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PIECEWISE ANALYTIC SUBACTIONS FOR ANALYTIC DYNAMICS

G. CONTRERAS (*), A. O. LOPES (**), E. R. OLIVEIRA (**) AND D. SMANIA (***)

Abstract. We consider a piecewise analytic expanding map f : [0, 1] → [0, 1] of degree d which preservesorientation, and an analytic positive potential g : [0, 1] → R.

We address the analysis of the following problem: for a given analytic potential β log g, where β is areal constant, it is well known that there exists a real analytic eigenfunction φβ for the Ruelle operator.One can ask: what happen with the function φβ , when β goes to infinity? The domain of analyticity

can change with β. The correct question should be: is 1β

log φβ analytic in the limit, when β → ∞?

Under a uniqueness assumption, this limit, when β →∞, is in fact a calibrated subaction V .Denote

m(log g) = maxν an invariant probability for f

∫log g(x) dν(x),

and µ∞, any probability which attains the maximum value. Any one of these probabilities µ∞ iscalled a maximizing probability for log g. The probability µ∞ is the limit of the Gibbs states µβ , forthe potentials β log g. In this sense one case say that µ∞ corresponds to the Statistical Mechanics attemperature zero.

We show that when µ∞ is unique, has support in a periodic orbit, the analytic function g is genericand satisfies the twist condition, then the calibrated sub-action V : [0, 1] → R for the potential log g ispiecewise analytic. We assume the twist condition only in some of the proofs we present here.

An interesting case where the theory can be applied is when log g(x) = − log f ′(x). In this case werelate the involution kernel to the so called scaling function.

Date: December 21, 2009 - (*) CIMAT, Guanajuato, Mexico (**) Instituto de Matematica, UFRGS, 91509-900 - PortoAlegre, Brasil. Partially supported by CNPq, PRONEX – Sistemas Dinamicos, Instituto do Milenio, and beneficiary ofCAPES financial support, (***) Departamento de Matematica, ICMC-USP 13560-970 Sao Carlos, Brasil. Partiallysupported by CNPq 310964/2006-7, and FAPESP 2008/02841-4 .

1

2 G. CONTRERAS (*), A. O. LOPES (**), E. R. OLIVEIRA (**) AND D. SMANIA (***)

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 analyticextension to a neighborhood of [0, 1] in the complex plane.

We denotem(log g) = max

ν an invariant probability for f

∫log g(x) dν(x),

and µ∞ (or, µ∞ log g) any probability which attains the maximum value. Any one of these probabilitiesµ∞ is called a maximizing probability for log g.

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) = log g(x), and a relation

with the dual potential A∗(w) = (log g)∗(w) defined in the Bernoulli space 1, ..., dN. In this caseW : 1, ..., dN × [0, 1] → R and by analytic we mean: for each w ∈ 1, ..., dN fixed, the function W (w, .)has a complex analytic extension to a neighborhood of [0, 1]. We refer the reader to [1] [24] for definitionsand properties related to the involution kernel.

An interesting case where the theory can be applied is when log g(x) = − log f ′(x). In this case werelate the involution kernel to the so called scaling function [18] [26].

By definition, a calibrated subaction for log g is a function V such that

supy such that f(y) = x

V (y) + log g(y) − m(log g) = V (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 φβ (with the normalization∫

φβ dνβ = 1, where νβ is the eigen-probability, associatedto the same eigenvalue as φβ , of the dual of the Ruelle operator associated to β A = β log g) the maineigenfunction of the Ruelle operator for the potential β log g, it is known (theorem 29 in [9]) that for anyconvergent (uniform convergence) subsequence of 1

β log φβ we have that the limit is a calibrated subaction.Under our hypothesis of uniqueness we have that the sequence 1

β log φβ , when β → ∞, converges to acalibrated subaction.

Note that, in the general case, not always the Gibbs invariant probability µβ , for β log(g), will convergeto a unique (maximizing) probability when β → ∞ [7]. This happens if the maximizing probability isunique.

Using the above results we show that under some conditions, when µ∞ is unique, has support in aperiodic orbit and log g is twist, then there exists a piecewise analytic calibrated sub-action (denoted byV : [0, 1] → R) for the potential log g. Our main result is Theorem 6.2. We will also show that the aboveconditions 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) isnot defined (via a critical path problem) from the potential A to be maximized. Sometimes the analogousstatements in each theory have to be proved under different techniques. A basic tool in Aubry-Mathertheory is the assumption: the Lagrangian is convex on the velocity v. Without this hypothesis the Mathergraph theorem (see [8]) is not true, etc.... In Ergodic Optimization a natural assumption, which in somesense play the role of convexity, is the twist condition on the involution kernel (it is a condition thatdepends 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 twistcondition appear there. The twist condition is an open property in the variation of the analytic potentialA = log g defined in a fixed open complex neighborhood of the interval [0, 1]. We assume f preservesorientation in order it can exist potentials A = log g which satisfy the twist condition (see [24]). We pointout that we do not need this hypothesis for the results of all sections from 2 to 5. In the case f reverse

PIECEWISE ANALYTIC SUBACTIONS FOR ANALYTIC DYNAMICS 3

orientation, a similar reasoning can be applied, but we have to consider another dynamics (not the shift)on the dual space Σ. The proof requires a lot of different technicalities and we avoid to consider this case(we address the question to [24]).

The main idea of the proof in a short is the following: denote by I∗ the deviation function of the familyof Gibbs probabilities µβA∗ which converges to µ∞A∗ (the maximizing probability for A∗, see [1]), andV ∗ the calibrated subaction for A∗. From our assumptions the maximizing probability µ∞A∗ is supportedin 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) = supw∈Σ

[ W (w, x)− V ∗(w)− I∗(w) ].

Therefore, for each x there exists wx such that

V (x) = supw∈Σ

[ W (w, x)− V ∗(w)− I∗(w) ] =

V (x) = [ W (wx, x)− V ∗(wx)− I∗(wx) ].

For each fixed w we proved in sections 4 and 5 that W (w, x) is analytic in x.As for a fixed wx, W (wx, x) is analytic on x (see corollary 5.2), a result on piecewise analyticity of V

can be obtained if we are able to assume conditions to assure that wx is a locally constant as a functionof x.

We need two nice conditions for getting that: 1) the set of w such that I∗(w) < K, for any givenconstant K is a finite set.

If this is true, the possible number of wx is finite.This is so because W (w, x) is bounded in Σ and V ∗(w) is bounded in Σ. Then points w with a large

value of I can not be a maximizing wx (otherwise, take wx in the periodic maximizing probability whereI 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∗ (herewe 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 → wx is monotonous (here we have to use a twist

condition for W , which is a condition to be checked just on A). This is proved in section 6.Therefore, values wx can not change up and down when we move x in a monotonous way. From the

above we get that wx is locally constant as a function of x and so V is piecewise analytic.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 theinitial problem.

In section 1 we present basic definitions and in section 2 we show the existence of a certain functionhw(x) = h(w, x) which defines by means of log(h(w, x)) an involution kernel for log g. In section 3 wepresent some basic results in Ergodic Optimization and we describe the main strategy for getting thepiecewise analytic sub-action V . Section 4 shows the relation of the scaling function (see [30] [18]) withthe involution kernel and the potential log g = − log f ′. In fact, we consider in this section a more generalsetting considering any given potential log g. In section 5 (and also 3) we consider Gibbs states for thepotential β log g, where β is a real parameter. In section 6 we show, under a natural, but technicalcondition, the existence of the piecewise complex analytic calibrated sub-action, when µ∞ is unique, hassupport in a periodic orbit and A = log g satisfies a twist condition. We also show that our technicalcondition is true for a generic g. In sections 7 and 8 a more general setting for generic propertiesof potentials is considered. The results about a generic g, which were used before, are obtained as aparticular application. Finally, in section 9 we present a result of independent interest for the case wherethe maximizing probability is not a periodic orbit: we consider properties of the involution kernel for ageneric x.

After this paper was written we discovered that some of the ideas described in section 2 appeared insome 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 closedintervals

(1) Iii∈1,2,..,d,

with pairwise disjoint interiors, such that

- For each i we have that f(Ii) = I,

- fi is monotone on each Ii.

We say that f is expanding if f is C1 on each Ii and there exists λ > 1 such that

infi

infx∈Ii

|Df(x)| ≥ λ.

Denote byψi : I → Ii

the inverse branch of f satisfyingψi f(x) = x

for each x ∈ Ii.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 ψi has a univalent extension

ψi : O → ψi(O).

Since f is expanding, we can choose O such that- ψi has a continuous extension

ψi : O → C.

- We haveψi(O) ⊂ O.

- Moreover

supi

supx∈O

|Dψi(x)| ≤ λ =λ−1 + 1

2< 1.

Consider a finite wordγ = (i1, i2, . . . , ik),

where ij ∈ 1, 2, .., d. Denote |γ| = k. Define the univalent maps

ψγ : O → Cas

ψγ = ψik ψik−1 · · · ψi1 ,

We will denoteIγ := ψγ(I).

Given either an infinite word

ω = (i1, i2, . . . , ik, . . . ) ∈ Σ := 1, 2, .., dN,or a finite word with |ω| ≥ k, define its k-truncation as

ωk = (i1, i2, . . . , ik).

Note that for k ≥ 1ψωk

= ψik ψωk−1 .

For every finite word γ we can define the cylinder

(2) Cγ = ω ∈ 1, 2, .., dN : ω|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 : ∪i int Ii → R

is a complex analytic potential if there are complex analytic functions gi : ψi(O) → C such that- The functions gi and g coincides in the interior of Ii.- The functions gi have a continuous extension to ψi(O).- There exists θ < 1 such that

0 < infx∈ψi(O)

|gi(x)| ≤ supx∈ψi(O)

|gi(x)| ≤ θ.

- We havegi(R ∩ ψi(O)) ⊂ R+.

Denotehi(x) = gi(ψi(x)).

For every finite word γ we will define by induction on the lenghts of the words the function

hγ : O → C

in the following way: Let γ = (i1, i2, . . . , ik+1). If |γ| = k + 1 = 1 define hγ(x) = gi1(ψi1(x)), otherwise

hγ(x) = hγk(x) · gik+1 ψγk+1(x) = hγk

(x) · hik+1 ψγk(x).

Define the Perron-Frobenious operator

Plog g : C(I) → C(I).

as(Plog g q)(x) =

∑

i

hi(x) q(ψi(x)).

Note that(Pn

log g q)(x) =∑

|γ|=n

hγ(x) q(ψγ(x)).

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) Pnlog gv = αnv, µ(v) = 1,

andµ(Pn

log gq) = αn µ(q)

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) =∫

A

v(x) dµ(x).

Then the probability vµ is f -invariant. Let ω be either an infinite word ω = (i1, i2, . . . , ik, . . . ) or a finiteword with |ω| ≥ k + n. Then

µ(Iωk+n) =

1αn

∫

Iωk

hωn+k−ωk(x) dµ(x),

where ωn+k − ωk is the word(ik+1, ik+2, . . . , ik+n).

For every finite word γ, define

hγ =hγ

α|γ|µ(Iγ).


Note that for |ω| ≥ k + 1

(4) hωk+1(x) = hωk(x) · gik+1 ψωk+1(x)

µ(Iωk)

α µ(Iωk+1)= hωk

(x) · hik+1 ψωk(x)

µ(Iωk)

α µ(Iωk+1).

Let U ⊂ C be a pre-compact open set. Consider the Banach space B(U) of all complex analyticfunctions

h : U → Cthat have a continuous extension on U , endowed with the sup norm.

Remark 2.1. These function spaces have the following remarkable property. If U,U1 ⊂ C are precompactopen sets and U1 ⊂ U then the inclusion ı : B(U) → B(U1) is a compact linear operator. So every boundedsequence fn ∈ B(U) has a subsequence fni

such that fniconverges uniformly on U1 to a continuous

function that is complex analytic in U1. Picking a sequence of open sets Un such that Un ⊂ U and

∪nUn = U,

we can use the a diagonal argument to show that we can find a subsequence fniand a complex analytic

function f on U such that fniconverges uniformly to f on each compact subset of U .

Theorem 2.1. There exists K > 0 with the following property: For every infinite word ω the sequencehωk

is a Cauchy sequence in B(O). Let hω be its limit. For every ω and x ∈ O we have1K≤ |hω(x)| ≤ K.

Proof. Indeed sinceψik+1(Iωk

) = Iωk+1 ,

we have

(5) α µ(Iωk+1) =∫

Iωk

gik+1 ψik+1(y) dµ(y).

Since gi is analytic anddiam ψωk+1(O) ≤ Cλk+1,

by Eq. (4) we have thatgik+1 ψik+1(y)gik+1 ψik+1(x)

= 1 + δk,x,y,

with|δk,x,y| ≤ Cλk+1.

for every x, y ∈ ψωk(O). Here C does not depend on either x, y ∈ O, k ≥ 1, or ω. In particular

gik+1 ψωk+1(x)µ(Iωk

)α µ(Iωk+1)

= 1 + δk,x,

with|δk,x| ≤ Cλk.

for x ∈ O. This implies that for m > n

hωm(x)hωn(x)

= 1 + εn,m,

with

(6) |εn,m| ≤ C1λn

for some C1. Here C1 does not depend on x, y ∈ O, k ≥ 1, or ω. Let m0 large enough such thatC1λ

m0 < 1. Then

infy∈O,|γ|<m0

|hγ(y)|∞∏

k=m0

(1− C1λk) ≤ |hωk

(x)| ≤ supy∈O,|γ|<m0

|hγ(y)|∞∏

k=m0

(1 + C1λk)


for every x ∈ O, infinite word ω and k ≥ 1. In particular there exists K > 0 such that

(7)1K≤ |hωk

(x)| ≤ K

for every k ≥ 1, x ∈ O and infinite word ω. By estimates Eq. (6) and (7) we conclude that hωkconverges.

Denotehω = lim

khωk

.

It follows from Eq. (7) that

(8)1K≤ |hω(x)| ≤ K

for every x ∈ O and infinite word ω. ¤

Corollary 2.1. For each ω ∈ Σ the function log hω(·) : I → R has a complex analytic extension to O.

Proof. Since O is a simply connected open set and hω(x) 6= 0 for every x ∈ O, this follows by classicalresults of complex analysis. ¤

We use the notation hω(x) = h(ω, x), hωk(x) = h(ωk, x), for x ∈ [0, 1] and ω ∈ 1, 2, .., dN, according

to convenience.For every µ-integrable function z : I → R we can define the signed measure zµ as

(zµ)(A) =∫

A

z(x)µ(x)

for every borelian set A ⊂ I.

Theorem 2.2. Letz : I → R

be a positive Holder-continuous function. Then the sequence

ρz(x) := limk

∑

|γ|=k

hγ(x) [ (z µ ) (Iγ)] = limk

∑

|γ|=k

hγ(x)∫

Iγ

z d µ,

converges for each x ∈ O. This convergence is uniform on compact subsets of O. Indeed

ρz(x) = v(x)∫

z dµ,

where v is the complex analytic extension of the function v defined in (3). Furthermore, there exists aborelian probability µ in the space of infinite words such that

v(x) = ρv(x) =∫

hω(x) dµ(ω).

Proof. Define ρk : O → C as

ρk(x) :=∑

|γ|=k

hγ(x)∫

Iγ

z d µ.

Firstly we will prove that

(9) ρk(x) →k v(x)∫

z dµ,

for each x ∈ I. Indeed for x ∈ I

∑

|γ|=k

hγ(x)∫

Iγ

z d µ =∑

|γ|=k

hγ(x) z(ψγ(x))(1 + εx,γ)µ(Iγ)

=∑

|γ|=k

hγ(x) z(ψγ(x))µ(Iγ) + εx,k


= α−k∑

|γ|=k

hγ(x) z(ψγ(x)) + εx,k

= α−k(P klog gz)(x) + εx,k.

Here,|εx,γ |, |εx,k| ≤ Cβk,

for some β < 1.It is a well know fact that

limk

α−k(P klog gz)(x) = v(x)

∫z dµ.

Solim

kρk(x) = v(x)

∫z dµ.

for x ∈ I.Next we claim that ρk converges uniformly on compact subsets of O to a complex analytic function

ρz. Note that by Eq. (7) we have

|∑

|γ|=k

hγ(x)∫

Iγ

z d µ| ≤ K supx∈I

|z(x)|∑

|γ|=k

µ(Iγ) ≤ K supx∈I

|z(x)|,

for every x ∈ O, so in particular the complex analytic functions ρk are uniformly bounded in O. ByRemark 5.2 every subsequence of ρk has a subsequence that converges uniformly on compact subsets ofO to a complex analytic function defined in O, so to prove the claim it is enough to show that everysubsequence of ρk that converges uniformly on compact subsets of O converges to the very same complexanalytic function. Indeed we already proved that such limit functions must coincide with

v(x)∫

z dµ

on I. Since the limit functions are complex analytic, if they coincide on I they must coincide everywherein O. This finishes the proof of the claim. In particular taking z(x) = 1 everywhere, this proves thatv : I → R has a complex analytic extension v : O → C. Consequently for every function z

ρz(x) = v(x)∫

z dµ,

once we already know that these functions coincide on I. For any given z we have that ρz(x) = v(x)∫

z dµis an eigenfunction of the Ruelle operator. So we have a natural spectral projection in the space ofeigenfunctions.

Now we will prove the second statement. Consider the unique probability µ defined on the space ofinfinite words such that on the cylinders Cγ , |γ| < ∞, it satisfies

µ(Cγ) = (vµ)(Iγ) =∫

Iγ

v d µ.

Note that µ extends to a measure on the space of infinite words because vµ is f -invariant and it has noatoms. For each fixed x ∈ O, the functions ω → hωk

(x) are constant on each cylinder Cγ , |γ| = k. So∫hωk

(x) dµ(ω) =∑

|γ|=k

hγ(x)µ(Cγ).

By the Dominated Convergence Theorem∫hω(x) dµ(ω) = lim

k

∫hωk

(x) dµ(ω) = limk

∑

|γ|=k

hγ(x) µ(Cγ) = limk

∑

|γ|=k

hγ(x) (vµ)(Iγ).

¤

Corollary 2.2. The function ρz = v(x)∫

z dµ is a α-eigenfunction of Plog g

Plog g(ρz) = α · ρz.


Therefore, any ρz is an eigenfunction for the Ruelle operator for A = log g. Later we will consider areal parameter β and we will denote by φβ(x) a specific normalized eigenfunction of the Ruelle operatorfor β log g.

The two results described above are in some sense similar to the ones in [1] and [26]. We explain thisclaim 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] withan analytic extension to a small complex neighborhood B ⊂ C of [0, 1].

By definition, the Bernoulli space is the set 1, 2, ..., dN = Σ. A general element w in Σ is denoted byw = (w0, w1, .., wn, ..).

We denote Σ the set Σ× [0, 1] and ψi indicates the i-th inverse branch of f . We also denote by σ∗ theshift on Σ. Finally, σ−1 is the backward shift on Σ given by σ−1(w, x) = (σ∗(w), ψw0(x))

Consider A(x) = log g(x).A sub-action for A is a function V : [0, 1] → R such that, for all x ∈ [0, 1] wehave

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

supy such that f(y) = x

V (y) + A(y) − m(A) = V (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] it is known that for a certain analytic functions f and A = log g, there is no analytic subaction.

Definition 3.1. Consider A : [0, 1] → R Holder. We say that W1 : Σ → R is a involution kernel for A,if there is a Holder function A∗ : Σ → R such that

A∗(w) = A σ−1(w, x) + W1 σ−1(w, x)−W1(w, x).

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.We point out that A∗ and W1 are not unique. It is also known (proposition 2 in [1]) that two involution

kernels for A differ by a function ϕ(w). It always true that

m(A) = maxν an invariant probability for f

∫A(x) dν(x) =

maxµ an invariant probability for σ∗

∫A∗(w) dµ(w) = m(A∗).

Remark: We point out that in section 5 we are going to consider two specific involution kernels forwhich 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 Bernoullispace 1, 2, ..., dZ. Here the infinite choice of the inverse branches is described by w. More precisely, they in Proposition 3 in [1] is the w here. Moreover, our x is in [0, 1] and not in 1, 2, ..., dN as in [1].

The results described here in last section correspond in [26] to the potential log g = A = − log f ′.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 Ww(x) = W1(w, x) is analytic on x, if f is considered analytic in [26].We fix from now on a certain W1 as the involution kernel for A = log g.


Remark: We point out that in the present moment we are consider fixed W1 and A∗ just for thepurpose of explaining the general theory for maximizing probabilities and large deviations. We willconsider an specific involution kernel W (and a A∗) later, and these ones are obtained in a unique wayfrom the procedure described here.

Given β one can consider βA and the associated Ruelle operator PβA. We will be interested here inThermodynamic 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 W1 is a involution kernel for A (weconsider W1 fixed as we just said) then βW1 is a involution kernel for βA.

Remark: In the notation of last section log(hw(x)) = log h(w, x) = W1(w, x). This will be shown inthe next section. Note that in last section the function h depends on A = log g in a natural way. In thisway, for a given β we get in a natural and unique way from log(hβ(w, x)), which is not necessarily equalto βW1, where W1 = h is fixed and associated to an initial A = log g (that is, β = 1). This is maindifference from the reasoning in our section 5 to the procedures in [1]. More precisely, If we know that

A∗(w) = A σ−1(w, x) + W1 σ−1(w, x)−W1(w, x),

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

β A∗(w) = β A σ−1(w, x) + βW1 σ−1(w, x)− βW1(w, x).

We are not doing this here. In our reasoning in section 5 we are deriving, for each β, a certain involutionkernel Wβ which is not necessarily equal to βW1. In this notation, just when β = 1 we have Wβ = 1 W1.

We describe briefly the main results in [1].As we said, given β, one can take the associated involution kernel (to βA) the function Wβ = β W1.

Moreover (β A)∗ = βA∗. The normalizing constant

c(β) = log∫ ∫

eβ W1(w,x) dνβA∗(w) dνβA(x),

is such thatφβA(x) =

∫eβW1(w,x)−c(β) dνβA∗(w),

where φβA is the normalized eigen-function associated to the Ruelle operator PβA and to the maximaleigenvalue λ(β), and finally νβA and νβA∗ are the associated eigen-probabilities for the dual of theRuelle operators P ∗βA and P ∗βA∗ (acting on probabilities) corresponding respectively to βA and βA∗ (seeproposition 3 in [1]). We denote by µβA = φβA dνβA and we note that,

∫φβA dνβA = 1. In analogous

way µβA∗ = φβA∗ dνβA∗ . Here, PβA∗(φβA∗) = λ(β) φβA∗ .Remember from the corollary of last section that given A = β log g, we have Pβ log g(ρv) = α ρv.

Therefore, the expression

ρv(x) =∫

hω(x) dµ(ω),

obtained in Theorem 1 is similar but slightly different from

φβ log g(x) =∫

eβ W1(w,x)−c(β) dν(β log g)∗(w),

because µ is an invariant probability for the shift and ν(β log g)∗ is an eigen-probability (not necessarilyinvariant for the shift) for P ∗β log g∗ . This point will be important in the last section.

Remark: Note that (using the above notation) log hω(x) is not necessarily equal to β W1(w, x) −c(β)− log φβ log g∗ .

It is known that in the analytic setting we consider before, given an analytic potential A = log g, theeigenfunction φβA for PβA is analytic in a neighborhood Cβ of [0, 1]. This can be also derived from theexpression above if we know that W1(w, x) = Ww(x) is analytic on x for any w fixed. A natural question


is: what happen with the domains Cβ of φβA when β →∞? The question that makes sense is to ask: isthere an analytic limit for

limβ→∞

log φβA

β?

Our purpose in this paper is to show that if the maximizing probability is unique and has supportin a periodic orbit, then certain subsequences βn → ∞ of above limit will define a piecewise analyticfunction V . The idea is to consider a fixed neighborhood C of [0, 1] on C and to show that we can selecta sequence of bounded complex analytic functions log φβnA

βn. Any of these limits will define a calibrated

sub-action (see [9] page 1404)We assume that the maximizing probability µ∞ for A is unique, and so, the maximizing probability for

µ∗∞ for A∗ is also unique (this follows from the cohomological equation for σ). In this case limβ→∞ µβ A∗ =µ∗∞ (see [9] [1])

In [1] is shown that for any cylinder C ∈ Σ

limβ→+∞

1β

log µβ A∗(C) = − infw∈C

I∗(w)

where

I∗(w) =∑

n≥0

(V ∗ σ∗ − V ∗ − (A∗ −m∗)

) (σ∗)n(w), m∗ =∫

A∗ dµ∗∞

where V ∗(x) is any calibrated subaction of A∗.That is, A∗ satisfies for all w

supw such that σ∗(w) = w

V ∗(y) + A∗(y) − m(A) = V ∗(w).

Adapting the proof of the Varadhan’s Theorem (theorem 4.3.1 in [11]) one can show that for a con-tinuous function G : Σ → R,

limβ→+∞

1β

log∫

eβG(w)µβ A∗(w) = supw∈Σ

(G(w)− I∗(w))

Note in our setting µβ A∗ → µ∞A∗ , the maximizing probability for A∗. In the same way for the caseof A (which has a deviation function denoted by I) the deviation function I∗ can have the value infinityfor some points w. The function I∗ is zero on the support of the maximizing probability for A∗. Anyway,in [24] a direct proof os this property is presented.

Moreover, for any x is true

φβA(x) =∫

eβW1(w,x)−c(β)−log φβA∗ (w) φβA∗(w) dνβA∗(w) =

∫eβ[ W1(w,x)− 1

β c(β)− 1β log φβA∗ (w)] dµβA∗(w),

where µβA∗ is the invariant probability which maximizes the pressure P (βA∗) and c(β) is the correspond-ing normalizing constant to such βW1. It is known that there exists γ, such that c(β)

β → γ, as β → ∞.

All these results are described in [1].

Remark: We point out that we will not follow the above strategy here because we have a procedurethat defines an involution kernel Wβ in a unique way (and it is not equal to βW1).


4. Scaling functions and dual potentials

Given a finite word γ = (i1, i2, . . . , ik), k > 1, define σ?(γ) = (i2, . . . , ik). For infinite words we defineσ? as the usual shift function. The scaling function s : Σ → R of the potential g is defined as

s(ω) = limk→∞

µ(Iωk)

µ(Iσ?(ωk)).

This definition is the natural generalization of the one in [30] and [18]. When log g = − log f ′ weget the usual one. In this section we show the existence of a natural involution kernel which providesa co-homology between the scaling function [ log(α s) ](w) and log g(x) = − log f ′(x). The constant α isthe eigenvalue defined before in section 1.

To verify that the above limit indeed exists, note that by Eq. (5) and since g is a Holder-continuousfunction we have that

µ(Iωk+1)µ(Iσ?(ωk+1))

=

∫Iωk

g ψik+1(y) dµ(y)∫

Iσ?(ωk)g ψik+1(y) dµ(y)

= (1 + εk)µ(Iωk)

µ(Iσ?(ωk)),

where |εk| ≤ Cλk. So s(ω) is well defined.Note that, since v > 0 is a Holder function and Iωk

⊂ Iσ(ωk),

s(ω) = limk→∞

(vµ)(Iωk)

(vµ)(Iσ?(ωk))= lim

k→∞µ(Cωk

)µ(Cσ?(ωk))

,

so the the scaling function s is the Jacobian of the measure µ.The dual potential g? is defined as

g?(ω) := αs(ω).For every ω = (i0, i1, . . . , ik, . . . ) and x ∈ I , define

σ−1(ω, x) := (σ∗(ω), ψi0(x)).

Proposition 4.1. We haveg?(ω)

g(ψi0(x))=

h(σ∗(ω), ψi0(x))h(ω, x)

.

Proof. Indeedh(σ∗(ω), ψi0(x))

h(ω, x)

= limk

h(σ∗(ωk), ψi0(x))h(ωk, x)

.

limk

h(σ∗(ωk), ψi0(x))h(ωk, x)

αkµ(Iωk)

αk−1µ(Iσ∗(ωk))

= limk

α

g(ψi0(x))µ(Iωk

)µ(Iσ∗(ωk))

=α

g(ψi0(x))s(ω)

¤

We finally get the following result:

Proposition 4.2. log hω(x) = log h(ω, x) is well defined and is an involution kernel for log g. For ωfixed, the function log h(ω, .) has a complex analytic extension to a complex neighborhood O of [0, 1].

The dual of A = log g is naturally associated to the scaling function s.We will need an special involution kernel H∞ later, not this one log hw(x). The reason is that we have

to consider a variable parameter β and moreover β log g. The complex neighborhood O of [0, 1] couldchange, in principle, very much with β.


5. When β →∞ we get an involution kernel which is analytic on x for w fixed

Given an analytic potential g, let gi : O → R, i = 1, . . . , d, be complex analytic functions satisfyingthe properties described in Section 2. Since O is simply connected and gi(x) 6= 0 for x ∈ O, reducing O alittle bit, if necessary, we have that log gi is a well defined continuous function in O and complex analyticin O. For each real value β, we consider the analytic potential gβ and the corresponding the functionsgβ

i (x) := exp(β log gi(x)). Then for each β these functions also satifies the assumptions of Section 2,so we can construct functions hβ(ω, x), as in Theorem 2.1, corresponding scaling functions sβ and dualpotentials g∗β = αβ sβ as in Section 4. The main goal of this section it to study how these functions areperturbed when we move β.

Note that we are going to consider g fixed, and for a variable β the potential gβ . We point outthat, in principle, the corresponding dual potential g∗β does not satisfy necessarily g∗β = (g∗)β , where g∗

corresponds to g by the procedure of last section.Note also that if g∗ is the one associated to g, then

(log g)∗(w) = log g σ−1(w, x) + W1 σ−1(w, x)−W1(w, x),

Therefore, given a real value β we have

β (log g)∗(w) = β log g σ−1(w, x) + β W1 σ−1(w, x)− β W1(w, x).

Therefore, β W1 is a involution kernel for β log g. This point was very important in [1].We consider here a new procedure that gives in a unique way (for each value β) an involution kernel

Wβ = log hβ for β log g.The important point we would like to stress here is that in [1] it is consider first ca fixed W1. Then,

it is known that one can get the main eigenfunction for the Ruelle operator for βA as

φβA(x) =∫

eβW1(w,x)−c(β) dνβA∗(w).

Here the procedure is different: we will get (for each value β another away different from [9])

φβA(x) =∫

eWβ(w,x) dνβA∗(w) =∫

hβ(w, x) dνβA∗(w),

for a Wβ which depends of the variable β (see the third remark of last section).hβ acting on the variable (w, x) is an integral kernel that transforms eigen-probabilities of the dual of

the Ruelle operator for β A∗ in eigen-functions of the Ruelle operator for βAFirst we want to show that there exists H∞(w, x) (complex analytic on x) such that hβ(w, x) ∼

eβ H∞(w,x) (in the sense that limβ→∞ 1β log hβ(w, x) = H∞(w, x)). In other words, we want to replace Wβ

by a βH∞ (in the notation that will be followed later). This will be useful to apply Varadhan’s Theoremlater from an expression in the form

φβA(x) =∫

eβH∞(w,x)−c(β) dνβA∗(w) =∫

eβH∞(w,x)−c(β) 1φβA∗(x)

d µβA∗(w).

Remember that for a given w ∈ Σ, we have hω = limk hωk.

Proposition 5.1. Let K ⊂ O be a compact. There exists C such that the following holds:A. For every β ≥ 1 and x ∈ K, ω ∈ Σ, we have

(10) e−βC ≤ |hβ(ω1, x)| ≤ eβC

B. For every β ≥ 1, x ∈ K, ω ∈ Σ and k ≥ 1 we have

(11) e−Cβλk ≤∣∣∣hβ(ωk+1, x)

hβ(ωk, x)

∣∣∣ ≤ eCβλk

.

C. For every finite word γ there is a function

qγ : R×O → C,


that is holomorphic on x, real valued for x ∈ R and which does not depend on K, such that forevery x ∈ O, β ≥ 1, ω ∈ Σ we have

(12) hβ(ωk+1, x) = eqωk+1 (β,x).

Furthermore

(13) |qω1(β, x)| ≤ Cβ

and

(14) |qωk+1(β, x)− qωk(β, x)| ≤ Cβλk

for every β ≥ 1, x ∈ K, ω ∈ Σ and k ≥ 1.

Proof of Claim A. Recall that for i ∈ 1, . . . , d

(15) hβ(i, x) =gβ

i (ψi(x))αµβ(Ii)

=gβ

i (ψi(x))∫Igβ

i (ψi(y))µβ(y),

so|hβ(i, x)| = 1

∫I

gβi (ψi(y))

|gβi (ψi(x))| µβ(y)

.

Since gi are holomorphic on ψi(O), gi 6= 0 in ψi(O), for every compact K ⊂ O there exists C such that

(16) e−C ≤ |gi(ψi(x))||gi(ψi(y))| ≤ eC

for every x, y ∈ K and i. Since µβ(I) = 1, it is now easy to obtain Eq. (10). ¤

Proof of Claim B. Since gi are holomorphic on ψi(O), gi 6= 0 in ψi(O), for every compact K ⊂ O thereexists C such that

(17) e−C|x−y| ≤∣∣∣gi(ψi(x))gi(ψi(y))

∣∣∣ ≤ eC|x−y|

for every x, y ∈ K and i. Note that every such compact is contained in a larger compact set K ⊂ O suchthat ψi(K) ⊂ K for every i, so we can assume that K has this property. Let x ∈ K. By Eq. (5)

hβ(ωk+1, x)hβ(ωk, x)

=hβ(ωk+1, x)hβ(ωk, x)

αkβµβ(Iωk

)

αk+1β µβ(Iωk+1)

=gβ

ik+1(ψωk+1(x))

αβ

µβ(Iωk)

µβ(Iωk+1)

=gβ

ik+1(ψωk+1(x))

αβ

αβµβ(Iωk)∫

Iωk

gβik+1

ψik+1(y) dµβ(y)

=gβ

ik+1(ψωk+1(x))µβ(Iωk

)∫

Iωk

gβik+1

ψik+1(y) dµβ(y)(18)

=µβ(Iωk

)∫

Iωk

gβik+1

ψik+1 (y)

gβik+1

(ψωk+1 (x))dµβ(y)

.(19)

In particular ∣∣∣hβ(ωk+1, x)hβ(ωk, x)

∣∣∣ =µβ(Iωk

)∫

Iωk

gβik+1

ψik+1 (y)

|gβik+1

(ψωk+1 (x))| dµβ(y).

For every y ∈ Iωkwe have

ψik+1(y), ψωk+1(x) ∈ ψωk+1(O)


From Eq. (17) we obtain

e−Cβλk ≤ e−Cβ diam ψωk+1 (O) ≤gβ

ik+1 ψik+1(y)

|gβik+1

(ψωk+1(x))|≤ eCβ diam ψωk+1 (O) ≤ eCβλk

So

e−Cβλk ≤ |hβ(ωk+1, x)||hβ(ωk, x)| ≤ eCβλk

.

¤

Proof of Claim C. Since gi ψi : O → C does not vanish and O is a simply connected domain, thereexists a (unique) function ri : O → C such that gi ψi = eri on O and Im ri(x) = 0 for x ∈ R. Sinceψγ(O) ∩ I 6= ∅ and diam ψγ(O) ≤ λ|γ| we have that

(20) |Im ri(ψγ(x))| ≤ Cλ|γ|

for every x ∈ O and every finite word γ.Define

qi(β, x) = βri(x) + log1∫

Iigβ

i ψi(y) dµβ(y).

and qγ , with γ = (i1, . . . , ik+1), by induction on k, as

qγ(β, x) = qγk(β, x) + βrik+1(ψγk

(x)) + logµβ(Iγk

)∫Iγk

gβik+1

ψik+1(y) dµβ(y).

It follows from Eq. (18) that qγ satisfies Eq. (12), so

Re qγ(β, x) = log |hβ(γ, x)|,in particular by Eq. (10) e (11) we have

(21) |Re qω1(β, x)| ≤ Cβ

and

(22) |Re qωk+1(β, x)−Re qωk(β, x)| ≤ Cβλk

for β ≥ 1. Furthermore for every β ∈ R, ω ∈ Σ and k ≥ 1

|Im qωk+1(β, x)− Im qωk(β, x)| = |β Im rik+1(ψωk

(x))| ≤ C|β|λk.

Moreover for β > 0 we have

|Im qi(β, x)| = |β||Im ri(ψωk(x))| ≤ C|β|.

¤

For every x ∈ O define

Hβ,k(ω, x) :=1β

qωk(β, x).

In particular, if x ∈ I we have that hβ(ωk, x) is a nonnegative real number by our choice of the branchesri, so

Hβ,k(ω, x) =1β

log hβ(ωk, x)

for x ∈ I. It follows from Proposition 5.1 that for every compact K ⊂ O there exists C such that

(23) |Hβ,1(ω, x)| ≤ C,

(24) |Hβ,k+1(ω, x)−Hβ,k(ω, x)| ≤ Cλk

for x ∈ K, and every k and ω. So there exists some constant C such that

|Hβ,k(ω, x)| ≤ C


for every k, ω, x ∈ K. This implies that the family of functions

F1 = Hβ,k(ω, ·)k,ω,β≥1

is a normal family on O, that is, every sequence of functions in this family admits a subsequence thatconverges uniformly on every compact subset of O. In Theorem 2.1 we showed that for every x ∈ I wehave

limk

hβ(ωk, x) = hβ(ω, x) > 0,

so

limk

Hβ,k(ω, x) =1β

log hβ(ω, x),

for x ∈ I. It follows from the normality of the family F that the limit

Hβ(ω, x) := limk

Hβ,k(ω, x)

exists for every x ∈ O and that this limit is uniform on every compact subset of O. Moreover

F2 = Hβ(ω, ·)ω,β≥1

is also a normal family on O.We consider in Σ the metric d, such that d(ω, γ) = 2−n, where n is the position of the first symbol in

which ω and γ disagree.

Corollary 5.1. For every compact K ⊂ O there exists C such that

(25) |Hβ(ω, x)−Hβ(γ, y)| ≤ C|x− y|+ Cd(ω, γ)

for every x, y ∈ K.

Proof. Since the family F2 is uniformly bounded on each compact set K ⊂ O, we have that the familyof functions

F3 := H ′β(ω, ·)ω,β≥1

has the same property, so it is easy to see that for every compact K ⊂ O there exists C such that

|Hβ(ω, x)−Hβ(ω, y)| ≤ C|x− y|.Note also that Eq. (24) implies

|Hβ(ω, x)−Hβ(ωk, x)| ≤ Cλk,

Let k + 1 = log(d(γ, ω))/ log λ. Then γk = ωk and we have

|Hβ(ω, y)−Hβ(γ, y)| ≤ |Hβ(ω, y)−Hβ(ωk, y)|+ |Hβ(γk, y)−Hβ(γ, y)| ≤ Cd(ω, γ).

¤

Corollary 5.2. There exists a sequence βn > 0 satisfying βn →∞ when n →∞ such that the limit

(26) H∞(ω, x) = limn→∞

Hβn(ω, x),

exists for every (ω, x) in1, . . . , dN ×O.

Moreover for every compact K ⊂ O there exist C such that

(27) |H∞(ω, x)−H∞(γ, y)| ≤ C|x− y|+ Cd(ω, γ)

and the limit in Eq. (26) is uniform with respect to (ω, x) on

(28) 1, . . . , dN ×K

In particular for each ω we have that x → H∞(ω, x) is holomorphic on O.


Proof. By Corollary 5.1, the family of functions Hβ is equicontinuous on each set of the form (28), whereK is a compact subset of O. So given a compact K ⊂ O and any sequence βj → +∞, as j → ∞, thereis a subsequence βji

such that the limitlim

i→∞Hβji

(ω, x)

exists and it is uniform on the set of the form (28). Then, choosing an exhaustion by compact sets of Oand using Cantor’s diagonal argument we can find a sequence βn → +∞ such that the limit

H∞(ω, x) = limn→+∞

Hβn(ω, x)

exists and it is uniform on every set of the form (28), with compact K ⊂ O. Eq. (27) follows directlyfrom Eq. (25). ¤

This shows the main result in this section:

Corollary 5.3. For any w fixed, H∞(ω, x) is analytic on x.

Given a continuous function G(w, x) = Gx(w), from Varadhan’s Integral Lemma (section 4.3 page 137[11])

limβ→+∞

1β

log∫

eβGx(w)µβ A∗(w) = supw∈Σ

(Gx(w)− I∗(w)).

Note that for each fixed x we are using Varadhan’s Integral Lemma. We will not use directly this. Weneed a small extra effort (I∗ is not continuous, see [24]).

From Corollary 5.2 (the convergence is uniform) and the fact that

ρv(x) =∫

hω(x) dµ(ω),

we get that for any x ∈ [0, 1]

V (x) = limβ→+∞

1βn

log φβn(x) = supw∈Σ

(H∞(w, x)− I∗(w)).

Remember that for every ω = (i0, i1, . . . , ik, . . . ) and x ∈ S1

σ−1(ω, x) := (σ∗(ω), ψi0(x)).

Proposition 5.2. The function H∞(w, x) is an involution kernel for g.

Proof. Consider g fixed. Let βn be a sequence as in Corollary 5.2. For any βn we haveg?

βn(ω)

gβn(ψi0(x))=

hβn(σ∗(ω), ψi0(x))hβn(ω, x)

.

Taking 1βn

log in both sides and taking the limit n → +∞ we get that

g(σ−1(ω, x)) + H∞(σ−1(ω, x)) − H∞(ω, x)

depends only in the variable w.Therefore, H∞(w, x) is an involution kernel. ¤Given the analytic involution kernel H∞(w, x) and a fixed calibrated V ∗ (unique up to additive con-

stant) define W (w, x) = H∞(w, x)−V ∗(w). We point out that W is also analytic on the variable x ∈ (0, 1)for each w fixed).

The reason for the introduction of such W (and not H∞) is that, in next section, instead of

V (x) = supw∈Σ

[H∞(w, x)− I∗(w)],

it is more convenient the expression

V (x) = supw∈Σ

[ W (w, x)− V ∗(w)− I∗(w) ].


6. The subaction is piecewise analytic when the potential A = log g is twist and g isgeneric

We sometimes denote σ∗ by σ. We believe will be no confusion: f acts on points x ∈ [0, 1] and σ∗ (or,σ) acts on w on Σ = 0, ..., d−1N. This avoids using ∗ ali the time. In any case σ acts on Σ× [0, 1] = Σ.

We suppose in this section that the maximizing probability for A∗ is unique (see [9]) in order wecan define the deviation function I∗. This property will follow from the uniqueness of the maximizingprobability for A = log g (which implies the same for A∗).

Adapting Varadhan’s Theorem one can show that that

V (x) = supw∈Σ

[ W (w, x)− V ∗(w)− I∗(w) ].

See also [24] for a direct proof of this result.For each x we get one (or, more) wx such attains the supremum above. Therefore,

V (x) = W (wx, x)− V ∗(wx)− I∗(wx) .

The main strategy in the present section is to find suitable hypothesis in such way that wx is uniqueand locally constant on x. Remember that for a fixed w, we have that W (w, x) is analytic on x. It seemsdifficult to us to imagine how one could be able to show that V (x) is locally analytic using a differentprocedure. But, we may be wrong.

One can consider on Σ = 0, ..., d−1N the lexicographic order. We will consider, by technical reasons,the case where f : (0, 1) → (0, 1) has positive derivative. In the most of the cases we will consider, d = 2,in order to avoid unnecessary complex notation.

Following [24] we define:

Definition 6.1. We say a continuous G : Σ = Σ × [0, 1] → R satisfies the twist condition on Σ, if forany (a, b) ∈ Σ = Σ× [0, 1] and (a′, b′) ∈ Σ× [0, 1], with a′ > a, b′ > b, we have

(29) G(a, b) + G(a′, b′) < G(a, b′) + G(a′, b).

Definition 6.2. We say a continuous A : [0, 1] → R satisfies the twist condition, if some (all) of itsinvolution kernels satisfies the twist condition.

Note that W satisfies the twist condition, if, and only if, W − V ∗ (or, W (w, x) − V ∗(w) − I∗(w))satisfies the twist condition.

We will assume later that A = log g satisfies the twist condition. We point out that in order to checkthat W is twist we just have to check properties of the potential A (see [24]). The property of been twistis stable by perturbations. Examples of twist potentials A are presented in [24].

We point out that in the case f reverse orientation (like ,−2x (mod 1)), then there is no potentialA = log g which is twist for the dynamics on Σ × [0, 1]. A careful analysis (for different types of Bakermaps) of when it is possible for A to be twist for a given dynamics f is presented in [24]. We will notconsider this case here.

Proposition 5 in [1] claims that if µmax is the natural extension of the maximizing probability µ∞,then for all (p, p∗) in the support of µmax we have

V (p) + V ∗(p∗) = W (p, p∗) − γ.

From this follows that if (p, p∗) in the support of µmax (then, p ∈ [0, 1] is in the support of µ∞ andp∗ ∈ Σ is in the support of µ∗∞), then

V (p) = supw∈Σ

(W (w, p)− γ − V ∗(w)− I∗(w) ) =

( W (p∗, p)− γ − V ∗(p∗)− I∗(p∗) ) = ( W (p∗, p)− γ − V ∗(p∗) ).If the potential log g is twist, then for any given p in the support of µ∞, there is only one p∗, such

that (p, p∗) is in the support of µmax (see [24]).In principle could exist another w ∈ Σ such that for such p we have

V (p) = W (w, p)− γ − V ∗(w)− I∗(w) .


The calibrated subaction will be analytic, if there exists w such that for all x

V (x) = supw∈Σ

(H∞(w, x)− I∗(w)) = H∞(w, x)− I∗(w) = W (w, x)− V ∗(w)− I∗(w).

This will not be always the case.Let’s consider for a moment the general case (A not necessarily twist) .We denote by M the support of µ∗∞.As I∗ is lower semicontinuous and W − V ∗ is continuous, then for each fixed x, the supremum of

H∞(w, x)− I(w) in the variable w is achieved, and we denote (one of such w) it by wx. In this case wesay wx is optimal for x. One can ask if this wx is independent of x, and equal to a fixed w. This wouldimply that V is analytic. If for all x in a certain open interval (a, b), the wx is the same, then V is analyticin this interval. We will show under some restrictions that given any x we can find a neighborhood (a, b)of x where this is the case.

Given x, this maximum at wx can not be realized where I(w) is infinity. Moreover, as W − V ∗ isbounded, there exists a constant K, such that, we know a priory that wx is such that I(wx) < K.

Consider for any x

K(x) = maxw

H∞(x,w)−minw

H∞(x,w)

Then, K = sup K(x).

Remark: We just have to consider w such that I(w) < K.

In order to simplify the notation we assume that m(A∗) = 0.If we denote R∗(w) = V ∗ σ(w)− V ∗(w)−A∗(w), then we know that R∗ ≥ 0.Consider the compact set of points P = w ∈ Σ, such that σ∗(w) ∈ M , and w is not on M.Assumption: We say that R∗ is good for A∗, if for each w ∈ P , we have that R∗(w) > 0.We point out that there are examples of potentials A∗ (with a unique maximizing probability) where

the corresponding R∗ is not good (see example 2 in the end of the present section) .We will assume first that R∗ is good, present some main results, and later we show that generically on

the analytic function g (not generically on A∗ which is much more easy) we have that the correspondingR∗ satisfies the assumption.

Example 1. We point out that in the example described in [24], for the potential A = −(1−x)2, and thetransformation T (x) = −2x (mod 1), we have that the maximizing probability µ∞ for A has support onx0 = 2/3. The pre-image of 2/3 outside the support of µ∞ is x1 = 1/6. That is, P = x ∈ [0, 1]−2/3,such that T (x) is in the support of µ∞ = 1/6. The explicit value of the calibrated sub-action isV (x) = −1/2 x2 + 2/9 x2. In this case R(1/6) = V (2/3)− V (1/6)− (A(1/6)−m(A) ) = 0.665.. > 0.

Therefore, the R corresponding to such A (not A∗) satisfies the property of been good for A.This potential A is not twist when we consider the question in Σ × [0, 1]. If we consider instead a

different kind of Baker map F , like the one that can be naturally defined F : [0, 1]× [0, 1] → [0, 1]× [0, 1],which satisfies F (x, f(y)) = (f(x), y),∀(x, y) ∈ [0, 1] × [0, 1], then the potential is twist (see [24] for theappropriate definition). All results we present in this section also applied to this last situation.

Remember that ,

I∗(w) =∑

n≥ 0

(V ∗ σ − V ∗ −A∗

) σn(w) =∑

n≥ 0

R∗(σn(w)).

In [23] section 5 it is shown that if I∗(w) is finite, then

limn→∞

1n

n−1∑

j=0

δ(σ∗)j(w) → µ∗∞.

Our main assumption says that R∗ is positive in the compact set P .


We consider in Σ the metric d, such that d(w1, w2) = 12n , where n is the first symbol in which w1 and

w2 disagree.

There exist a fixed 0 < δ < 2−(p+1), (in the case M is a periodic orbit, the p can be taken the period)for some p > 0, such that, if

Ωδ = w ∈ Σ | d(w, P ) < δ ,then

cδ = minw∈Ωδ

R∗(w) > 0.

Consider a small neighborhood Aδ of the set M such that σ∗(Ωδ) = Aδ.We can assume the above δ is such that any point in Aδ has a distance smaller that 2−p to a point of

M , where p is the period.Note that in order that the orbit of point w by σ∗ enter (a new time) the set Aδ, it has to pass before

by Ωδ.As µ∗∞(M) > 0, then considering the continuous function χAδ

(indicator of Aδ), we have that, ifI∗(w) < ∞, then

limn→∞

1n

n−1∑

j=0

χAδ((σ∗)j(w)) > 0.

Therefore, (σ∗)j(w) visits Aδ for infinitely many values of j.Given w, suppose there exist a N > 0, such that for all j > N , we have that (σ∗)j(w) ∈ Aδ. In this

case, there exist a k such that (σ∗)k(w) ∈ M.Now, we consider the other case.Denote by m1 the total amount of time the orbit (σ∗)k(w) remains in Aδ for the first time, then the

trajectory goes out of Aδ, and m2 is the total amount of time the orbit (σ∗)k(w) remains in Aδ for thesecond time it returns to Aδ, and so on...

We suppose from now on that the maximizing probability for A∗ has support in a unique periodicorbit of period p denoted by M = w1, w2, .., wp ⊂ Σ.

We have two possibilities:a) The times mn, n ∈ N, of visits to Aδ, satisfies 2−mn < δ, for infinitely many values of n. In this

case, the orbit visits Ωδ an infinite number of times, and I∗(w) = ∞, and we reach a contradiction.b) The times mn, n ∈ N, are bounded by a constant N . We can consider now a new set Aδ, which is

a smaller neighborhood of M , in such way that any point in Aδ has a distance smaller that 2−N , to apoint of M .

As,

limn→∞

1n

n−1∑

j=0

χAδ((σ∗)j(w)) > 0,

we reach a contradiction.We are interested only in the case the support is periodic orbit. The shift is expanding, then by the

shadowing property there is an ε such if the corresponding forward orbits of two points are ε close, for alln, then the points are the same. From this it follows that the orbit we are considering (which eventuallyremains indefinitely within Aδ) should be eventually periodic.

Therefore, if w is such that I∗(w) < ∞, then, there exists a k such that (σ∗)k = w ∈ M.From [1] [24] it follows that the support of the maximizing probability for A is a periodic orbit

M1 = x1, x2, .., xp ⊂ [0, 1].We are going to show in this case that if R∗ is good for A∗ and the twist condition for A∗ is also true,

then the subaction V is piecewise analytic.Remark: The function I∗ is lower semi-continuous, that is, if wn → w, then lim inf I(wn) ≥ I(w).

From this follows that given K > 0, if I(wn) < K, and wn → w, then I(w) ≤ K.We claim that if R∗ is good for A∗, then given K > 0 there exist just a finite number of points w

with I(w) < K. This is so, because the times to arrival in the set M are bounded. Indeed, if there was


an infinite number of such wn they would accumulate in a point w, such that I(w) < K, but this pointcannot reach the set M by forward iteration in a finite number of steps.

In this way, the above claim, applied to the situation we consider here, says that the set of all possiblewx is a finite set (all points in in the pre-orbit of the periodic maximizing probability), when we considerall the possible x ∈ [0, 1].

Remark: We point out that if A∗ depends on a finite number of coordinates and the maximizingprobability is unique, then R∗ is good for A∗.

For each w such that is in the pre-orbit of a point of M , denote by k(w), the smaller non-negativeinteger such that (σ∗)k(w)(w) ∈ M. Denote by o(w), this point in M , such that (σ∗)k(w)(w) = o(w). Aswe said before, the possible k(w) are uniformly bounded by a uniform constant N .

Remark: We point out that the above property is not necessarily true if we do not assume that R∗

is good for A∗.

The conclusion is that if R∗ is good for A∗, then

V (x) = supw∈Σ , σj(w)∈M, for some 0≤j≤N

(H∞(w, x)− I∗(w)).

For such kind of w we have

I∗(w) =∑

n≥ 0

(V ∗ σ − V ∗ −A∗

) σn(w) =∑

n≥ 0

R∗(σn(w)) =

k(w)−1∑n=0

(V ∗ σ − V ∗ −A∗

) σn(w) =k(w)−1∑

n=0

R∗(σn(w)) =

[ V ∗( o(w) )− V ∗(w) ]− ( A∗(w) + A∗(σ(w)) + ... + A∗(σk(w)−1(w)) ).In this way, for w satisfying σk(w) = o(w) ∈ M (where k is the smallest possible) we have that

H∞(w, x)− I∗(w) = W (w, x)− V ∗(w)− I∗(w) =

( W (w, x)− V ∗(o(w))) + ( A∗(w) + ... + A∗(σk(w)−1(w) ).The above expression is the main reason for considering W − V ∗ instead of H∞.The k above could be eventually equal to zero when w ∈ M . In this particulary case H∞(w, x)− I(w) =

W (w, x)− V ∗(w).We assume from now on that A = log g satisfies the twist condition.It is known (see [2] [23]) that x → wx is monotonous decreasing.

Indeed, as

V (x) = supw∈Σ

(W (w, x)− V ∗(w)− I∗(w)) = W (wx, x)− V ∗(wx)− I∗(wx),

thenW (w, x)− V ∗(w)− I∗(w) ≤ W (wx, x)− V ∗(wx)− I∗(wx)

for any w, and we also have that

V (x′) = supw∈Σ

(W (w, x′)− V ∗(w)− I∗(w)) = W (wx′ , x′)− V ∗(wx′)− I∗(wx′).

Therefore,W (w, x′)− V ∗(w)− I∗(w) ≤ W (wx′ , x

′)− V ∗(wx′)− I∗(wx′).for any w.

Suppose, x < x′. Substituting wx′ in the first one, and wx in the second one we get

∆(x, x′, wx′) ≤ ∆(x, x′, wx),

where W (x,w)−W (x′, w) = ∆(x, x′, w). So the twist property implies that wx′ ≤ wx.


Theorem 6.1. Consider the transformation T (x) = 2x. Suppose A satisfies the twist condition, R∗ isgood for A∗, and the maximizing probability for A has support in a periodic orbit, then the subaction Vfor A is piecewise analytic.

Proof. Consider a point x0 ∈ [0, 1] and a variable x in a small interval (x0− ε, x0) on the left of x0. Notethat x → wx is monotonous decreasing and can reach just a finite number of values.

Remember that from a previous remark the possible values of optimal wx are in a finite set.This shows that wx is constant for a certain interval (x0 − ε, x0), with ε > 0. Moreover, the above

argument shows that there exist t > 0 and a certain finite number of points zj , such that 0 = z1 < z2 <z3 < ... < zt = 1, t ∈ N, such that wx is constant in each interval (zj , zj+1) . Furthermore,

V (x) = H∞(wx, x)− I(wx),

is analytic, for x ∈ (zj , zj+1), j ∈ 1, .., t− 1.¤

It is easy to see from the above that if A is monotonous increasing on x, then the maximizing probabilityis in the fixed point 1 and V (x) is analytic.

Examples in which the hypothesis of the above theorem are true appear in [24].Similar results are true for a general f (as considered before) with positive derivative (if there exists d

branches for f , then one have to consider the space Σ = 0, 1, 2, ..., d− 1N, the lexicographic order anda similar definition for the twist condition).

Theorem 6.2. For a fixed transformation f and for a generic analytic potential g (with a unique max-imizing periodic probability) the corresponding R∗ is good for A∗, where A∗ is the dual potential ofA = log g. Then, in this case, the subaction V for A = log g is piecewise analytic.

The proof of the generic this result will be done in Theorem 8.1.

The final conclusion is that for a generic A = log g satisfying the twist condition, if the maximizingprobability is supported in a unique periodic orbit, then the corresponding subaction for A is piecewiseanalytic.

In the next sections we will show the proof of some results we used before.

Now we will provide a counterexample.

Example 2. The following example is due to R. Leplaideur.We will show an example on the shift where the maximizing probability for a certain Lipchitz poten-

tial A∗ : 0, 1N → R is a unique periodic orbit γ of period two, denoted by p0 = (01010101...), p1 =(10101010...), but for a certain point, namely, w0 = (110101010..), which satisfies σ(w0) = p1, we havethat R∗(w0) = 0.

The potential A∗ is given by A∗(w) = −d(w, γ ∪ Γ), where d is the usual distance in the Bernoullispace. The set Γ is described later.

For each integer n, we define a 2n + 3-periodic orbit zn, σ(zn), . . . , σ2n+2(zn) as follows:we first set

bn = (01010101 . . . 01︸︷︷︸2 n

1 01),

and the point zn is the concatenation of the word bn: zn = (bn, bn, ...)The main idea here is to get a sequence of periodic points which spin around the periodic orbit p0, p1

during the time 2 n, and then pass close by w0 (note that d(σ2 n(zn), w0) = 2−2(n+1)).Denote γn the periodic orbit γn = zn, σ(zn), σ2(zn), ..., σ2n+2(zn) .Consider the sequence of Lipchitz potentials A∗n(w) = −d(w, γn ∪ γ). The support of the maximizing

probability for A∗n is γn ∪ γ. Moreover

0 = m(A∗n) = maxν an invariant probability for σ

∫A∗n(w) dν(w).


Denote by V ∗n a Lipchitz calibrated subaction for A∗n such that V ∗

n (w0) = 0. In this way, for all w

R∗n(w) = (V ∗n σ − V ∗

n −A∗n) (w) ≥ 0,

and for w ∈ γn ∪ γ we have that R∗n(w) = 0.We know that R∗n is zero on the orbit γn, because γn is included in the Masur set.Note that we not necessarily have R∗n(w0) = 0.By construction, the Lipchitz constant for A∗n is 1. This is also true for V ∗

n . Hence the family ofsubactions (V ∗

n ) is a family of equicontinuous functions. Let us denote by V ∗ any accumulation point for(V ∗

n ) for the C0-topology. Note that V ∗ is also 1-Lipschitz continuous. For simplicity we set

V ∗ = limk→∞

V ∗nk

.

We denote by Γ the set which is the limit of the sets γn (using the Hausdorff distance). γ ∪ Γ is acompact set. Note that Γ is not a compact set, but the set of accumulation points for Γ is the set γ. Wenow consider A∗(w) = −d(w, γ ∪ Γ).

As any accumulation point of Γ is in γ, any maximizing probability for the potential A∗ has supportin γ. On the contrary, the unique σ-invariant measure with support in γ is maximizing for A∗.

Remember that for any n we have V ∗n (w0) = 0.We also claim that we have A∗nk

(w0) → 0 andV ∗

nk(σ(w0)) → 0, as k →∞.

For each fixed w we set

R∗nk(w) = (V ∗

nk σ − V ∗

nk−A∗nk

) (w) ≥ 0.

The right hand side terms converge (for the C0-topology) as k goes to +∞. Then R∗nkconverge, and we

denote by R∗ its limit. Then for every w we have:

R∗(w) = (V ∗ σ − V ∗ −A∗) (w) ≥ 0.

This shows that V ∗ is a subaction for A∗. Note that R∗(w0) = 0. From the uniqueness of themaximizing probability for A∗ we know that there exists a unique calibrated subaction for A∗ (up to anadditive constant).

Consider a fixed w and its two preimages wa and wb. For any given n, one of the two possibilitiesoccur R∗n(wa) = 0 or R∗n(wb) = 0, because V ∗

n is calibrated for A∗n.Therefore, for an infinite number of values k either R∗nk

(wa) = 0 or R∗nk(wb) = 0.

In this way the limit of V ∗nk

is unique (independent of the convergent subsequence) and equal to V ∗,the calibrated subaction for A∗ (such that V ∗(w0) = 0).

Therefore,R∗(w0) = (V ∗ σ − V ∗ −A∗) (w0) = 0,

and V ∗ is a calibrated subaction for A∗(w) = d(w, γ ∪ Γ).

7. Generic continuity of the Aubry set.

In sections 7 and 8 we will present the proof of the generic properties we mention before.We will present our main results in great generality. First, in this section, we analyze the main

properties of sub-actions and its dependence on the potential A.We refer the reader to [25] [9] [1] [15] [17] for related results in (eventually) different settings.First we will present the main definitions we will consider here.We denote by K a compact metric space and T : K→ K continuous expandig map (that is, there exists

ε > 0, and 1 > λ > 0, such that, if d(x, y) < ε, then d(T (x), T (y)) > 1/λ) such that supx∈K T−1x < ∞(that is, each point has a finite number of pre-images by T ). λ is called the contraction constant for T .

Typical examples are:1) the shift transformation acting on the Bernoulli space 1, 2, .., dN,and2)T : S1 → S1, of class C1+α, such that there exists λ > 1, such that, |T ′(x)| > λ, for all x ∈ S1.


A function g(x), defined locally on an open set Z, such that T (g(z)) = z, ∀z ∈ Z, is called an inversebranch. Most of the time we consider maximal inverse branches.

The constant λ gives an upper bound for the rate of contraction in each inverse branch.F ⊂ C0(K,R) denotes a complete metric space with a (topology finer than) metric larger than

dC0(f, g) = ‖f − g‖0 := supx∈K |f(x)− g(x)|; (for instance, Holder functions, Lispchitz functions orCk(K,R)) AND such that

(30) ∀K ⊂ K compact , ∃ψ ∈ F s.t. ψ ≤ 0, [ψ = 0] = x |ψ(x) = 0 = K.

Property (30) does not hold for real analytic functions on [0, 1] unless K is a finite set, like, for instance,a periodic orbit.

We will assume in the rest of the paper at least that F ∈ Cα(K,R), unless we explicit something elseless.

When the Aubry set (see definiton bellow) is one periodic orbit, the arguments below should apply forF = Cω([0, 1],R) (real analytic functions) with the C0 topology. This will be enough for the purpose ofour main result on piecewise analytic subactions which was stated before.

Given A ∈ F , a calibrated sub-action for A is F : K→ R continuous such that

F (x) = maxy∈T−1x

[F (y) + A(y)−mA] ∀x ∈ K

wheremA := sup

µ∈M(T )

∫A dµ, M(T ) := Borel T -invariant probabilities;

its error is denoted by R = RA : K→ [0,+∞[:

R(x) := F (T (x))− F (x)−A(x) + mA ≥ 0.

S(A) denotes the set of Holder calibrated sub-actions (it is not empty [10] [9])The expression

F (T (x))− F (x) + mA ≥ A(x), ∀x ∈ K,

is the discrete time version of the sub-solution of Hamilton-Jacobi equations [12] [8] [9] [14]. The nextdefinitions were also consider and explored in these references. We denote by

The Mane action potential is:

SA(x, y) := limε→0

[sup

n−1∑

i=0

[A(T i(z))−mA

] ∣∣∣ n ∈ N, Tn(z) = y, d(z, x) < ε]

.

Given x and y the above value describe the A-cost of going from x to y following the dynamics.The Aubry set is A(A) := x ∈ K |SA(x, x) = 0 .The terminology is borrowed from the Aubry-Mather Theory [8].For any x ∈ A(A), we have that SA(x, .) is a sub-action (in particular, in this case, SA(x, y) > −∞,

for any y), see Proposition 23 in [9].The set of maximizing measures is

M(A) := µ ∈M(T ) |∫

Adµ = mA .

If F ∈ Cα(K,R) is a Holder function define

‖F‖α := supx6=y

|F (x)− F (y)|d(x, y)α

.

Define the Mane set asN(A) :=

⋃

F∈S(A)

I−1F 0,

where the union is among all the α-Holder callibrated sub-actions F for A and

IF (x) =∞∑

i=0

RF (T i(x)).


IF (x) is the deviation function we considered before.For A ∈ F define the Mather set as

M(A) :=⋃

µ∈M(A)

supp(µ).

The Peierls barrier is

hA(x, y) : = limε→0

lim supk→+∞

SA(x, y, k, ε),

where SA(x, y, k, ε) := sup n−1∑

i=0

[A(T i(z))−mA

] ∣∣∣ n ≥ k, Tn(z) = y, d(z, x) < ε

.

Several properties of the Mane potential and the Peierls barrier are similar (but not all, see section 4in [16]). We will present proofs for one of them and the other case is similar.

Lemma 7.1.(1) If µ is a minimizing measure then

supp(µ) ⊂ A(A) = x ∈ K | SA(x, x) = 0 .(2) SA(x, x) ≤ 0 for every x ∈ K.(3) For any z ∈ K the function F (y) = hA(z, y) is Holder continuous.(4) If a ∈ A(A) then hA(a, x) = SA(a, x) for all x ∈ K.

In particular, F (y) = SA(a, x) is continuous if a ∈ A(A).(5) If SA(w, y) = hA(w, y) then the function F (y) = SA(w, y) is continuous at y.(6) If S(x0, T

nk(x0)) =∑nk−1

j=0 A(T j(x0)), limk Tnk(x0) = b and limk nk = +∞, then

limk

S(x0, Tnk(x0)) = S(x0, b)

.

Item (1) follows from Atkinson-Mane’s lemma which says that if µ is ergodic for µ-almost every x andevery ε > 0, the set

N(x, ε) :=

n ∈ N∣∣∣

∣∣∣∣∣∣

n−1∑

j=0

A(T j(x))− n

∫Adµ

∣∣∣∣∣∣≤ ε

is infinite (see Lemma 2.2 [25] (which consider non-invertible transformation, [8], [9] or [15] for the proof).We will show bellow just the items which are not proved in the mentioned references.

The problem with the discontinuity of F (y) = SA(w, y) is when the maximum is obtained at a finiteorbit segment (i.e. when SA(w, y) 6= hA(w, y)), the hypothesis in item 5.

Proof.By adding a constant we can assume that mA = 0.(2). Let F be a continuous sub-action for A. Then

−RF = A + F − F T ≤ 0.

Given x0 ∈ K, let xk ∈ K and nk ∈ N be such that Tnk(xk) = x0, limk xk = x0 and

S(x0, x0) = limk

nk−1∑

j=0

A(T j(xk)).

We havenk−1∑

j=0

A(T j(xk)) =[ nk−1∑

j=0

(A + F − F T

)(T j(xk))

]+ F (x0)− F (xk)

≤ F (x0)− F (xk).


Then

S(x0, x0) = limk

nk−1∑

j=0

A(T j(xk)) ≤ limk

[F (x0)− F (xk)

]= 0.

The proofs of (3) (4) (5) can be found in [9] [15]

(6). Let σk be the branch of the inverse of Tnk such that σk(Tnk(x0)) = x0. Let bk = σk(b) for ksufficiently large. Then

d(x0, bk) ≤ λnk d(Tnk(x0), b)k−→ 0.

∣∣∣∣∣nk−1∑

i=0

A(T i(x0))−nk−1∑

i=0

A(T i(bk))

∣∣∣∣∣ ≤‖A‖α

1− λαd(Tnk(x0), b)α.

Write Q := ‖A‖α

1−λα , then

S(x0, b) ≥ lim supk

nk−1∑

i=0

A(T i(bk))

≥ lim supk

S(x0, Tnk(x0))−Qd(Tnk(x0), b)α

≥ lim supk

S(x0, Tnk(x0)).

Now for ` ∈ N let b` ∈ K and m` ∈ N be such that lim b` = x0, Tm`(b`) = b and

lim`

m`−1∑

j=0

A(T j(b`)) = S(x0, b).

Let σ` be the branch of the inverse of Tm` such that σ`(b) = b`. Let x` := σ`(Tnk(x0)). Then

d(x`, x0) ≤ d(x0, b`) + d(b`, x`)

≤ d(x0, b`) + λ` d(Tnk(x0), b)`−→ 0.

∣∣∣∣∣∣

m`−1∑

j=0

A(T j(x`))−m`−1∑

j=0

A(T j(b`))

∣∣∣∣∣∣≤ Qd(Tnk(x0), b)α.

Since x` → x0 and Tm`(x`) = Tnk(x0), we have that

S(x0, Tnk(x0)) ≥ lim sup

`

m`−1∑

j=0

A(T j(x`))

≥ lim sup`

m`−1∑

j=0

A(T j(b`))−Qd(Tnk(x0), b)α

≥ S(x0, b)−Qd(Tnk(x0), b)α.

And hencelim inf

kS(x0, T

nk(x0)) ≥ S(x0, b).

¤

Proposition 7.1. The Aubry set is

A(A) =⋂

F∈S(A)

I−1F 0,

where the intersection is among all the α-Holder callibrated sub-actions for A.


Proof.By adding a constant we can assume that mA = 0.We first prove that A(A) ⊂ ⋂

F∈S(A) I−1F 0.

Let F ∈ S(A) be a Holder sub-action and x0 ∈ A(A). Since SA(x0, x0) = 0 then there is xk → x0 andnk ↑ ∞ such that limk Tnk(xk) = x0 and limk

∑nk−1j=0 A(T j(xk)) = 0. If m ∈ N we have that

F (Tm+1(x0)) ≥ F (Tm(x0)) + A(Tm(x0))

≥ F (Tm+1(xk)) +nk+m−1∑

j=m+1

A(T j(xk)) + A(Tm(x0))

≥ F (Tm+1(xk)) +nk−1∑

j=0

A(T j(xk))−m∑

j=0

|A(T j(xk))−A(T j(x0))|(31)

When k →∞ the right hand side of (31) converges to F (Tm+1(x0)), and hence all those inequalities areequalities. Therefore RF (Tm(x0)) = 0 for all m and hence IF (x0) = 0.

Now let x0 ∈ ⋂F∈S(A) I−1

F 0. Sinc K is compact there is nkk→ +∞ such that the limits b =

limk Tnk(x0) ∈ K and µ = limk µk ∈ M(T ), µk := 1nk

∑nk−1i=0 δT i(x0) exist and b ∈ supp(µ). Let G be a

Holder callibrated sub-action. For m ≥ n we have

G(Tn(x0)) + SA(Tn(x0), Tm(x0)) ≥ G(Tn(x0)) +m−1∑

j=n

A(T j(x0))

= G(Tm(x0)) [because IG(x0) = 0 ]

≥ G(Tn(x0)) + SA(Tn(x0), Tm(x0)).

Then they are all equalities and hence for any m ≥ n

SA(Tn(x0), Tm(x0)) =m−1∑

j=n

A(T j(x0)).

Since

0 = limk

1nk

SA(Tn(x0), Tm(x0)) = limk

1nk

m−1∑

j=n

A(T j(x0)) =∫

Adµ,

µ is a minimizing measure. By lemma 7.1.(1), b ∈ A(A).Let F : K → R be F (x) := SA(b, x). Then F is a Holder callibrated sub-action. By hypothesis

IF (x0) = 0 and then

F (Tn(x0)) = F (x0) + SA(x0, Tnk(x0)).

SA(b, Tnk(x0)) = SA(b, x0) + SA(x0, Tnk(x0)).

By lemma 7.1.(4) and Lemma 7.1.(6), taking the limit on k we have that

0 = SA(b, b) = SA(b, x0) + SA(x0, b) = 0.

0 ≥ SA(x0, x0) ≥ SA(x0, b) + SA(b, x0) = 0

Therefore x0 ∈ A(A).¤

We want to show the following result which will require several preliminary results.

Theorem 7.1. The set

(32) R := A ∈ Cα(K,R) | M(A) = µ, A(A) = supp(µ) contains a residual set in Cα(K,R).

The proof of the bellow lemma (Atkinson-Mane) can be found in [25] and [8].


Lemma 7.2. Let (X, B, ν) be a probability space, f an ergodic measure preserving map and F : X → Ran integrable function. Given A ∈ B with ν(A) > 0 denote by A the set of points p ∈ A such that for allε > 0 there exists an integer N > 0 such that fN (p) ∈ A and

∣∣∣N−1∑

j=0

F(f j(p)

)−N

∫F dν

∣∣∣ < ε .

Then ν(A) = ν(A).

Corollary 7.3. If besides the hypothesis of lemma 7.2, X is a complete separable metric space, and Bis its Borel σ-algebra, then for a.e. x ∈ X the following property holds: for all ε > 0 there exists N > 0such that d(fN (x), x) < ε and

∣∣∣∣∣∣

N−1∑

j=0

F(f j(x)

)−N

∫F dν

∣∣∣∣∣∣< ε

Proof. Given ε > 0 let Vn(ε) be a countable basis of neighbourhoods with diameter < ε and let Vn

be associated to Vn as in lemma 7.2. Then the full measure subset ∩m∪n

Vn( 1m ) satisfies the required

property. ¤

Lemma 7.4. Let R be as in Theorem 7.1. Then if A ∈ R, F ∈ S(A) we have

(1) If a, b ∈ A(A) then SA(a, b) + SA(b, a) = 0.(2) If a ∈ A(A) = supp(µ) then F (x) = F (a) + SA(a, x) for all x ∈ K.

Proof.(1). Let a, b ∈ A(A) = supp(µ). Since µ is ergodic, by Corollary 7.3 there are sequences αk ∈ K,

mk ∈ N such that limk mk = ∞, limk αk = a, limk d(Tmk(αk), αk) = 0,

mk−1∑

j=0

A(T j(αk)) ≥ 1k

, and writing µk :=1

mk

mk−1∑

j=1

δT mk (αk), limk

µk = µ.

Since b ∈ supp(µ) there are nk ≤ mk such that limk Tnk(αk) = b.Let σk be the branch of the inverse of Tnk such that σk(Tnk(αk)) = αk. Let bk := σk(b). Then

Tnk(bk) = b and

d(bk, a) ≤ d(bk, αk) + d(αk, a)

≤ λnk d(b, Tnk(αk)) + d(αk, a)

≤ d(b, Tnk(αk)) + d(αk, a) k−→ 0.

We have that ∣∣∣∣∣∣

nk−1∑

j=0

A(T j(bk))−nk−1∑

j=0

A(T j(αk))

∣∣∣∣∣∣≤ ‖A‖α

1− λαd(Tnk(αk), b)α.

S(a, b) ≥ lim supk

nk−1∑

j=0

A(T j(bk))

≥ lim supk

nk−1∑

j=0

A(T j(αk))−Q d(Tnk(αk), b)α.


Let τk be the branch of the inverse of Tmk−nk such that τk(Tmk(αk)) = Tnk(αk). Let ak := τk(a)Then Tmk−nk(ak) = a and

d(b, ak) ≤ d(b, Tnk(αk)) + d(Tnk(αk), ak)

≤ d(b, Tnk(αk)) + λmk−nk d(Tmk(ak), a)

≤ d(b, Tnk(αk)) + d(Tmk(ak), a) k−→ 0.

Also ∣∣∣∣∣∣

mk−nk−1∑

j=0

A(T j(ak))−mk−1∑

j=nk

A(T j(αk))

∣∣∣∣∣∣≤ ‖A‖α

1− λαd(a, Tmk(αk)).

S(a, b) ≥ lim supk

mk−nk−1∑

j=0

A(T j(ak))

≥ lim supk

mk−1∑

j=nk

A(T j(αk)−Q d(a, Tnk(αk))

Therefore

0 ≥ S(a, a) ≥ S(a, b) + S(b, a)

≥ lim supk

nk−1∑

j=0

A(T j(αk)) + lim supk

mk−1∑

j=nk

A(T j(αk))

≥ lim supk

[ nk−1∑

j=0

A(T j(αk)) +mk−1∑

j=nk

A(T j(αk))]

≥ lim supk

1k

≥ 0.

(2). We first prove that if for some x0 ∈ K and a ∈ A(A) we have

(33) F (x0) = F (a) + SA(a, x0),

then equation (33) holds for every a ∈ A(A). If b ∈ A(A), using item 1 we have that

F (x0) ≥ F (b) + S(b, x0)

≥ F (a) + SA(a, b) + SA(b, x0)

≥ F (a) + SA(a, b) + SA(b, a) + SA(a, x0)

= F (a) + SA(a, x0)

= F (x0).

Therefore F (x0) = F (b) + S(b, x0).It is enough to prove that given any x0 ∈ K there is a ∈ A(A) such that the equality (33) holds. since

F is callibrated there are xk ∈ K and nk ∈ N such that Tnk(xk) = x0, ∃ limk xk = a and for every k ∈ N,

F (x0) = F (xk) +nk−1∑

j=0

A(T j(xk)).


We have that

SA(a, x0) ≥ lim supk

nk−1∑

j=0

A(T j(xk))

= lim supk

F (x0)− F (xk)

= F (x0)− F (a)

≥ S(a, x0).

Therefore equality (33) holds.It remains to prove that a ∈ A(A), i.e. that SA(a, a) = 0. We can assume that the sequence nk is

increasing. Let mk = nk+1 − nk. Then Tmk(xk+1) = xk. Let σk be the branch of the inverse of Tmk

such that σk(xk) = xk+1 and ak+1 := σk(a). We have that∣∣∣∣∣∣

mk−1∑

j=0

A(T j(ak+1))−mk−1∑

j=0

A(T j(xk+1))

∣∣∣∣∣∣≤ ‖A‖α

1− λαd(a, xk)α.

Since xk → a we have that

d(ak+1, a) ≤ d(ak+1, xk+1) + d(xk+1, a)

≤ λmk d(xk, a) + d(xk+1, a)

≤ d(xk, a) + d(xk+1, a) k−→ 0.

Therefore

0 ≥ SA(a, a) ≥ lim supk

mk−1∑

j=0

A(T j(ak+1))

≥ lim supk

mk−1∑

j=0

A(T j(xk+1))−Q d(a, xk)α

= lim supk

F (xk)− F (xk+1)−Q d(a, xk)α

= 0.

¤The above result (2) is true for F only continuous.

Corollary 7.5. Let R be as in Theorem 7.1. Then if A ∈ R, F ∈ S(A) we have(1) If x /∈ A(A) then IF (x) > 0.(2) If x /∈ A(A) and T (x) ∈ A(A) then RF (x) > 0.

Proof.(1). By lemma 7.4.(2) modulo adding a constant there is only one Holder callibrated sub-action F in

S(A). Then by proposition 7.1, A(A) = [IF = 0]. Since IF ≥ 0, this proves item 1.(2). Since T (x) ∈ A(A)

IF (x) =∑

n≥1

RF (Tn(x)) = 0.

Since x /∈ A(A), by item 2 and proposition 7.1, A(A) = [IF = 0]. Then

IF (x) =∑

n≥0

RF (Tn(x)) > 0.

Hence RF (x) > 0. ¤


Lemma 7.6.(1) A 7→ mA has Lipschitz constant 1.(2) Fix x0 ∈ K. The set S(A) of α-Holder calibrated sub-actions F for A with F (x0) = 0 is an

equicontinuous family. In factsup

F∈S(A)

‖F‖α < ∞.

(3) The set S(A) of α-Holder continuous callibrated sub-actions is closed under the C0 topology.(4) If #M(A) = 1, An

n→ A uniformly, supn ‖An‖α < ∞ and Fn ∈ S(A),then limn Fn = F uniformly.

(5) A ≤ B & mA = mB =⇒ SA ≤ SB.(6) lim sup

B→AN(B) ⊆ N(A), where

lim supB→A

N(B) =

limn

xn

∣∣ xn ∈ N(Bn), Bnn→ A, ∃ lim

nxn

(7) If A ∈ R thenlim

B→AdH(A(B),A(A)) = 0,

where dH is the Hausdorff distance.(8) If A ∈ R with M(A) = µ and νB ∈M(B) then

limB→A

dH

(supp(νB), supp(µ)

)= 0.

(9) If A ∈ R thenlim

B→AdH

(M(B),A(A)

)= 0.

If X, Y are two metric spaces and F : X → 2Y = P(Y ) is a set valued function, define

lim supx→x0

F(x) =⋂ε>0

⋂

δ>0

⋃

d(x,x0)<δ

Vε(F(x)),

lim infx→x0

F(x) =⋂ε>0

⋃

δ<0

⋂

d(x,x0)<δ

Vε(F(x)),

whereVε(C) =

⋃

y∈C

z ∈ Y | d(z, y) < ε .

Proof.(1). We have that A ≤ B + ‖A−B‖0, then

∫A dµ ≤

∫B dµ + ‖A−B‖0 , ∀µ ∈M(T ),

∫A dµ ≤ sup

µ∈M(T )

∫B dµ + ‖A−B‖0 = mB + ‖A−B‖0 ,

mA ≤ mB + ‖A−B‖0 .

Similarly mB ≤ mA + ‖A−B‖0 and then |mA −mB | ≤ ‖A−B‖0.See also [20] and [9] for a proof.

(2). Let ε > 0 and 0 < λ < 1 be such that for any x ∈ K there is an inverse branch τ of T which isdefined on the ball B(T (x), ε) := z ∈ K | d(z, T (x)) < ε , has Lipschitz constant λ and τ(T (x)) = x.

Let F ∈ S(A). Let

K := ‖F‖α := supd(x,y)<ε

|F (x)− F (y)|d(x, y)α

, a := ‖A‖α := supd(x,y)<ε

|A(x)−A(y)|d(x, y)α


be Holder constants for F and A. Given x, y ∈ K with d(x, y) < ε let τi, i = 1, . . . , m(x) ≤ M be theinverse branches for T about x and let xi = τi(x), yi = τi(y). We have that

|F (xi)− F (yi)|K; λα d(x, y)α, |A(xi)−A(yi)|α λα d(x, y)α.

F (xi) + A(xi) ≤ F (yi) + A(yi) + (K + a) λα d(x, y)α,

maxi

[F (xi) + A(xi)−mA

] ≤ maxi

[F (yi) + A(yi)−mA

]+ (K + a) λα d(x, y)α,

F (x) ≤ F (y) + (K + a) λα d(x, y)α,

Then ‖F‖α ≤ λα (‖F‖α + ‖A‖α) and hence

(34) ‖F‖α ≤λα

(1− λα)‖A‖α .

This implies the equicontinuity of S(A).The proof of the above result could be also get if we just assume that F is continuous.

(3). It is easy to see that uniform limit of callibrated sub-actions is a sub-action, and it is callibratedbecause the number of inverse brancehs of T is finite, i.e. supy∈K#T−1y < ∞. By (2) all Cα callibratedsub-actions have a common Holder constant, the uniform limits of them have the same Holder constant.(4). The family Fn satisfies Fn(x0) = 0 and by inequality (34)

‖Fn‖α <λα

(1− λα)sup

n‖An‖α < ∞.

Hence Fn is equicontinuous. By Arzela-Ascoli theorem it is enough to prove that there is a uniqueF (x) = SA(x0, x) which is the limit of any convergent subsequence of Fn. Since sup ‖An‖α < ∞, byinequality (34), any such limit is α-Holder. Since by lemma 7.4.(2), S(A) ∩ [F (x0) = 0] = F (x) =SA(x0, x) , it is enough to prove that any limit of a subsequence of Fn is a calibrated sub-action. Butthis follows form the continuity of A 7→ mA, the equality

Fn(x) = maxT (y)=x

Fn(y) + An(y)−mAn

and the fact supx∈K

#(T−1x) < ∞.

(5). The proof follows from the expression

SA(x, y) := limε→0

[sup

n−1∑

i=0

[A(T i(z))−mA

] ∣∣∣ n ∈ N, Tn(z) = y, d(z, x) < ε]

.

(6). Let xn ∈ Bn → A be such that xn → x0. Let Fn ∈ S(A) be such that IFn(xn) = 0. Addinga constant we can assume that Fn(x0) = 0 for all n. By (2), taking a subsequence we can assumethat ∃F = limn Fn in the C0 topology. Then F is a Cα callibrated sub-action for A. Also RFn → RF

uniformly and there is a common Holder constant C for all the RFn . We have that

|RFn(T k(xn))−RF (T k(x0))| ≤∣∣RFn(T k(xn))−RFn(T k(x0))∣∣ +

∣∣RFn(T k(x0))−RF (T k(x0))∣∣

≤ C d(T k(xn), T k(x0))α + ‖RFn −RF ‖ n−→ 0

Since for all n, k, RFn(T k(xn)) = 0, we have that RF (T k(x0)) = 0 for any k. Hence IF (x0) = 0 and thenx0 ∈ N(A).

(7). By Lemma 7.4.(2), there is only one callibrated sub-action modulo adding a constant. Then byProposition 7.1, A(A) = N(A). Then by (6) lim supB→AA(B) ⊂ A(A). It is enough to prove that forany x0 ∈ A(A) and Bn → A, there is xn ∈ A(Bn) such that limn xn = x0. Let µn ∈ M(Bn). Thenlimn µn = µ in the weak* topology. Given x0 ∈ A(A) = supp(µ) we have that

∀ε > 0 ∃N = N(ε) > 0 ∀n ≥ N : µn(B(x0, ε)) > 0.


We can assume that for all m ∈ N, N( 1m ) < N( 1

m+1 ). For N( 1m ) ≤ n < N( 1

m+1 ) choose xn ∈ supp(µn)∩B(x0,

1m ). Then xn ∈ A(Bn) and limn xn = x0. .

(8). For any B ∈ F We have that

supp(νB) ⊆ A(B) ⊆ N(B).

By item 7,lim sup

B→Asupp(νB) ⊆ A(A) = supp(µ).

It remains to prove thatlim infB→A

supp(νB) ⊇ supp(µ).

But this follows from the convergence limB→A νB = µ in the weak* topology.

(9). Write M(A) = µ . By items (7) and (8) we have that

lim supB→A

M(B) ⊆ lim supB→A

A(B) ⊆ A(B),

A(A) = supp(µ) ⊆ lim infB→A

M(B).

¤

Proof of Theorem 7.1.The set

D := A ∈ F |#M(A) = 1 is dense (c.f. [9]). We first prove that D ⊂ R, and hence that R is dense.

Given A ∈ D with M(A) = µ and ε > 0, let ψ ∈ F be such that ‖ψ‖0 + ‖ψ‖α < ε ψ ≤ 0,[ψ = 0] = supp(µ). It is easy to see that M(A + ψ) = µ = M(A). Let x0 /∈ supp(µ). Given δ > 0,write

SA(x0, x0; δ) := sup n−1∑

k=0

A(T k(xn)) | Tn(xn) = x0, d(xn, x0) < δ

.

If Tn(xn) = x0 is such that d(xn, x0) < δ thenn−1∑

k=0

(A + ψ)(T k(xn)) ≤ SA(x0, x0; δ) +n−1∑

k=0

ψ(T k(xn))

≤ SA(x0, x0; δ) + ψ(xn).

Taking lim supδ→0,SA+ψ(x0, x0) ≤ SA(x0, x0) + ψ(x0) ≤ ψ(x0) < 0.

Hence x0 /∈ A(A+ψ). Since by lemma 7.1.(1), supp(µ) ⊂ A(A+ψ), then A(A+ψ) = supp(µ) and henceA + ψ ∈ R.

LetU(ε) := A ∈ F | dH(A(A),M(A)) < ε .

From the triangle inequality

dH(A(B),M(B)) ≤ dH(A(B),A(A)) + dH(A(A),M(B))

and items (7) and (9) of lemma 7.6, we obtain that U(ε) contains a neighbourhood of D. Then the set

R =⋂

n∈NU( 1

n )

contains a residual set.¤


8. Duality.

In this section we have to consider properties for A∗ which depends of the initial potential A. This isa main step in the reasoning of the main section 6.

We will consider now the specific example described before. We point out that the results presentedbellow should hold in general for natural extensions.

We will assume that T and σ are topologically mixing.So we take, K = [0, 1], T (x) = 2x mod 1, Σ = Πn∈N0, 1 , σ : Σ ← the shift map σ(x)n = xn+1 and

T : K× Σ → K× Σ,

T(x, ω) = (T (x), τx(ω)) , T−1(x, ω) = (τω(x), σ(ω))

τx(ω) = (ν(x), ω) ∈ Σ , τω(x) : =x

2+

ω0

2

ν(x) =

0 x ∈ [0, 1

2 [,1 x ∈ [ 12 , 1].

Given A ∈ F define ∆A : K×K× σ → R as

∆A(x, y, ω) :=∑

n≥0

A(τn,ω(x))−A(τn,ω(y))

whereτn,ω(x) = τσnω τσn−1ω · · · τω(x).

Fix x ∈ K and w ∈ Σ.Define the involution W -kernel as WA : K × Σ → R, WA(x, ω) = ∆A(x, x, ω). Writing A := A π1 :

K× Σ → R, we have that

W (x, ω) =∑

n≥0

A(T−n(x, ω))− A(T−n(x, ω))

Define the dual function A∗ : Σ → R as

A∗(ω) := (WA T−1 −WA + A π1)(x, ω).

Define a metric on Σ by

d(ω, ν) := λN , N := min k ∈ N |ωk 6= νk Then λ is a Lipschitz constant for both τx, and τω and also for T|x×Σ and T−1|K×ω

Write F := Cα(K,R) and F∗ := Cα(Σ,R). Let B and B∗ be the set of coboundaries

B : = u T − u | u ∈ Cα(K,R) ,B∗ : = u σ − u | u ∈ Cα(Σ,R) .

Define[z]α := ‖z‖0 + ‖z‖α .

Lemma 8.1.(1) z ∈ B ⇐⇒ z ∈ Cα(K,R) & ∀µ ∈M(T )

∫z dµ = 0.

(2) The linear subspace B ⊂ Cα(K,R) is closed.(3) The function

[z + B]α = infb∈B

[z + b]α

is a norm in F/B.

Proof.(1). This follows1 from [5], Theorem 1.28 (ii) =⇒ (iii).

1Theorem 1.28 of R. Bowen [5] asks for T to be topologically mixing.


(2). We prove that the complement Bc is open. If z ∈ Cα(K,R) \ B, by item (1), there is µ ∈ M(T )such that

∫z dµ 6= 0. If u ∈ Cα(K,R) is such that

‖u− z‖0 <12

∣∣∣∣∫

z dµ

∣∣∣∣

then∫

u dµ 6= 0 and hence u /∈ B.(3). This follows from item (2).

¤

Lemma 8.2.

(1) If A is Cα then A∗ is Cα.(2) The linear map L : Cα(K,R) → Cα(Σ,R) given by L(A) = A∗ is continuous.(3) B ⊂ kerL.(4) The induced linear map L : F/B → F∗/B∗ is continuous.(5) Fix one ω ∈ Σ. Similarly the corresponding linear map L∗ : F∗ → F , given by

L∗(ψ) = W ∗ψ T−Wψ + ψ π2

=∑

n≥0

Ψ(Tn(x, ω))−Ψ(Tn(Tx, ω))

= Ψ(x, ω) +∑

n≥0

Ψ(Tn(Tx, τxω))−Ψ(Tn(Tx, ω)),

with Ψ = ψ π2, is continuous and induces a continuous linear mapL∗ : F∗/B∗ → F/B, which is the inverse of L : F/B → F∗/B∗.

Proof.(1) and (2). We have that

A∗(ω) =∑

n≥0

A(T−n(x, ω))− A(T−n(x, σ ω))

= A(x) +∑

n≥0

A(T−n(τω x, σ ω))− A(T−n(x, σ ω))

Since d(T−n(τω x, σ ω),T−n(x, σ ω)) ≤ λn d(τω x, x) ≤ λn and ‖A‖α = ‖A π1‖α = ‖A‖α, we have that

‖A∗‖0 ≤ ‖A‖0 +‖A‖α

1− λα.

Also if m := min k ≥ 0 |wk 6= νk A∗(ω)−A∗(ν) =

∑

n≥m−1

A(T−n(τω x, σ ω))− A(T−n(x, σ ω))

−∑

n≥m−1

A(T−n(τν x, σ ν))− A(T−n(x, σ ν))

|A∗(ω)−A∗(ν)| ≤ 2 ‖A‖α

λ(m−1)α

1− λα=

2 ‖A‖α λ−α

1− λαd(ω, ν)α.

‖A∗‖α ≤2 ‖A‖α

λα(1− λα).

(3). If u ∈ F and U := u π1 from the formula for L (in the proof of item [1]) we have that

L(u T − u) = U(T(x, ω))− U(T(x, σω))

= u(T x)− u(T x) = 0.

(4). Item (4) follors form items (2) and (3).


(5). We only prove that for any A ∈ F , L∗(L(A)) ∈ A + B. Write

L∗(L(A)) =(W ∗

A∗ T−W ∗A∗ + A∗

)( · , ω)

= (W ∗A∗ T−W ∗

A∗ + WA T−1 −WA) + A

Write

(35) B := W ∗A∗ T−W ∗

A∗ + WA T−1 −WA.

Since A,L∗(L(A)) ∈ F = Cα(K,R), then B ∈ Cα(K,R).Following Bowen, given any µ ∈ M(T ) we construct an associated measure ν ∈ M(T). Given z ∈

C0(K× Σ,R) define z] ∈ C0(K,R) as z](x) := z(x, ω). We have that∥∥(z Tn)] Tm − (z Tn+m)]

∣∣0≤ varnz

n−→ 0,

where

varnz = sup |z(a)− z(b)| | ∃x ∈ K, a, b ∈ Tn(x × Σ) ≤ sup |z(a)− z(b)| | dK×Σ(a, b) ≤ λn n−→ 0,

dK×Σ = dK (π1, π1) + dΣ (π2, π2).Then ∣∣µ((z Tn)])− µ((z Tn+m)]

∣∣ =∣∣µ((z Tn)] Tm)− µ((z Tn+m)]

∣∣ ≤ varnz.

Therefore µ((z Tn)]) is a Cauchy sequence in R and hence the limit

ν(z) := limn

µ((z Tn)])

exists. By the Riesz representation theorem ν defines a Borel probability measure in K × Σ, and it isinvariant because

ν(z T) = limn

µ((z Tn+1)]) = ν(z).

Now let B := L∗(L(A)) − A and B := B π1. By formula (35) we have that B is a coboundary inK× Σ. Since π1 Tn = Tn we have that

0 = ν(B) = limn

µ((B Tn)])

= limn

µ(B Tn)

= µ(B).

Since this holds for every µ ∈ M(T ), by lemma 8.1.(1), B ∈ B and then(L+ L)(A + B) ⊂ A + B.

¤

Theorem 8.1.There is a residual subset Q ⊂ Cα(K,R) such that if A ∈ Q and A∗ = L(A) then

(36)M(A) = µ , A(A) = supp(µ),

M(A∗) = µ∗ , A(A∗) = supp(µ∗).

In particular

IA(x) > 0 if x /∈ supp(µ),

IA(ω) > 0 if ω /∈ supp(µ∗).

and

RA(x) > 0 if x /∈ supp(µ) and T (x) ∈ supp(µ),

RA(ω) > 0 if ω /∈ supp(µ∗) and σ(ω) ∈ supp(µ).


Proof. Observe that the subset R defined in (32) in theorem 7.1 is invariant under translations bycoboundaries, i.e. R = R+ B. Indeed if B = u T − u ∈ B, we have that∫

(A + B) dµ =∫

Adµ, ∀µ ∈ B,

SA+B(x, y) = SA(x, y) + B(x)−B(y), ∀x, y ∈ K.

Then the Aubry set and the set of minimizing measures are unchanged:M(A + B) = M(A), A(A + B) = A(A).

For the dynamical system (Σ, σ) let

R∗ = ψ ∈ Cα(Σ,R) | M(ψ) = µ, A(ψ) = supp(ν) By theorem 7.1, the subset R∗ contains a residual set in F∗ = Cα(Σ,R) and it is invariant undertranslations by coboundaries: R∗ = R∗ + B∗

By lemma 8.2 the linear map L : F/B → F∗/B∗ is a homeomorphism with inverse L∗. Then the setQ := R∩ L−1(R∗) = R∩ (L∗(R∗) + B) contains a residual subset and satisfies (36).

By Corollary 7.5 the other properties are automatically satisfied.¤

From this last theorem it follows our main result about the generic analytic g, by adapting the proofand taking T = f , where f is the transformation defined in section 2.

9. The optimal solution when the maximizing probability is not a periodic orbit

We are going to analyze now the variation of the optimal point when the support of the maximizingprobability is not necessarily a periodic orbit. What can be said in the general case?

Consider the subaction defined by,

V (x) = supw∈Σ

(H∞(w, x)− I∗(w))

Remember that as I∗ is lower semicontinuous and H∞ = W − V ∗ is continuous, then for each fixedx, the supremum of H∞(w, x)− I(w) in the variable w is achieved, and we denote (one of such w) it bywx. In this case we say wx is an optimal point for x.

We want to show that wx is unique for the generic xDefine the multi-valuated function U : [0, 1] → Σ given by:

U(x) = wx|x ∈ [0, 1]As graph(U) is closed in each fiber, and Σ is compact we can define:

u+(x) = max U(x), and u−(x) = min U(x).

Since the potential A is twist we know that U is a monotone not-increasing multi-valuated function,that is,

u−(x) ≥ u+(x + δ),when x < x + δ. In particular are monotone not-increasing single-valuated functions.

S

X1x x+d

u+(x)

u+(x + d)

u-(x)

u-(x + d)

0

The graph of U


We claim that u+ is left continuous. In order to conclude that, take a sequence xn → x on the leftside. Consider, the sequence u+(xn) ∈ Σ, so its set of accumulation points is contained in U(x). Indeed,suppose lim inf u+(xn) → w ∈ Σ. In one hand, we have, V (xn) = H∞(u+(xn), xn)− I∗(u+(xn)). Takinglimits on this equation and using the continuity of V and H∞ and the lower semicontinuity of I∗ we get,

V (x) ≤ H∞(w, x)− I∗(w).

Because lim inf I∗(u+(xn)) ≥ I∗(w). So w ∈ U(x). On the other hand, u+ is monotone not-increasing,so u+(xn) ≥ u+(x). From the previous we get

lim sup u+(xn) ≥ u+(x) ≥ w = lim inf u+(xn),

that is,lim

xn→x−u+(xn) = u+(x).

Now consider a sequence xn → x on the right side. Take, the sequence u+(xn) ∈ Σ, so its setof accumulation points is not necessarily contained in U(x). However it is the case. Let xnk

be asubsequence such that, u+(xnk

) → w.We know that V (xnk

) = H∞(u+(xnk), xnk

) − I∗(u+(xnk)). Taking limits on this equation and using

the uniform continuity of V and H∞ we get

I∗(w) ≤ lim infk→∞

I∗(u+(xnk)) =

= lim infk→∞

H∞(u+(xnk), xnk

)− V (xnk) = H∞(w, x)− V (x).

In other words, V (x) ≤ H∞(w, x)− I∗(w), that is, w ∈ U(x). So

cl(u+(xn)) ⊆ U(x).

Since u+ is monotone not-increasing, u+(xn) ≤ u+(x), thus

lim supu+(xn) ≤ u+(x),

that is, u+ is right upper-semicontinuous.It is known that for any USC function defined in a complete metric space the set of points of continuity

is generic.Therefore, we get that:

Theorem 9.1. For a generic x we have that U(x) = u+(x) = u−(x) and wx is unique.

The graph of u+

Proof. Indeed, suppose that there is a point in the set of continuity of u+(x) such that, u+(x) > u−(x)so the monotonicity of U implies that

u+(x) > u−(x) ≥ u+(x + δ),

for all δ > 0. Contradicting the continuity. ¤

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