Piecewise Analytic Subactions for Analytic Dynamics

0.44 MB·English

by  Gonzalo Contreras

PIECEWISE ANALYTIC SUBACTIONS FOR ANALYTIC DYNAMICS
G.CONTRERAS(*),A.O.LOPES (**),E.R.OLIVEIRA(**)ANDD.SMANIA (***)
0
1 Abstract. Weconsiderapiecewiseanalyticexpandingmapf :[0,1]→[0,1]ofdegreedwhichpreserves
0 orientation,andananalyticpositivepotentialg:[0,1]→R.
2 Weaddresstheanalysisofthefollowingproblem: foragivenanalyticpotentialβ logg,whereβ isa
n realconstant,itiswellknownthatthereexistsarealanalyticeigenfunctionφβ fortheRuelleoperator.
a One can ask: what happen with the function φβ, when β goes to infinity? The domain of analyticity
J can change with β. The correct question should be: is β1 logφβ analytic in the limit, when β → ∞?
4 Underauniquenessassumption,thislimit,whenβ→∞,isinfactacalibratedsubactionV.
Denote

] m(logg) = max logg(x)dν(x),
S ν aninvariantprobabilityforfZ
D and µ∞, any probability which attains the maximum value. Any one of these probabilities µ∞ is
called a maximizing probability for logg. The probability µ∞ is the limit of the Gibbs states µβ, for
h. the potentials βlogg. In this sense one case say that µ∞ corresponds to the Statistical Mechanics at
t temperaturezero.
a Weshowthatwhenµ∞ isunique,hassupportinaperiodicorbit,theanalyticfunctiong isgeneric
m and satisfies thetwistcondition, then thecalibrated sub-action V :[0,1]→R for thepotential logg is
[ piecewiseanalytic. Weassumethetwistconditiononlyinsomeoftheproofswepresenthere.
An interesting case where the theory can be applied is when logg(x)=−logf′(x). In this case we

3 relatetheinvolutionkerneltothesocalledscalingfunction.
v
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9
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Date: January 4, 2010-(*) CIMAT, Guanajuato, Mexico (**) Instituto de Matem´atica, UFRGS, 91509-900 - Porto
Alegre, Brasil. Partially supported by CNPq, PRONEX – Sistemas Dinaˆmicos, Instituto do Milˆenio, and beneficiary of
CAPES financial support, (***) Departamento de Matem´atica, ICMC-USP 13560-970 S˜ao Carlos, Brasil. Partially
supportedbyCNPq310964/2006-7,andFAPESP2008/02841-4.
1
0. INTRODUCTION
We consider a piecewise analytic expanding map f : [0,1] → [0,1] of degree d which preserves orien-
tation and an analytic positive potential g :[0,1]→R. Here analytic means to have a complex analytic
extension to a neighborhood of [0,1] in the complex plane.
We denote
m(logg) = max logg(x)dν(x),
ν aninvariantprobabilityforf
Z
and µ (or, µ ) any probability which attains the maximum value. Any one of these probabilities
∞ ∞logg
µ is called a maximizing probability for logg.
∞
In general these probabilities are not positive in open sets [9].
We refer the reader to [20] [10] [19] [28] [14] [22] [6] [7] and [9] for general references and definitions on
Ergodic Optimization.
We show the existence of W(w,x), an analytic involution kernel for A(x) = logg(x), and a relation
with the dual potential A∗(w) = (logg)∗(w) defined in the Bernoulli space {1,...,d}N. In this case
W :{1,...,d}N×[0,1]→R and by analytic we mean: for each w ∈{1,...,d}N fixed, the function W(w,.)
hasacomplexanalyticextensiontoaneighborhoodof[0,1]. Wereferthereaderto[1][24]fordefinitions
and properties related to the involution kernel.
An interesting case where the theory can be applied is when logg(x) = −logf′(x). In this case we
relate the involution kernel to the so called scaling function [18] [26].
By definition, a calibrated subaction for logg is a function V such that
sup {V(y) + logg(y) − m(logg)} = V(x).
ysuchthatf(y)=x
If the maximizing probability is unique the calibrated subaction is unique, up to an additive constant
[3] (Lemme C) or [1] (Proposition 5).
If we denote φ (with the normalization φ dν = 1, where ν is the eigen-probability, associated
β β β β
to the same eigenvalue as φ , of the dual of the Ruelle operator associated to βA = βlogg) the main
β
R
eigenfunction of the Ruelle operator for the potential βlogg, it is known (theorem 29 in [9]) that for any
convergent(uniformconvergence)subsequenceof 1 logφ wehavethatthelimitisacalibratedsubaction.
β β
Under our hypothesis of uniqueness we have that the sequence 1 logφ , when β → ∞, converges to a
β β
calibrated subaction.
Notethat,inthegeneralcase,notalwaystheGibbsinvariantprobabilityµ ,forβlog(g),willconverge
β
to a unique (maximizing) probability when β → ∞ [7]. This happens if the maximizing probability is
unique.
Using the above results we show that under some conditions, when µ is unique, has support in a
∞
periodic orbit and logg is twist, then there exists a piecewise analytic calibrated sub-action (denoted by
V :[0,1]→R) for the potential logg. Our main result is Theorem 6.2. We will also show that the above
conditions are generic on the analytic potential g. Explicit examples of piecewise analytic subactions
(which are not analytic) for analytic potentials are presented in [4] and [24].
The ergodic optimization setting has a main difference to the twist maps theory [13] or to the La-
grangian Aubry-Mather problem [27] [8] [12] [25]: the dynamics of the shift (or the transformation f) is
notdefined(viaacriticalpathproblem)fromthepotentialAtobemaximized. Sometimestheanalogous
statements in each theory have to be proved under different techniques. A basic tool in Aubry-Mather
theoryistheassumption: theLagrangianisconvexonthevelocityv. WithoutthishypothesistheMather
graph theorem (see [8]) is not true, etc.... In Ergodic Optimization a natural assumption, which in some
sense play the role of convexity, is the twist condition on the involution kernel (it is a condition that
depends just on A). Here we will assume this hypothesis which was first considered in this context in
[23] and [24]. Examples of potentials A such that the corresponding involution kernel satisfies the twist
conditionappearthere. Thetwistconditionisanopenpropertyinthevariationoftheanalyticpotential
A = logg defined in a fixed open complex neighborhood of the interval [0,1]. We assume f preserves
orientationinorderitcanexistpotentialsA=logg whichsatisfythetwistcondition(see[24]). Wepoint
out that we do not need this hypothesis for the results of all sections from 2 to 5. In the case f reverse
orientation, a similar reasoning can be applied, but we have to consider another dynamics (not the shift)
onthedualspaceΣ. Theproofrequiresalotofdifferenttechnicalitiesandweavoidtoconsiderthiscase
(we address the question to [24]).
Themainideaoftheproofinashortisthefollowing: denotebyI∗ thedeviationfunctionofthefamily
of Gibbs probabilities µβA∗ which converges to µ∞A∗ (the maximizing probability for A∗, see [1]), and
V∗ thecalibratedsubactionforA∗. Fromourassumptionsthemaximizingprobabilityµ∞A∗ issupported
in a periodic orbit where I∗ is zero. Note, however, that I∗ could be also zero outside this support.
We show in section 5 and 6 that
V(x)= sup[W(w,x)−V∗(w)−I∗(w)].
w∈Σ
Therefore, for each x there exists w such that
x
V(x)= sup[W(w,x)−V∗(w)−I∗(w)]=
w∈Σ
V(x)=[W(w ,x)−V∗(w )−I∗(w )].
x x x
For each fixed w we proved in sections 4 and 5 that W(w,x) is analytic in x.
As for a fixed w , W(w ,x) is analytic on x (see corollary 5.2), a result on piecewise analyticity of V
x x
can be obtained if we are able to assume conditions to assure that w is a locally constant as a function
x
of x.
We need two nice conditions for getting that: 1) the set of w such that I∗(w) < K, for any given
constant K is a finite set.
If this is true, the possible number of w is finite.
x
This is so because W(w,x) is bounded in Σˆ and V∗(w) is bounded in Σ. Then points w with a large
value of I can not be a maximizing w (otherwise, take w in the periodic maximizing probability where
x x
I is zero). Note that I is a non-negative lower semi-continuous function (that can attain the value ∞).
Condition 1) above will follow if I∗ is strictly positive on the pre-images of the support of µA∗ (here
we have to use a generic condition on A, not on A∗, which is much more easier).
To show this statement is the purpose of section 7 and 8 (see theorem 8.1).
2) For the lexicographic order in Σ the function x → w is monotonous (here we have to use a twist
x
condition for W, which is a condition to be checked just on A). This is proved in section 6.
Therefore, values w can not change up and down when we move x in a monotonous way. From the
x
above we get that w is locally constant as a function of x and so V is piecewise analytic.
x
The proof above is very indirect. Some problems in mathematics are of such kind: you have a primal
problem that you are not able to solve, then you take the dual problem, and somehow, you solve the
initial problem.
In section 1 we present basic definitions and in section 2 we show the existence of a certain function
h (x) = h(w,x) which defines by means of log(h(w,x)) an involution kernel for logg. In section 3 we
w
present some basic results in Ergodic Optimization and we describe the main strategy for getting the
piecewise analytic sub-action V. Section 4 shows the relation of the scaling function (see [30] [18]) with
theinvolutionkernelandthepotentiallogg =−logf′. Infact,weconsiderinthissectionamoregeneral
setting considering any given potential logg. In section 5 (and also 3) we consider Gibbs states for the
potential βlogg, where β is a real parameter. In section 6 we show, under a natural, but technical
condition, the existence of the piecewise complex analytic calibrated sub-action, when µ is unique, has
∞
support in a periodic orbit and A = logg satisfies a twist condition. We also show that our technical
condition is true for a generic g. In sections 7 and 8 a more general setting for generic properties
of potentials is considered. The results about a generic g, which were used before, are obtained as a
particular application. Finally, in section 9 we present a result of independent interest for the case where
the maximizing probability is not a periodic orbit: we consider properties of the involution kernel for a
generic x.
After this paper was written we discovered that some of the ideas described in section 2 appeared in
some form in [29] [18] (but, as far as we can see, not exactly like here).
We would like to thanks R. Leplaideur for a nice example which is described in section 6.
1. Onto analytic expanding maps
Denote I =[0,1]. We say that f: I →I is an onto map if there exists a finite partition of I by closed
intervals
(1) {I } ,
i i∈{1,2,..,d}
with pairwise disjoint interiors, such that
- For each i we have that f(I )=I,
i
- f is monotone on each I .
i i
We say that f is expanding if f is C1 on each I and there exists λ˜ >1 such that
i
inf inf |Df(x)|≥λ˜.
i x∈Ii
Denote by
ψ : I →I
i i
the inverse branch of f satisfying
ψ ◦f(x)=x
i
for each x∈I .
i
We will say that an expanding onto map is analytic if there exists an simply connected, precompact
open set O ⊂C, with I ⊂O, such that, each ψ has a univalent extension
i
ψ : O →ψ (O).
i i
Since f is expanding, we can choose O such that
- ψ has a continuous extension
i
ψ : O →C.
i
- We have
ψ (O)⊂O.
i
- Moreover
λ˜−1+1
supsup|Dψ (x)|≤λ= <1.
i
2
i x∈O
Consider a finite word
γ =(i ,i ,...,i ),
1 2 k
where i ∈{1,2,..,d}. Denote |γ|=k. Define the univalent maps
j
ψ : O →C
γ
as
ψ =ψ ◦ψ ◦···◦ψ ,
γ ik ik−1 i1
We will denote
I :=ψ (I).
γ γ
Given either an infinite word
N
ω =(i ,i ,...,i ,...)∈Σ:={1,2,..,d} ,
1 2 k
or a finite word with |ω|≥k, define its k-truncation as
ω =(i ,i ,...,i ).
k 1 2 k
Note that for k ≥1
ψ =ψ ◦ψ .
ωk ik ωk−1
For every finite word γ we can define the cylinder
N
(2) C ={ω ∈{1,2,..,d} : ω =γ}.
γ |γ|
2. Analytic potentials, spectral projections and invariant densities
Some of the results presented in this section extend some of the ones in [26]. We say that a function
g: ∪ int I →R
i i
is a complex analytic potential if there are complex analytic functions g : ψ (O)→C such that
i i
- The functions g and g coincides in the interior of I .
i i
- The functions g have a continuous extension to ψ (O).
i i
- There exists θ <1 such that
0< inf |g (x)|≤ sup |g (x)|≤θ.
i i
x∈ψi(O) x∈ψi(O)
- We have
g (R∩ψ (O))⊂R+.
i i
Denote
h˜ (x)=g (ψ (x)).
i i i
For every finite word γ we will define by induction on the lenghts of the words the function
h˜ : O →C
γ
in the following way: Let γ =(i ,i ,...,i ). If |γ|=k+1=1 define h˜ (x)=g (ψ (x)), otherwise
1 2 k+1 γ i1 i1
h˜ (x)=h˜ (x)·g ◦ψ (x)=h˜ (x)·h˜ ◦ψ (x).
γ γk ik+1 γk+1 γk ik+1 γk
Define the Perron-Frobenious operator
P : C(I)→C(I).
logg
as
(P q)(x)= h˜ (x) q(ψ (x)).
logg i i
i
X
Note that
(Pn q)(x)= h˜ (x) q(ψ (x)).
logg γ γ
|γX|=n
It is well known that there exists a probability µ˜, with no atoms and whose support is I, a Holder-
continuous and positive function v and α>0 such that
(3) Pn v =αnv, µ˜(v)=1,
logg
and
µ˜(Pn q)=αn µ˜(q)
logg
for every q ∈ C(I). Let vµ˜ be the measure absolutely continuous with respect to µ˜ and whose Radon-
Nikodyn derivative with respect to µ˜ is v, that is, for every borelian A we have
vµ˜(A)= v(x) dµ˜(x).
ZA
Then the probability vµ˜ is f-invariant. Let ω be either an infinite word ω =(i ,i ,...,i ,...) or a finite
1 2 k
word with |ω|≥k+n. Then
1
µ˜(I )= h˜ (x) dµ˜(x),
ωk+n αn ωn+k−ωk
ZIωk
where ω −ω is the word
n+k k
(i ,i ,...,i ).
k+1 k+2 k+n
For every finite word γ, define
h˜
γ
h = .
γ α|γ|µ˜(I )
γ
Note that for |ω|≥k+1
µ˜(I ) µ˜(I )
(4) h (x)=h (x)·g ◦ψ (x) ωk =h (x)·h˜ ◦ψ (x) ωk .
ωk+1 ωk ik+1 ωk+1 α µ˜(I ) ωk ik+1 ωk α µ˜(I )
ωk+1 ωk+1
Let U ⊂ C be a pre-compact open set. Consider the Banach space B(U) of all complex analytic
functions
h: U →C
that have a continuous extension on U, endowed with the sup norm.
Remark 2.1. Thesefunctionspaceshavethefollowingremarkableproperty. IfU,U ⊂Careprecompact
1
opensetsandU ⊂U thentheinclusionı: B(U)→B(U )isacompactlinearoperator. Soeverybounded
1 1
sequence f ∈ B(U) has a subsequence f such that f converges uniformly on U to a continuous
n ni ni 1
function that is complex analytic in U . Picking a sequence of open sets U such that U ⊂U and
1 n n
∪ U =U,
n n
we can use the a diagonal argument to show that we can find a subsequence f and a complex analytic
ni
function f on U such that f converges uniformly to f on each compact subset of U.
ni
Theorem 2.1. There exists K > 0 with the following property: For every infinite word ω the sequence
h is a Cauchy sequence in B(O). Let h be its limit. For every ω and x∈O we have
ωk ω
1
≤|h (x)|≤K.
ω
K
Proof. Indeed since
ψ (I )=I ,
ik+1 ωk ωk+1
we have
(5) α µ˜(I )= g ◦ψ (y) dµ˜(y).
ωk+1 ik+1 ik+1
ZIωk
Since g is analytic and
i
diam ψ (O)≤Cλk+1,
ωk+1
by Eq. (4) we have that
g ◦ψ (y)
ik+1 ik+1 =1+δ ,
k,x,y
g ◦ψ (x)
ik+1 ik+1
with
|δ |≤Cλk+1.
k,x,y
for every x,y ∈ψ (O). Here C does not depend on either x,y ∈O, k ≥1, or ω. In particular
ωk
µ˜(I )
g ◦ψ (x) ωk =1+δ˜ ,
ik+1 ωk+1 α µ˜(I ) k,x
ωk+1
with
|δ |≤Cλk.
k,x
for x∈O. This implies that for m>n
h (x)
ωm =1+ǫ ,
n,m
h (x)
ωn
with
(6) |ǫ |≤C λn
n,m 1
for some C . Here C does not depend on x,y ∈ O, k ≥ 1, or ω. Let m large enough such that
1 1 0
C1λm0 <1. Then
∞ ∞
inf |h (y)| (1−C λk)≤|h (x)|≤ sup |h (y)| (1+C λk)
y∈O,|γ|<m0 γ kY=m0 1 ωk y∈O,|γ|<m0 γ kY=m0 1
for every x∈O, infinite word ω and k ≥1. In particular there exists K >0 such that
1
(7) ≤|h (x)|≤K
K ωk
foreveryk ≥1,x∈O andinfinitewordω. ByestimatesEq. (6)and(7)weconcludethath converges.
ωk
Denote
h =limh .
ω k ωk
It follows from Eq. (7) that
1
(8) ≤|h (x)|≤K
ω
K
for every x∈O and infinite word ω. (cid:3)
Corollary 2.1. For each ω ∈Σ the function logh (·): I →R has a complex analytic extension to O.
ω
Proof. Since O is a simply connected open set and h (x) 6= 0 for every x ∈ O, this follows by classical
ω
results of complex analysis. (cid:3)
We use the notation h (x)=h(ω,x), h (x)=h(ω ,x), for x∈[0,1] and ω ∈{1,2,..,d}N, according
ω ωk k
to convenience.
For every µ˜-integrable function z: I →R we can define the signed measure zµ˜ as
(zµ˜)(A)= z(x)µ˜(x)
ZA
for every borelian set A⊂I.
Theorem 2.2. Let
z: I →R
be a positive Holder-continuous function. Then the sequence
ρ (x):=lim h (x)[(zµ˜)(I )]=lim h (x) zdµ˜,
z γ γ γ
k |γX|=k k |γX|=k ZIγ
converges for each x∈O. This convergence is uniform on compact subsets of O. Indeed
ρ (x)=v(x) z dµ˜,
z
Z
where v is the complex analytic extension of the function v defined in (3). Furthermore, there exists a
borelian probability µ in the space of infinite words such that
v(x)=ρ (x)= h (x) dµ(ω).
v ω
Z
Proof. Define ρk: O →C as
ρk(x):= h (x) zdµ˜.
γ
|γX|=k ZIγ
Firstly we will prove that
(9) ρk(x)→ v(x) z dµ˜,
k
Z
for each x∈I. Indeed for x∈I
h (x) zdµ˜ = h (x) z(ψ (x))(1+ǫ )µ˜(I )
γ γ γ x,γ γ
|γX|=k ZIγ |γX|=k
= h (x) z(ψ (x))µ˜(I )+ǫ˜
γ γ γ x,k
|γX|=k
=α−k h˜ (x) z(ψ (x))+ǫ˜
γ γ x,k
|γX|=k
=α−k(Pk z)(x)+ǫ˜ .
logg x,k
Here,
|ǫ |,|ǫ˜ |≤Cβk,
x,γ x,k
for some β <1.
It is a well know fact that
limα−k(Pk z)(x)=v(x) z dµ˜.
logg
k
Z
So
limρk(x)=v(x) z dµ˜.
k
Z
for x∈I.
Next we claim that ρk converges uniformly on compact subsets of O to a complex analytic function
ρ . Note that by Eq. (7) we have
z
| h (x) zdµ˜|≤K sup|z(x)| µ˜(I )≤K sup|z(x)|,
γ γ
|γX|=k ZIγ x∈I |γX|=k x∈I
for every x ∈ O, so in particular the complex analytic functions ρk are uniformly bounded in O. By
Remark 5.2 every subsequence of ρk has a subsequence that converges uniformly on compact subsets of
O to a complex analytic function defined in O, so to prove the claim it is enough to show that every
subsequenceofρk thatconvergesuniformlyoncompactsubsetsofO convergestotheverysamecomplex
analytic function. Indeed we already proved that such limit functions must coincide with
v(x) z dµ˜
Z
on I. Since the limit functions are complex analytic, if they coincide on I they must coincide everywhere
in O. This finishes the proof of the claim. In particular taking z(x) = 1 everywhere, this proves that
v: I →R has a complex analytic extension v: O →C. Consequently for every function z
ρ (x)=v(x) z dµ˜,
z
Z
oncewealreadyknowthatthesefunctionscoincideonI. Foranygivenzwehavethatρ (x)=v(x) zdµ˜
z
is an eigenfunction of the Ruelle operator. So we have a natural spectral projection in the space of
R
eigenfunctions.
Now we will prove the second statement. Consider the unique probability µ defined on the space of
infinite words such that on the cylinders C , |γ|<∞, it satisfies
γ
µ(C )=(vµ˜)(I )= vdµ˜.
γ γ
ZIγ
Note that µ extends to a measure on the space of infinite words because vµ˜ is f-invariant and it has no
atoms. For each fixed x∈O, the functions ω →h (x) are constant on each cylinder C , |γ|=k. So
ωk γ
h (x) dµ(ω)= h (x)µ(C ).
ωk γ γ
Z |γX|=k
By the Dominated Convergence Theorem
h (x) dµ(ω)=lim h (x) dµ(ω)=lim h (x) µ(C )=lim h (x) (vµ˜)(I ).
ω k ωk k γ γ k γ γ
Z Z |γX|=k |γX|=k
(cid:3)
Corollary 2.2. The function ρ =v(x) z dµ˜ is a α-eigenfunction of P
z logg
R Plogg(ρz)=α·ρz.
Therefore, any ρ is an eigenfunction for the Ruelle operator for A = logg. Later we will consider a
z
real parameter β and we will denote by φ (x) a specific normalized eigenfunction of the Ruelle operator
β
for βlogg.
The two results described above are in some sense similar to the ones in [1] and [26]. We explain this
claim in a more precise way in the next section.
3. Maximizing probabilities, subactions and the involution kernel
In this section we review some definitions and properties of Ergodic Optimization (see [20] [9] [1] [3])
We compare the setting and notation of [1] with the one described here.
We consider here f an expanding real analytic transformation of degree d on the interval [0,1] with
an analytic extension to a small complex neighborhood B ⊂C of [0,1].
By definition, the Bernoulli space is the set {1,2,...,d}N =Σ. A general element w in Σ is denoted by
w =(w ,w ,..,w ,..).
0 1 n
We denote Σˆ the set Σ×[0,1] and ψ indicates the i-th inverse branch of f. We also denote by σ∗ the
i
shift on Σ. Finally, σˆ−1 is the backward shift on Σˆ given by σˆ−1(w,x)=(σ∗(w),ψ (x))
w0
Consider A(x)=logg(x).
A sub-action for A is a function V :[0,1]→R such that, for all x∈[0,1] we
have
V(f(x)) ≥ V(x) + A(x) − m(A).
A calibrated sub-action V :[0,1]→R for the potential A, is a function V such that
sup {V(y) + A(y) − m(A)} = V(x).
ysuchthatf(y)=x
A calibrated sub-action is a particular case of sub-action.
If we assume the maximizing probability for A is unique, then there is just one calibrated sub-action
up to an additive constant (see [9] [1]).
From[4]itisknownthatforacertainanalyticfunctionsf andA=logg,thereisnoanalyticsubaction.
Definition 3.1. Consider A:[0,1]→R Holder. We say that W :Σˆ →R is a involution kernel for A,
1
if there is a Holder function A∗ :Σ→R such that
A∗(w)=A◦σˆ−1(w,x)+W ◦σˆ−1(w,x)−W (w,x).
1 1
We say that A∗ is a dual potential of A, or, that A and A∗ are in involution.
In [1] it was used the terminology W-kernel instead of involution kernel.
WepointoutthatA∗ andW arenotunique. Itisalsoknown(proposition2in[1])thattwoinvolution
1
kernels for A differ by a function ϕ(w). It always true that
m(A)= max A(x)dν(x)=
ν aninvariantprobabilityforf
Z
max A∗(w)dµ(w)=m(A∗).
µ aninvariantprobabilityforσ∗
Z
Remark: We point out that in section 5 we are going to consider two specific involution kernels for
which we will reserve the notation H and W.
∞
The definition of involution kernel is basically the same as in [1] where it’s considered the Bernoulli
space {1,2,...,d}Z. Here the infinite choice of the inverse branches is described by w. More precisely, the
y in Proposition 3 in [1] is the w here. Moreover, our x is in [0,1] and not in {1,2,...,d}N as in [1].
The results described here in last section correspond in [26] to the potential logg =A=−logf′.
In this way eW1(w,x) coincides with the function |Dψw(x)| on the variables (x,w) of [26]. Note that in
this case for a fixed w the function W (x)=W (w,x) is analytic on x, if f is considered analytic in [26].
w 1
We fix from now on a certain W as the involution kernel for A=logg.
1
Remark: We point out that in the present moment we are consider fixed W and A∗ just for the
1
purpose of explaining the general theory for maximizing probabilities and large deviations. We will
consider an specific involution kernel W (and a A∗) later, and these ones are obtained in a unique way
from the procedure described here.
Given β one can consider βA and the associated Ruelle operator P . We will be interested here in
βA
Thermodynamic Formalism properties for the potential βA, when β →∞ (the zero temperature case).
The point of view of [1] is the following: it is easy to see that if W is a involution kernel for A (we
1
consider W fixed as we just said) then βW is a involution kernel for βA.
1 1
Remark: In the notation of last section log(h (x)) = logh(w,x) = W (w,x). This will be shown in
w 1
the next section. Note that in last section the function h depends on A=logg in a natural way. In this
way, for a given β we get in a natural and unique way from log(h (w,x)), which is not necessarily equal
β
to βW , where W = h is fixed and associated to an initial A = logg (that is, β = 1). This is main
1 1
difference from the reasoning in our section 5 to the procedures in [1]. More precisely, If we know that
A∗(w)=A◦σˆ−1(w,x)+W ◦σˆ−1(w,x)−W (w,x),
1 1
then, an easy way to get an involution kernel for βA is just multiply the above equation by β. Then, in
[9] it is used the relation
βA∗(w)=βA◦σˆ−1(w,x)+βW ◦σˆ−1(w,x)−βW (w,x).
1 1
We are not doing this here. In our reasoning in section 5 we are deriving, for each β, a certain involution
kernel W which is not necessarily equal to βW . In this notation, just when β =1 we have W =1W .
β 1 β 1
We describe briefly the main results in [1].
As we said, given β, one can take the associated involution kernel (to βA) the function W = βW .
β 1
Moreover (βA)∗ =βA∗. The normalizing constant
c(β)=log eβW1(w,x)dνβA∗(w)dνβA(x),
Z Z
is such that
φβA(x)= eβW1(w,x)−c(β)dνβA∗(w),
Z
where φ is the normalized eigen-function associated to the Ruelle operator P and to the maximal
βA βA
eigenvalue λ(β), and finally νβA and νβA∗ are the associated eigen-probabilities for the dual of the
Ruelle operators P∗ and P∗ (acting on probabilities) corresponding respectively to βA and βA∗ (see
βA βA∗
proposition 3 in [1]). We denote by µ = φ dν and we note that, φ dν = 1. In analogous
βA βA βA βA βA
way µβA∗ =φβA∗dνβA∗. Here, PβA∗(φβA∗)=λ(β)φβA∗.
R
Remember from the corollary of last section that given A = β logg, we have P (ρ ) = αρ .
β logg v v
Therefore, the expression
ρ (x)= h (x) dµ(ω),
v ω
Z
obtained in Theorem 1 is similar but slightly different from
φβ logg(x)= eβW1(w,x)−c(β)dν(β logg)∗(w),
Z
becauseµisaninvariantprobabilityfortheshiftandν(β logg)∗ isaneigen-probability(notnecessarily
invariant for the shift) for P∗ . This point will be important in the last section.
β logg∗
Remark: Note that (using the above notation) logh (x) is not necessarily equal to βW (w,x)−
ω 1
c(β)−logφβ logg∗.
It is known that in the analytic setting we consider before, given an analytic potential A=logg, the
eigenfunction φ for P is analytic in a neighborhood C of [0,1]. This can be also derived from the
βA βA β
expression above if we know that W (w,x)=W (x) is analytic on x for any w fixed. A natural question
1 w