation
threshold), when it exists Wein99 , is above the relevant non-existence lower
bounds similar to $R_{\mathrm{crit}}$ proved herein by contraction mapping-
arguments. However, for certain parametric and frequency regimes, the
aforementioned studies verified that they can be even sharp estimates of the
actual power of the DNLS breathers, particularly in the cases where the
excitation threshold (whose emergence depends on the dimension of the lattice
Wein99 ) is not existing. In what follows, we discuss features of the
estimates, particularly in some limiting cases for their parameters and
identify those regimes requiring a coherent change of the parameters. We also
point to potential “sharpness” properties of the estimates in specific
parametric regimes. We conclude by discussing other properties of solutions
which follow from the existence Theorem III.2.

  1. 1.

Let us fix $\Omega^{2}>4$. Considering the limits as
$\lambda\rightarrow\infty$ for fixed $\overline{K}$, or as
$\overline{K}\rightarrow 0$, for fixed $\lambda$, solely for the upper bound
$R_{\mathrm{max}}$, we observe that

$\displaystyle\lim_{\lambda\rightarrow\infty}R_{\mathrm{max}}$
$\displaystyle=$ $\displaystyle 0,$ (3.28)
$\displaystyle\lim_{\overline{K}\rightarrow 0}R_{\mathrm{max}}$
$\displaystyle=$ $\displaystyle\infty.$ (3.29)

The limit (3.28) seems to be physically rather relevant in a “continuous”
regime for the system, than for a discrete regime: In a continuous regime the
limit (3.28) suggests that the more localised the solution becomes, the less
is its contribution to its continuous $L^{2}$-norms. We recall that

$\displaystyle||u||_{l^{2}}\leq||u||_{l^{2}_{w}},$ (3.30)

for the sequence spaces, which remains valid as we approximate the continuous
case, and is also valid in the continuous limit due to the weight
$w_{x}=\exp[\lambda|x|]\geq 1$. However, in the discrete regime, the limit
(3.28) is counter-intuitive, since the embedding relation

$\displaystyle||u||_{l^{\infty}}\leq||u||_{l^{2}}\leq||u||_{l^{2}_{w}},$
(3.31)

would imply that the $l^{2}$-norm of the solution, and consequently, its
amplitude, vanishes.

On the other hand, the limit (3.29) seems to be relevant in both the discrete
and the continuous regime. Since for $\overline{K}\rightarrow 0$, the system
approximates its linear limit, spatially extended ”almost harmonic” modes
result instead of localised ones, implying the ”unboundness” of the weighted
norms.

Both of the above examples implicate a coherent dependence of $\Omega$ on $R$
and the other parameters. Evidentially the functional dependence of $R$ on all
the parameters in the inequality (3.22) leads to the derivation of the
estimates.

  2. 2.

When $\Omega$ satisfies (3.14) instead of only assuming (3.12), we may deduce
some information concerning the coherent dependence of the parameters
discussed above being physically relevant.

First, in the limit of large $\lambda$, condition (3.14) shows that $\Omega$
becomes arbitrarily large. From the functional expressions of
$R_{\mathrm{max}}$ and $R_{\mathrm{crit}}$ we have that

$\displaystyle\lim_{(\Omega,\lambda)\rightarrow(\infty,\infty)}R_{\mathrm{crit}}=\lim_{(\Omega,\lambda)\rightarrow(\infty,\infty)}R_{\mathrm{max}}=\infty.$
(3.32)

The limit (3.32) seems to be in accordance with the physical intuition that,
at least for hard interaction potentials, in the limit of arbitrary large
frequency, a type of “energy” of the solution, measured herein in the norm of
$X_{0}$, should become also arbitrarily large. That the weighted norm grows
upon increasing the localisation rate seems to be consistent with energy
concentration phenomena due to enhanced localisation, which may lead
accordingly to phenomena of quasi-collapse. Such a phenomenology for the
increase of energy was already observed in the formation of a localised state
in the $\beta$-FPU model His . Note also that in His  an energy-threshold for
the formation of breathers is defined, induced by modulational instability.

  3. 3.

The functional forms of the upper and lower bounds in (3.32), which differ
only on the $\alpha$-dependent exponents, suggest that they may provide sharp
estimates for the actual $X_{0}$-norm of the localised modes in certain
parametric regimes of finite values for the involved parameters. We elaborate
further on this, by discussing the features of the assumption (3.14) for small
values of $\lambda$ as a potential effective condition for existence of
anharmonic oscillations. Assuming (3.12), together with the violation of the
condition (3.14), implies as $\lambda\rightarrow 0$, that

$4<\Omega^{2}\lesssim 4+2\overline{K}$

When $\overline{K}\rightarrow 0$, then $\Omega\rightarrow 2^{+}$. Hence, as
the frequency approaches the upper edge of the phonon band, we do not expect
existence of localised modes. Regarding the limit of small $\lambda$, we
remark the following: localised solutions on the infinite lattice
${\mathbb{Z}}$ are represented by (infinite) square-summable sequences, while
exponential decay of the solutions for $|n|\rightarrow\infty$ takes place in
the sense of the exponentially weighted $l^{2}$ norm. Notably, for weight
function $w_{n}\sim 1$, i.e. $\lambda\rightarrow 0$, our proof establishes the
existence of general localised patterns (e.g. multi-site breathers).

  4. 4.

Since the obtained time-periodic $H^{2}$ fixed-point-solutions are by Sobolev
embeddings $C^{1}$ in time and since the operator $x\mapsto W^{\prime}(x)$
maps $C^{1}$ into itself, one concludes from Eq. (3.16) that $\ddot{x}\in
C^{1}$ are classical solutions.

##  IV Conclusions

We have proved that the Schauder’s Fixed Point Theorem can be used to
establish the existence of non-trivial, exponentially localised breather
solutions of both even-parity and odd-parity symmetry for general systems of
nonlinearly coupled particles with general interaction potentials, on the
infinite lattice ${\mathbb{Z}}$. The existence problem has been reformulated
as a fixed point equation (involving the relevant linear and nonlinear
operators associated to the system), on weighted sequence spaces, to directly
associate the existence problem with a prescribed rate of spatial
localisation, under physically relevant non-resonant conditions, providing in
addition, energy bounds for the solutions. Future extensions may consider
systems with other physically important classes of interaction potentials, and
importantly (as mentioned in the introductory section), multi-dimensional
systems. Another direction may investigate Discrete Nonlinear Schrödinger
models and different degrees of localisation such as algebraic, which can be
relevant for the existence of discrete rational solutions akhm_AL ,akhm_AL2 .
An important task may concern the extension of the method as suitable for the
proof of traveling wave solutions FWattis ; Smets ; APB . Such investigations
are in progress and will be documented elsewhere.

## Acknowledgment

We would like to thank the referee for his/her constructive comments and
suggestions.

Authors Declarations
The authors have no conflicts to disclose.

Data availability
The data that supports the findings of this study are available within the
article.

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