Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach
Eduardo A. Silva
math-ph math.MP 37D35

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[NO \title GIVEN]
    [NO \author GIVEN]
    December 30, 2023
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In this paper we use a recent version of the Ruelle-Perron-Frobenius 
Theorem to compute, in terms of the maximal eigendata of the Ruelle
 operator, 
the pressure derivative of translation invariant spin systems
taking values on a general compact metric space. 
On this setting the absence of metastable states for continuous potentials
on one-dimensional one-sided lattice is proved.  
We apply our results, to show that the pressure 
of an essentially one-dimensional 
Heisenberg-type model, on the lattice ℕ×ℤ, 
is Fréchet differentiable, on a suitable Banach space. Additionally, 
exponential decay of the two-point function, for this model, 
is obtained for any positive temperature.


 Mathematics Subject Classification: 37D35.
Keywords:  Thermodynamic Formalism, pressure derivative, transfer operator.





§ INTRODUCTION

The Ruelle operator  was introduced by David Ruelle in the seminal 
paper  <cit.>, in order to  prove 
the  existence and uniqueness of the Gibbs measures 
for some long-range Statistical Mechanics models in the 
one-dimensional lattice. Ever since  
the Ruelle operator  has become a standard tool 
in a variety  of mathematical fields, for instance, in Dynamical Systems, 
and other branches of Mathematics and Mathematical Physics.
The Ruelle operator was generalized in several 
directions and its generalizations are commonly 
called transfer operators.  Transfer operators 
appear in IFS theory, Harmonic Analysis and 
C^*-algebras, see for instance
 <cit.> respectively.
 



In Dynamical Systems the existence 
of Markov Partitions allows one to conjugate
uniform hyperbolic maps on compact  differentiable manifolds
with the shift map in the   Bernoulli space.
For more details see, for example, <cit.> and references therein.


A field in which the Ruelle operator formalism  has also been proved useful  
is  the Multifractal Analysis. Bowen, in the seminal work
<cit.>, has established a relationship between the  Hausdorff dimension 
of certain fractal sets and  topological pressure, 
for more
details see <cit.> 
and also the introductory texts <cit.>.
 


The classical Thermodynamic Formalism was  originally developed 
in the Bernoulli space M^ℕ, with M being a finite alphabet, see <cit.>. 
The motivation to consider more general alphabets from the 
dynamical system point of view is given
in <cit.>, where proposed models with 
infinite alphabet M=ℕ are used to 
describe some  non-uniformly hyperbolic maps, 
for instance, the Manneville-Pomeau maps.
Unbounded alphabets as general standard Borel spaces, which includes 
compact and non-compact, are considered in details in <cit.>.


In <cit.> the authors considered the alphabet M=S^1 and a Ruelle operator formalism is developed. 
Subsequently, in <cit.>, this formalism was extended to general compact metric alphabets.
Those alphabets do not fit in the classical theory, since the number of preimages 
under the shift map may be not countable. To circumvent this 
problem the authors considered an a priori measure μ
defined on M, and so a generalized Ruelle operator can be defined  
and a version of the Ruelle-Perron-Frobenius Theorem is proved. 
In this general setting concepts of entropy and pressure are also introduced. 
A variational principle is obtained in <cit.>. The authors also show that their 
theory can also be used to recovery some results of Thermodynamic Formalism for countable 
alphabets, by taking the one-point compactification of ℕ, 
and choosing a suitable a priori measure μ. 




In Classical Statistical Mechanics uncountable alphabets 
shows up, for example, in the so-called O(n) models
with n≥ 2. These  are models 
on a d-dimensional lattice for which a vector 
in the (n-1)-dimensional  sphere is assigned   
to every lattice site  and the vectors  at 
adjacent sites interact  ferromagnetically 
via their inner product.
More specifically, let n≥ 1 be an integer  
and let G=(V(G), E(G)) a finite graph.  A configuration 
of the Spin O(n) model on G is an 
assignment σ:V(G)→ S^n-1, 
we denote by Ω:=(S^n-1)^V(G)  the 
space of configurations. At 
inverse temperature β∈ (0, ∞), 
configurations are randomly chosen from 
the probability measure μ_G,n, β 
given by 

    dμ_G,n, β(σ):=1Z_G, n, βexp ( β∑_u,v ∈ E(G)σ_u·σ_v)  dσ,

where σ_u·σ_v denotes the inner product 
in ℝ^n of the assignments σ_u and σ_v, 
Z_G, n, β=∫_Ωexp( 
β∑_u,v ∈ E(G)σ_u·σ_v)  dσ
and dσ is the uniform probability measure 
on Ω.

Special cases of these models have names of their own:
 when n=0 this model is 
   the Self Avoiding Walking (SAW); when n=1, 
this model is  the Ising model,
when n=2,  the model is 
called the XY model; finally  
for n=3 this model is called  the Heisenberg model,
see <cit.> for details.




In <cit.> the authors generalize the previous 
Ruelle-Perron-Frobenius Theorem for a more general class of potentials, 
satisfying what the authors called weak and  strong Walters conditions, 
which in turn are natural generalizations for the classical 
Walters condition.
The regularity properties of the pressure functional are studied 
and its analyticity is proved on the space of the Hölder 
continuous potentials. 
An exponential decay rate for correlations are obtained, in the case where 
the Ruelle operator exhibits the spectral gap property.
An example, derived from the long-range Ising model,
of a potential in the Walters class where 
the associated Ruelle operator has no
spectral gap is  given.


One of the main results of this work provide an 
explicit expression for the derivative 
of the pressure P:C^α(Ω)→ℝ, 
where Ω≡ M^ℕ and M is a general compact metric space.
To be more precise, we show that  

    1
    P^'(f)φ=∫_Ωφ h_f  dν_f,

where h_f and ν_f are eigenfunction 
and  eigenmeasure of the associated Ruelle operator.

The proof follows closely the one given 
in <cit.>, where the expression (<ref>) is obtained 
in the context of finite alphabet.
We also prove the existence of the limit

    P(f)=lim_n→∞1nlogℒ_f^n 1(x)

in the uniform sense, for any continuous potential f. We would
like  to point out  that the existence of this limit in this setting has been  proved in 
<cit.>. 
We give a new proof of this fact here for two reasons: 
first, it is different from the proof presented in <cit.> 
and we believe it is more flexible to be adapted to other contexts;
second, some pieces of it are used to compute the pressure 
derivative. 

In the last section we apply our results in cases where M=(S^2)^ℤ 
endowed with a suitable a priori DLR-Gibbs measure. 
We introduce a Heisenberg-type model on the lattice ℕ×ℤ, 
depending on a real parameter α, and 
use the Ruelle operator to obtain differentiability of the pressure and 
exponential decay rate for the two-point function for any positive temperature. 











§ BASIC DEFINITIONS AND RESULTS


In this section we set up the notation and present 
some preliminaries results. 
Let M=(M,d) be a compact metric space, equipped 
with a Borel probability measure μ:ℬ(M)→ [0,1], 
having the whole space M as its support. 
In this paper, the set of positive integers is denoted by ℕ.  
We shall denote by Ω=M^ℕ 
the set of all sequences x=(x_1,x_2,… ), where 
x_i ∈ M, for all i∈ℕ. 
We denote the left shift mapping by σ:Ω→Ω, 
 which is given 
by σ(x_1,x_2,…)=(x_2,x_3,…).
We consider the metric d_Ω on Ω given by

    d_Ω (x,y)
    =
    ∑_n=1^∞1/2^nd(x_n,y_n).

The metric d_Ω induces the product topology
and therefore it follows from  Tychonoff's theorem that 
(Ω,d_Ω) is a compact metric space.















The space of all  continuous real functions C(Ω, ℝ)
is denoted simply by C(Ω) and will be endowed with the norm
·_0 defined  by
f_0=sup_x∈Ω |f(x)|, which in turn is  a Banach space. 
For any fixed 0< α< 1 we denote by C^α(Ω) 
the space of all α-Hölder continuous functions, 
that is, the set of all functions f:Ω→ℝ satisfying 

    Hol_α(f)
    =
    sup_x,y∈Ω: x≠ y|f(x)-f(y)|d_Ω(x,y)^α
    <+∞.
 

We equip C^α(Ω), 0< α< 1, with the norm 
 given by  
f_α= f_0+ Hol(f).
We recall that 
(C^α(Ω),·_α) is a Banach space for
any 0< α< 1.




Our potentials will be elements of C(Ω)
and, in order to have a well defined Ruelle operator 
when (M,d) is a general compact metric space, we need 
to consider an a priori measure which is simply a 
Borel probability measure μ:ℬ(M)→ℝ,
where ℬ(M) denotes the Borel σ-algebra 
of M. For many of the most popular choices of an uncountable
space M, there is a natural a priori measure μ. 
Throughout this paper the a priori measure μ 
is supposed to have the whole space M as its support. 
The Ruelle operator 
ℒ_f: C(Ω) → C(Ω)
is the mapping sending φ to ℒ_f(φ) 
defined for any x∈Ω by the expression

    ℒ_f(φ)(x)
    =
    ∫_M e^f(ax)φ(ax)dμ(a),

where ax denotes the sequence 
ax=(a, x_1, x_2, …)∈Ω. 





The classical Ruelle operator can be recovered on this  
setting by considering M={0,1,…,n} and 
the  a priori measure  μ as the normalized counting measure. 





Let (M,d) be a compact metric space, μ a 
Borel probability measure on M having full support and 
f a potential in C^α(Ω), 
where 0<α<1. Then
ℒ_f: C^α(Ω) → C^α(Ω)
has a simple positive eigenvalue of maximal modulus λ_f, 
and there are a strictly positive function h_f 
and a Borel probability measure ν_f on Ω such that,


	 

  (i)
ℒ_f h_f=λ_f h_f, ℒ^*_fν_f=λ_fν_f; 


  (ii)  
the remainder of the spectrum of 
ℒ_f: C^α(Ω) → C^α(Ω) is con­tai­ned in 
a disc of radius strictly smaller than λ_f;


  (iii)
for all continuous functions φ∈ C(Ω) we have

    lim_n→∞λ_f^-nℒ^n_fφ-h_f∫φ   dν_f_0
    =
    0.





See <cit.> for the case M=S^1 and <cit.> 
for a general compact metric space.


















Following the references <cit.> we define 
the entropy of a shift invariant measure
and the pressure of the potential f, respectively, as follows

    h(ν)
    =
    inf_f ∈ C^α(Ω){-∫_Ωf dν+ logλ_f} and  
    P(f)
    =
    sup_ν∈ℳ_σ{h(ν)+∫_Ωf  dν},

where ℳ_σ is the set of all shift invariant Borel 
probability measures. 


For each f ∈ C^α(Ω) we have for all 
x∈Ω that 

    P(f)
    =
    lim_n→∞1/nlog[ ℒ_f^n( 1)(x)]
    =
    logλ_f
    =sup_ν∈ℳ_σ{h(ν)+∫_Ωf  dν}.

Moreover the  supremum is attained by m_f=h_f ν_f.  


See <cit.> Corollary 1.




The function defined by C^α(Ω)∋ f↦ P(f)∈ℝ 
is a real analytic function.



See <cit.> for a proof.



Let f∈ C^α(Ω) be a Hölder continuous potential, ν_f
be the measure given by Theorem <ref> and
φ, ψ∈ C^α(Ω). For each n∈ℤ
we define the correlation function

    C_φ,ψ,m_f(n)
    =
    ∫ (φ∘σ^n)ψ  dm_f
    -∫φ dm_f ∫ψ  dm_f.

We have that the above correlation function  has exponential decay. 
More precisely we have the following proposition:

For each n∈ℕ let C(n) denote the correlation function 
defined by (<ref>). Then there exist constants
K>0 and 0<τ<1 such that |C(n)|≤ K τ^n.



The proof when M is finite is given in <cit.>. Due to 
Theorem <ref> this proof can be easily adapted to
the case where M is compact metric space. 
We will include the details here for completeness.
Before prove the above theorem, we present two auxiliary lemmas.

Let f∈ C^α(Ω), and ∂ D the boundary of a disc D with  center in λ_f,
then the spectral projection 

    π_f≡π_ℒ_f=∫_∂ D(λ I-ℒ_f)^-1dλ.
 
 is given by 
π_f(φ) 
= 
(∫φ  dν_f) · h_f.



Let be φ,ψ∈ C^α(Ω) then
ℒ_f^n(φ∘σ^n ·ψ· h_f)
=
φℒ_f^n( ψ h_f).

The proof of both lemmas are 
straightforward computation so they will be omitted.

Proof of  Proposition <ref>.
Since m_f=h_f dν_f it follows from  
the definition of the correlation function that

    |C_φ,ψ,m_f (n)|
       =
    | 
    ∫ (φ∘σ^n)ψ h_f  dν_f
    - 
    ∫φ  h_f  dν_f∫ψ h_f  dν_f|.

Notice that 
(ℒ^*_f)^nν_f
=
λ_f^nν_f 
and therefore the rhs above is equal to 

    |
    ∫λ_f^-nℒ_f^n((φ∘σ^n)ψ h_f)
      dν_f
    -
    ∫φ  h_f  dν_f∫ψ h_f   dν_f|.

By using the Lemma <ref> and performing simple 
algebraic computations we get 

    |C_φ,ψ,m_f (n)|
    ≤(   ∫|φ|  dν_f)
    λ^-n_fℒ_f^n
    (
    	ψ h_f-  h_f∫ψ h_f  dν_f) 
    _0.






By Theorem <ref> we know that the spectrum of 
ℒ_f: C^α(Ω)→ C^α(Ω) 
consists in a simple eigenvalue λ_f>0 and a subset 
of a disc of radius strictly smaller than λ_f.  
Set 
τ=sup{|z|; |z|<1 and z·λ_f∈Spec(ℒ_f)}. 
The existence of the spectral gap guarantees that τ<1. 
Let π_f 
the spectral projection associated to eigenvalue λ_f,  
then, 
the spectral radius of the operator 
ℒ_f(I-π_f) is exactly τ·λ_f. 
Since the commutator [ℒ_f,π_f]=0,
we get ∀ n ∈ℕ that
[ℒ_f(I-π_f)]^n
=
ℒ_f^n(I-π_f).
From the spectral radius formula  
it follows that for each choice of τ>τ 
there is n_0≡ n_0(τ)∈ℕ 
so that for all n≥ n_0 we have
ℒ_f^n(φ-π_fφ)_0
≤λ_f^nτ^n
φ_0, ∀φ∈  C^α(Ω).
Therefore there is a constant 
C( τ)>0 
such that for every n≥ 1

    ℒ_f^n(φ-π_fφ)_0
    ≤ 
    C( τ)  λ_f^n τ^n φ_0
        ∀φ∈  C^α(Ω).

By using the Lemma <ref> 
and the above upper bound 
in the inequality (<ref>)
we obtain 

    |C_φ,ψ,m_f(n)|
       ≤( ∫ |φ|  dν_f)
    C τ^n ψ h_f_0
    
       ≤
    C(τ) 
    h_f_0( ∫ |φ|dν_f)
    ψ_0  τ^n.
    *








§ MAIN RESULTS





Proposition <ref> ensures for any Hölder potential f
that the limit
P(f)
=
lim_n→∞
n^-1log[ ℒ_f^n( 1)(x)]
always exist and is independent of x∈Ω. 
In what follows, we will extend this result for all continous potentials
f∈ C(Ω). This is indeed a surprising result and it does not have a
counter part on one-dimensional two-sided lattices, 
due to the existence of metastable states discovered by Sewell in <cit.>.
Before present the proof of this fact, 
we want to explain what is the mechanism behind the absence of metastable 
states for one-dimensional systems on the lattice ℕ.
The absence of metastable states for continuous potentials and finite state
space M, on one-dimensional one-sided lattices, as far as we known 
was first proved by Ricardo  Mañé in <cit.>, but apparently he did not realize it.
A generalization of this result for M being a compact metric space appears 
in <cit.> and again no mention to metastable states is made in this paper. 
The proof of this result presented in <cit.> is completely different from ours and we
believe that one presented here is more suitable to be adapted to other context. 
An alternative explanation of this fact can be found in <cit.> Remark 2.2,
and following the first author of <cit.> the first person to realize this
fact was Aernout van Enter. 

Let (ℬ,·) and (ℬ,|||·||| ) the classical 
Banach spaces of interactions defined as in <cit.>. 
In case of free boundary conditions, we have for any Φ∈ℬ
and a finite volume Λ_n⊂ℤ that
|P_Λ_n(Φ)-P_Λ_n(Ψ)| ≤ |||Φ-Ψ|||
and therefore the finite volume pressure with free boundary conditions 
is 1-Lipschitz function 
from (ℬ,|||·|||) to ℝ. 
On the lattice ℤ, when boundary conditions are considered 
the best we can prove is 
|τ_nP_Λ_n(Φ)-τ_nP_Λ_n(Ψ)|
≤Φ-Ψ,
for interactions Φ,Ψ∈ℬ.
From this we have that finite volume pressure with boundary conditions 
is 1-Lipschitz function from the smaller Banach space 
(ℬ,·) to ℝ. The last inequality can not be 
improved for general boundary conditions and interactions in the big Banach space
(ℬ,|||·|||), because 
as we will see in the proof of Theorem <ref> 
it would imply the independence of the boundary 
conditions of the infinite volume pressure which is a contradiction with 
Sewell's theorem. The mechanism that prevents existence of metastable 
states for continuous potential on the lattice ℕ is the 
possibility of proving that the analogous of the finite volume pressure
with boundary conditions on the lattice ℕ is indeed 1-Lipschitz
function. To be more precise.



For each continuous potential  f∈ C(Ω) there is a real number P(f) such that 

    lim_n→∞1nlogℒ_f^n 1 -P(f)_0
    =0.





 
It is sufficient to prove that  
Φ_n:C(Ω) → C(Ω), given by 

    Φ_n(f)(x)=1/nlogℒ_f^n 1(x)

converges to a Lipschitz continuous function Φ:C(Ω) → C(Ω) 
in the following sense Φ_n(f)-Φ(f)_0→ 0, when n→∞.
Indeed, by Proposition <ref>, for any fixed 0<α< 1 we have   
Φ(C^α(Ω))⊂⟨ 1⟩,
where ⟨ 1 ⟩ denotes the subspace generated by the constant functions
in C(Ω). Since C^α(Ω) is a dense subset of (C(Ω),·_0)
and Φ is Lipschitz, then 
Φ(C(Ω))=Φ(C^α(Ω))⊂⟨ 1 ⟩.




In order to deduce the convergence of
(Φ_n)_n∈ℕ,  
it is more convenient to identify 
Φ_n:C(Ω)→ C(Ω)
with the function  
Φ_n:C(Ω)×Ω→ℝ,
given by  
Φ_n(f,x)=(1/n)logℒ_f^n 1(x). 



For any fixed x∈Ω, 
follows from the Dominated Convergence Theorem
that the Fréchet derivative of 
Φ_n:C(Ω)×Ω→ℝ, 
evaluated  at f and computed  in φ, is given by  

    ∂/∂ fΦ_n(f,x)·φ
    =
    1/n∫_M^n(S_nφ)( x)exp(S_nf)( x)  dμ()/ℒ_f^n 1(x),

where (S_nφ)(x)≡∑_j=0^n-1φ∘σ^j(x).
Clearly,  
n^-1S_nφ_0≤φ_0,
and therefore we have the following estimate




    |∂/∂ fΦ_n(f,x)·φ|
       =
    |∫_M^n1/n(S_nφ)( x)exp(S_n f)( x)  dμ()/ℒ_f^n 1(x)|
    
       ≤∫_M^n|1/n(S_nφ)( x)exp(S_nf)( x) | dμ()/|∫_M^nexp(S_nf)( x)  dμ()|
       ≤φ_0,

for any f∈ C(Ω) independently of x∈Ω.
Taking the supremum in (<ref>) over x∈Ω we obtain   

    ∂/∂ fΦ_n(f,·)·φ_0
    ≤φ_0,

for all  n∈ℕ. The above inequality 
allows to conclude  that 
∂/∂ fΦ_n(f)≤ 1, 
where · means the operator norm.

Fix f and f̃ in C(Ω) 
and define for each  n∈ℕ the map  
Φ̂_n(t)=Φ_n(α(t),x), 
where
α(t)=t f+(1-t)f̃
with   0≤ t≤ 1. Obviously, Φ̂_n  
is a differentiable map when  seen as 
a map from [0,1] to  ℝ and we have that 
|Φ̂_n(1)-Φ̂_n(0)|=|d/dtΦ̂_n(t̂)(1-0)| 
for some 
t̂∈ (0,1). 
Using the above estimative of the Fréchet derivative norm
we have that 

    |Φ_n(f,x)-Φ_n(f̃,x)|
    =
    |∂/∂ fΦ_n(f,x)(f-f̃)|≤f-f̃_0.

As an outcome for any fixed f∈ C(Ω) the sequence  (Φ_n(f))_n∈ℕ 
is  uniformly equicontinuous.
Moreover, sup_n∈ℕΦ_n(f)_0<∞. 
Indeed, from inequality (<ref>), the triangular inequality 
and existence of the limit  lim_n→∞Φ_n( 1),  
it follows that

    |Φ_n(f)|
     =
     |Φ_n(f)-Φ_n( 1)|+|Φ_n( 1)|
        ≤f- 1_∞ +|Φ_n( 1)|
     
       ≤f- 1_∞ +sup_n∈ℕ|Φ_n( 1)|
     
       ≡ 
     M(f).
 
 
 
 
Now, we are able  to apply the  Arzelà-Ascoli's Theorem 
to obtain a  subsequence
(Φ_n_k(f))_k∈ℕ, 
which  converges to a function Φ(f)∈ C(Ω). 




 
We now show that   Φ_n(f)→Φ(f), when n→∞.
Let ε>0 
and  g∈ C^α(Ω) such that f-g_0<ε.
Choose n_k and n sufficiently large so that  
the inequalities  
Φ_n_k(f)- Φ(f)_0<ε,
Φ_n(g)-Φ_n_k(g)_0<ε
and
Φ_n_k(g)-Φ_n_k(f)_0<ε
are satisfied.
For these choices of  g,n and  n_k, we have by the triangular inequality
and inequality (<ref>) that

    Φ_n(f)- Φ(f)_0   ≤Φ_n(f)-Φ_n_k(f)_0
    +
    Φ_n_k(f)- Φ(f)_0
       <
    Φ_n(f)-Φ_n_k(f)_0
    +
    ε
       ≤Φ_n(f)-Φ_n(g)_0
    +
    Φ_n(g)-Φ_n_k(f)_0
    +
    ε
       < 
    Φ_n(g)-Φ_n_k(f)_0
    +
    2ε
       ≤Φ_n(g)-Φ_n_k(g)_0
    +
    Φ_n_k(g)-Φ_n_k(f)_0
    +
    2ε
       <
    4ε,

thus proving the desired convergence. 

To finish the proof it is enough to observe 
that the inequality <ref>
implies that Φ is a Lipschitz continuous function.




For each fixed  0<α<1 and f∈ C^α(Ω) 
the  Fréchet derivative of the pressure  functional 
P:C^α(Ω)→ℝ
is given by

    P'(f)φ=∫φ h_f  dν_f





For each  fixed f ∈  C^α(Ω), 0<α< 1, there exists 
the limit

    lim_n→∞1/nℒ_f^n(S_nφ)/ℒ^n_f 1
    -
    ∫φ h_f  dν_f_0
    =
    0

for every  φ∈ C(Ω), and  the convergence is uniform.




 
A straightforward calculation shows that   

    1/nℒ_f^n(S_nφ)/ℒ^n_f 1
    -
    ∫φ h_f  dν_f_0   =
    λ_f^n/ℒ^n_f 11/nλ_f^-nℒ_f^n(S_nφ)
    -
    ∫φ h_f  dν_f_0
       ≤sup_n∈ℕλ_f^n/ℒ^n_f 1_01/nλ_f^-nℒ_f^n(S_nφ)-
    (λ_f^-nℒ^n_f 1)∫φ h_f  dν_f_0.

Therefore to get the desired result it is 
sufficient to show that:


  (a) 
sup_nλ_f^n/ℒ^n_f 1_0 is finite;


  (b)
(λ_f^-nℒ^n_f 1)∫φ h_f  dν_f
converges to  
h_f∫φ h_f  dν_f;


  (c) 
1/nλ_f^-nℒ_f^n(S_nφ)
converges to
h_f∫φ h_f  dν_f.




The first two items 
are immediate consequences of the Ruelle-Perron-Frobenius Theorem. Indeed,  the 
convergence 

λ_f^-nℒ_f^n  1·_0⟶h_f,

 immediately  give that


    (λ_f^-nℒ^n_f 1)
    ∫φ h_f  dν_f·_0⟶  
    h_f∫φ h_f  dν_f.



Since  h_f is a continuous strictly positive function, 
it follows from compactness of Ω that  h_f is bounded away  from  
zero, 
and consequently
 λ_f^n/ℒ_f^n 1·_0⟶ 1/h_f.

Once  h_f  is strictly positive 1/h_f is also positive and 
bounded away from zero, which gives  that  



    sup_nλ_f^n/ℒ^n_f 1_0<∞.



The third expression  in  (c)  is  harder to analyze
than the previous  two, so  we will split  the analysis 
in three claims.



Claim 1. 
For all φ∈ C(Ω) and   n∈ℕ we have 
that,

    λ_f^-nℒ_f^n(S_nφ)
    =
    ∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j
    (φλ_f^-jℒ^j_f 1).

 





We first observe that,
ℒ_f^n(φ∘σ^n)
=
φℒ^n_f 1, 
which  is an easy consequence of the definition of the  Ruelle operator.  
From that and  the linearity of the Ruelle operator it follows,

    ℒ_f^n(S_nφ)
       =
    ℒ_f^n(φ)+ℒ_f^n-1(ℒ_f(φ∘σ))+
    …+ℒ_f(ℒ_f^n-1φ∘σ^n-1)
    
       
    =
    ℒ_f^n(φ)+ℒ_f^n-1(φℒ_f 1)+
    …+ℒ_f(φℒ^n-1_f 1)
    =
    ∑_j=0^n-1ℒ_f^n-j(φℒ^j_f 1)


















finishing the proof.  
From  the linearity of the Ruelle operator 
we easily get  

    λ_f^-nℒ_f^n(S_nφ)
    =
    ∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j
    (φλ_f^-jℒ^j_f 1).

 
Claim  2. For each  φ∈ C(Ω) 
we have 

    lim_n→∞1/n(
    λ_f^-nℒ_f^n(S_nφ)-∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-jφ h_f)
    _0
    =
    0.

To verify (<ref>)  we use  (<ref>)  to obtain the following estimate,

    1/n{λ_f^-nℒ_f^n(S_nφ)
    		-
    		∑_j=0^n-1λ_f^-(n-j)ℒ_f^(n-j)φ h_f
    	}_0
    
    
    
    
    
    
    
    
    
    
    
       =
    1/n∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j(φλ_f^-jℒ^j_f 1-φ h_f)_0
    
    
    
    
    
    
    
    
       ≤const./n ∑_j=0^n-1φλ_f^-jℒ^j_f 1-φ h_f_0.

The last term in the above inequality converges 
to zero, when n→∞, because it  is a Cesàro summation associated to 
the sequence  φλ_f^-jℒ^j_f 1-φ h_f,
which converges to  zero in the uniform norm by    the 
Ruelle-Perron-Frobenius Theorem, so the claim is proved. 



Claim 3. For each  φ∈ C(Ω) there exists  
the following limit

    lim_n→∞1/n∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-jφ h_f-
    h_f∫φ h_fdν_f_0=0.

Define
A_n,j:=λ_f^-(n-j)ℒ_f^n-jφ h_f 
and 
B:=h_f∫φ h_f  dν_f. 
In one hand,
 by  the  Ruelle-Perron-Frobenius Theorem,
 we must have for any
fixed  
j∈ℕ that lim_n→∞A_n,j-B_0=0.
On the other hand, we have by the triangular inequality 
and the convergence in the Cesàro sense that 

    1n∑_j=0^n-1A_n,j-B_0≤1n∑_j=0^n-1A_n,j-B_0⟶ 0,

when  n→∞,   finishing the proof of  Claim 3.
Therefore the proof of the Lemma is established.

 
  
Proof of Theorem <ref> 

 Fix x∈Ω and define  function
 Φ_n:C^α(Ω)→ℝ 
 as 

    Φ_n(f)≡1/nlog(ℒ_f^n 1)(x).


As we have seen in the proof of Theorem <ref>  
the  Fréchet derivative
of  Φ_n at f evaluated in  φ∈ C^α(Ω) is 
given  by  

    Φ_n'(f)φ
    =
    1/nℒ_f^n(S_nφ)/ℒ^n_f 1,

Since  we have  the analyticity of the pressure
functional in  C^α(Ω) (Theorem <ref>) it follows from 
the previous  Lemma that 

    P'(f)φ
    =
    lim_n→∞Φ_n'(f)φ
    =
    ∫φ h_f  dν_f.







§ A HEISENBERG TYPE MODEL 

The aim of this section is to  introduce a Heisenberg type model on
the half-space ℕ×ℤ, 
prove absence of phase transition and exponential decay of correlations 
for this model. 


The construction of this model is split in two steps. 
Firts step. We consider  (S^2)^ℤ as the configuration space. At 
inverse temperature β∈ (0, ∞), 
the configurations  are randomly chosen according 
to the following probabilities 
  measures μ_n, β 

    dμ_n, β(σ):=1Z_n, βexp (
    β∑_i,j∈Λ_nσ_i·σ_j
    )  dσ,

where Λ_n denotes the symmetric interval of 
integers [-n,n], σ_i·σ_j denotes the inner product 
 in ℝ^3 of  the first neighbors σ_i and σ_j, 

    Z_n, β=∫_(S^2)^Λ_nexp
    (
    β∑_i,jσ_i·σ_j 
    )  dσ

and dσ is the uniform probability measure 
on (S^2)^Λ_n.  Let measure ν̂ be 
the unique accumulation point of the sequence of probability measures
given by (<ref>), see <cit.> for details.
The measure ν̂ will be used as the a priori measure 
in the second step.



Second step. Now we introduce a Heisenberg type model. 
We begin with the compact metric space (S^2)^ℤ, where S^2 is the 
2-dimensional unit sphere in  ℝ^3, as our alphabet. 
Now the configuration space is the Cartesian product 
Ω =((S^2)^ℤ)^ℕ, 
that is, a configuration is a point 
σ=(σ(1),σ(2), ⋯)∈Ω,
where each σ(i) is of the form 
σ(i)=(…,σ_(i,-2),σ_(i,-1),σ_(i,0), σ_(i,1), …), 
and each σ_(i,j)∈ S^2. We denote by · the ℝ^3
Euclidean norm, and v· w the inner product of two elements of ℝ^3.


Fix a summable ferromagnetic translation invariant 
 interaction  J on ℤ, that is,  
 a function J:ℤ→ (0,∞) and assume that 
J(n)= e^-|n|α, for some α>0. Of course, we have  
∑_n∈ℤ J(n)<∞.
Now we consider the potential f:Ω→ℝ given by 

    f(σ)= ∑_ n∈ℤ J(n) σ_(1,n)·σ_(2,n).

Note that this potential has only first nearest neighbors interactions.





The  potential f given by (<ref>) is actually 
an α-Hölder continuous function. 
Indeed, 

    |f(σ)-f(ω)|
       ≤ |∑_n∈ℤJ(n)σ_(1,n)· (σ_(2,n)-ω_(2,n))|
        +
    |∑_n∈ℤJ(n)ω_(2,n)· (σ_(1,n)-ω_(1,n))|
    
       ≤∑_n∈ℤJ(n)σ_(2,n)-ω_(2,n)+ 
    ∑_n∈ℤJ(n)σ_(1,n)-ω_(1,n)
 

From the very definition of the distance we have 


    d(σ,ω)
    ≥12^n∑_j∈ℤ12^|j|σ_(n,j)-ω_(n,j)≥12^n+|j|σ_(n,j)-ω_(n,j).

By using (<ref>) and (<ref>) we get that 

    |f(σ)-f(ω)|d(σ,ω)^α≤  
    K_1∑_n∈ℕJ(n)2^α n
    +
    K_2∑_n∈ℕJ(n)2^α n.

Since for all n∈ℕ we have 
J(n)2^α n≤exp(-n(α(1-log 2))), 
and the constant α (1-log 2) is  
positive, follows that the series ∑_n∈ℕJ(n)2^α n is  convergent.
Therefore, f is an α-Hölder continuous function.

From Theorem <ref>, we have that there is a 
unique probability measure ν_f so that 
ℒ_f^*ν_f=λ_fν_f. This probability 
measure, following <cit.>, is a unique DLR-Gibbs measure associated 
to a quasilocal specification associated to f, see <cit.> for the 
construction of this specification. 

By observing that the horizontal interactions, in our model, goes fast to zero in the y-direction 
is naturally to expect that the model is essentially a one-dimensional model.
This feature allow us to obtain the following result 



Let β f be a potential, where f is given by 
(<ref>) and the inverse temperature β∈ (0,∞).
Consider the a priori measure ν̂ on (S^2)^ℤ, 
constructed from (<ref>).
Then for any fixed β>0 and m∈ℤ 
there are positive constants 
K(β) and c(β) such that for all n∈ℕ we have

    ∫_Ω (σ_(1,m)·σ_(n+1,m)) dν_β f≤ 
    K(β)e^-c(β)n.

Furthermore, the pressure functional is differentiable at β f and 
its derivative is given by expression (<ref>).




Fix m∈ℤ and let 
σ_(n,m)≡(σ_(n,m)^x,σ_(n,m)^y, σ_(n,m)^z).
Consider the following continuous potentials φ^u, ξ^u given by

    φ^u(σ) = σ_(1,m)^u    and    ξ^u(σ) = σ_(1,m)^u/h_β f(σ),    u=x,y,z.

Note that 

    ∫_Ωξ  dm_β f
    =
    ∫_Ωσ_(1,m)^u  dν_β f(σ) 
    =
    0,   u=x,y,z,

where the last equality comes from the O(3)-invariance of 
the eingemeasure. 

Since h_β f(σ)=h_β f(-σ), see <cit.>, it follows again from the 
O(3)-invariance of ν_f that 

    ∫_Ωσ_(1,m)^u  dm_β f 
    = 
    ∫_Ωσ_(1,m)^u h_f(σ)   dν_β f 
    =
    0,    u=x,y,z.

Therefore 

    ∫_Ω (σ_(1,m)^uσ_(n+1,m)^u) dν_β f   =
    ∫ (φ^u∘σ^n)ξ^u   dm_β f
       =
    C_φ^u,ξ^u,m_β f(n)
    =
    O(e^-c(β)n),

u=x,y,z.
By summing (<ref>) with u=x,y,z we get the claimed exponential decay.


The last statement follows from the fact that (S^2)^ℤ is a compact metric space, 
ν̂ is a full support a priori measure in
(S^2)^ℤ and β f is an α-Hölder.
Therefore the Proposition <ref> applies and 
corollary follows.





§ ACKNOWLEDGMENTS

The author would like to thank  Leandro Cioletti, Artur Lopes
and Andréia Avelar for fruitfull discussions and comments. 




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