Entropy and Variational principles for holonomic probabilities of IFS

Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$, defined by $\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$ were $\Sigma=\{0,1, ..., d-1\}^{\mathbb{N}}$, $\sigma: \Sigma \to \Sigma$ is given by$\sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : \Sigma \to \{0,1, ..., n-1\}$ is the projection on the coordinate $k$. A $\rho$-weighted system, $\rho \geq 0$, is a weighted system $([0,1], \tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \to \mathbb{R}$ and probability $\nu $ on $[0,1]$ satisfying $ P_{u}(h)=\rho h, \quad P_{u}^{*}(\nu)=\rho\nu$. A probability $\hat{\nu}$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma}$ if $ \int g \circ \hat{\sigma} d\hat{\nu}= \int g d\hat{\nu}, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by ${\cal H}$. Via disintegration, holonomic probabilities $\hat{\nu}$ on $[0,1]\times \Sigma$ are naturally associated to a $\rho$-weighted system. More precisely, there exist a probability $\nu$ on $[0,1]$ and $u_i, i\in\{0, 1,2,..,d-1\}$ on $[0,1]$, such that is $P_{u}^*(\nu)=\nu$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(\phi)=$$

There is no meaning to ask if the probabilities ν on [0, 1] arising in IFS are invariant for a dynamical system, but, we can ask if probabilitiesν on [0, 1] × Σ are holonomic forσ.
Finally, we analyze the problem: given φ ∈ B + , find the solution of the maximization problem We show na example where such supremum is attained by a holonomic not-invariant probability.

IFS and holonomic probabilities
We want to analyze, in the setting of holonomic probabilities [6] associated to an IFS, the concepts of entropy and pressure. We point out that this is a different problem from the usual one among invariant probabilities (see remarks 3 and 4 in section 7).
The present work is part of the PhD thesis of the second author [13].
If we consider a IFS as a multiple dynamical systems (several maps) then, for a single point x there exists several combinations of "orbits" on the IFS (using different τ i ). Considering the mapσ one can describe the global behavior of iterates of x. Moreover, one can think the IFS, as a branching process with index in Σ. More precisely, we define the n-branch from x ∈ [0, 1] by w ∈ Σ, as With this notation, we havê σ n (x, w) = (Z n (x, w), σ n (w)).
Definition 2. A weighted system (see [15], Pg. 6) is a triple, The function φ is called weight function, (in the literature this function is also called g-function, see [10] for example). Note that φ can attain the value 0. This is useful for some applications of IFS to wavelets [3] In this case, we write the IFS as The above definition is a strong restriction in the weighted system. Several problems in the classical theory of Thermodynamic Formalism for the shift or for a d to 1 continuous expanding transformations T : S 1 → S 1 can be analyzed via a IFS with a weight function φ (see [14]). In this case the τ i , i ∈ {0, 2, .., d − 1}, are the inverse branches of T .
We will consider later the pressure problem for a weight function φ which is not necessarily uniformly normalized.
We now return to the general case.
Definition 5. Given a weighted system, ([0, 1], τ i , u i ), we will define de Transference Operator (or Ruelle Operator) P u by , The correct approach to analyze an IFS [5] [1] [9] [7] [8] with probabilities ([0, 1], τ i , u i ), is to consider for each x ∈ [0, 1], the sequence of random variables (Z n (x, .) : Σ → [0, 1]) n∈N as a realization of the Markov process associated to the Markov chain with initial distribution δ x and transitions of probability P u . Moreover, we have a probability P x in the space of paths, Σ, given by when g = g(x, w) depends only of the n first coordinates (see [9], for a proof of the Komolgorov consistence condition).
The probability on path space and the transference operator are connected by when g(x, w) = f (τ wn ...τ w 1 (x)) for some continuous f .
such that there exists a positive bounded function h : [0, 1] → R and ν probability satisfying Note that a IFS with probabilities is a 1-weighted system (see [4], [15] or [17] for the existence of P u -invariant probabilities and [15], Theorem 4, or [16] for non-uniqueness of this probabilities). Also, a weighted system, Thus the set of ρ-weighted systems is as big class of weighted systems.
Moreover, from a ρ-weighted system ([0, 1], τ i , u i ) one can get a normalization ([0, 1], τ i , v i ), in the following way Then P v (1) = 1 and P * v (µ) = µ. We thanks an anonymous referee for some comments on a previous version of the present paper. We would like to point out that there exists some similarities of sections 1, 2, 3 and 5 of our paper with some results in [7] and [8]. We would like to stress that we consider here the holonomic setting which can not be transfer by some coding to the usual shift case (see remark 4 in section 7). We introduce for such class of probabilities in IFS (which is different from the set of invariant probabilities forσ) the concept of entropy and pressure. It is not the same same concept of entropy as for a measure invariant for the shiftσ (see remarks 3 and 4 in section 7). Also, in our setting, it is natural to consider the all set of possible potentials u. In this way our results are of different nature than the ones in [7] [8] where the dynamical concepts are mainly consider for the shiftσ acting on [0, 1] × Σ.
In sections 1 to 6 we consider the basic definitions and results. In sections 7 and 8 we introduce entropy and pressure for holonomic probabilities of IFS.

Holonomic probabilities
For IFS we introduce the concept of holonomic probability on [0, 1] × Σ (see [6] for general definitions and properties in the setting of symbolic dynamics of the two-sided shift in Σ × Σ). Several results presented in [6] can be easily translated for the IFS setting. In [6] the main concern was maximizing probabilities. Here we are mainly interested in the variational principle of pressure.
By the other hand, some of the new results we presented here can also be translated to that setting.
Then the set of holonomic probabilities can be viewed as the set of probabilities on [0, 1] × Σ such that From this point of view it is clear that the set of holonomic probabilities is bigger (see section 4) than the set ofσ-invariant probabilities (because C([0, 1]) can be viewed as a subset of C([0, 1] × Σ)).

Characterization of holonomic probabilities
Disintegration of probabilities for IFS have been previously consider but for a different purpose [3], [4], [8]. Therefore, all spaces considered here are Radon spaces.
, Pg 78, (70-III) or [19], Theorem 5.3.1) Let X and X be a separable metric Radon spaces,μ probability on X, π : X → X Borel mensurable and µ = π * μ . Then there exists a Borel family of probabilities {μ} x∈X on X, uniquely determined µ-a.e, such that, This decomposition is called the disintegration of the probabilityμ.

Theorem 11. (Holonomic Disintegration) Consider a holonomic probabilitŷ
be the disintegration given by Theorem 10. Then ν is P u -invariant for the IFS with probabilities ([0, 1], τ i , u i ) i=0..d−1 , were the u i 's are given by, Proof. Consider a continuous function f : [0, 1] → R and defines Now applying the disintegration for both integrals we get On the other hand

Invariance of Holonomic probabilities on IFS
As we said before, holonomic probabilities are not necessarily invariant for the mapσ. On the other hand allσ-invariant probability is holonomic. Now we show an example of holonomic probability which is notσ-invariant (see [6] for the case of the two sided shift).
Suppose that x 0 ∈ [0, 1], is such that Z n (x 0 ,w) = x 0 , for somew ∈ Σ, n ∈ N. Then, one can obtain a holonomic probability in the following waŷ Then, Note that this probability is holonomic but notσ-invariant. In fact, it is enough to see that , and it is clearly not identical to 0, ∀g.
However,ν is holonomic because given any continuous function f :
Let Z n (·) : [0, 1] ←֓, n ∈ N, be a sequence of random variables on [0, 1]. Then, we obtain a Markov process with transition of probabilities P u and initial distribution ν, that we will denote by (Z n , P u , ν).
This process is a stationary process by construction, thus does make sense to ask if (Z n , P u , ν) is ergodic( [5] for details of this process and definition of ergodicity).

Definition 12.
A holonomic probabilityν is called ergodic, if the associated Markov process (Z n , P u , ν) is an ergodic process.
Theorem 14. (Elton, [5]) Let ([0, 1], τ i ) be a contractive IFS (contractiveness means that τ i is a contraction for all i) andν be a ergodic holonomic probability with holonomic disintegration ([0, 1], τ i , u i ). Suppose that Proof. The proof is a straightforward modification of the one presented in Elton's ergodic theorem (see [5] [8]). In fact, the contractiveness of ([0, 1], τ i ) its stronger that Dini condition that appear in Elton's proof (see [5] and [17]) and the ergodicity ofν can replace the uniqueness of the initial distribution in the last part of the argument. The other parts of the proof are the same as in [5].
We point out that Elton's Theorem is not the classical ergodic theorem forσ. The claim is: ∀x ∈ [0, 1] there exists G x ⊆ Σ such that P x (G x ) = 1 and for each w ∈ G x ... Moreover, f : [0, 1] → R.
This theorems fits well for holonomic probabilities in the IFS case. We just mention it in order to give to the reader a broader perspective of the holonomic setting. We do not use it in the rest of the paper.
Remark 1. We point out that that the holonomic liftingμ of a given µ (as above) is aσ-invariant probability (one can see that by taking functions that depends only of finite symbols and applying de definition of a P x probability). So We will consider in the next sections the concept of pressure. The value of pressure among holonomic or among invariant will be the same. However one cannot reduce the study of variational problems involving holonomic probabilities to the study ofσ-invariant probabilities. This will be explained in remark 4 in example 3 after Theorem 23.
Then, we have u i (y) =ν y (y,ī) = (δ y × P y )(y,ī) = P y (ī) = v i (x), ν − a.e . The central idea in this section is to consider a generalization of the definition of entropy for the case of holonomic probabilities via the concept naturally suggested by Theorem 4 in [11]. We will show that under such point of view the classical results in Thermodynamic Formalism are also true.

From this argument we get the following proposition
Givenν ∈ H we can define the functional αν : Let αν be the functional defined above. Observe that αν doesn't depend of ψ.

From above we get
The above definition agrees with the usual one for invariant probabilites when it is consider a transformation of degree d, its d-branches and the naturally associated IFS (see [11]). This lemma follows from the choice ε = exp( 1 d−1 (ln( d−1 i=0 a i ))/( d−1 i=0 ln(a i )) and the fact that a i ≥ 1 + α.
Proof. Asν is holonomic, then we have This can be written as Proof. Initially, consider ψ = 1. We know that h(ν) = inf f ∈B + ln( P 1 f f )dν ≤ ln( P 1 1 1 )dν = ln(d). Now, in order to prove the inequality f •τ i f )dν and suppose, without lost of generality, that 1+α ≤ f ≤ β (because this integral is invariant under the projective function f → λf ). Then, we can write this integral as In order to use the inequality obtained from Lemma 17, denote (for each fixed x) Then, we get by the compactness of [0, 1]. From this choice we get Using (2) in (1) we obtain Moreover, using the inequality from Lemma 18 applied to the function ln(εf ) (note that ln(ε 0 f ) ∈ B + , because ε 0 ≥ 1), we get Using the formula for entropy we get a characterization of topological pressure as
Remember that Letν 0 be a fixed holonomic probability such that the normalized dual operator verifies P * u (π * ν0 ) = π * ν0 (always there exists if P u is the normalization of P φ ), were π(x, ω) = x. Thus we can write Note that, from the normalization property we get Moreover, we know that ln(P u g) ≥ P u ln(g), ∀g, by concavity of logarithmic function. Now, considering an arbitrary f ∈ B + , we get So, inf f ∈B + ln( , that is, p(φ) ≥ ln(ρ). From this we get p(φ) = ln(ρ).
In order to obtain the second part of the claim it is enough to see that p(φ) = ln(ρ), for all ρ, thus the eigenvalue is unique.
Remark 4. This shows that suchν 0 is a maximal entropy holonomic probability which is notσ-invariant. This also shows that the holonomic setting can not be reduced, via coding, to the analysis ofσ-invariant probabilities in a symbolic space. Otherwise, in the symbolic case a probability with support in two points would have positive entropy.
From this example one can see that there exists equilibrium states which are notσ-invariant probabilities.
Definition 24. Two functions ψ 1 , ψ 2 ∈ B + will be called holonomic-equivalent (or, co-homologous) if there exists a function h ∈ B + , such that It is clear that, two holonomic equivalent potentials ψ 1 , ψ 2 will have the same equilibrium states.
8 An alternative point of view for the concept of entropy and pressure for IFS Definition 25. Givenν ∈ H and let ([0, 1], τ i , v i ) be the IFS with probabilities arising in the holonomic disintegration ofν (see Theorem 11). We can also define the Entropy ofν by Proposition 26. Considerν ∈ H. Then, Proof. Firstly consider u 0 On the other hand, u i ≤ 1 so ln(u i ) ≤ 0, and then Thus, 0 ≤ h(ν).
Remember that the normalization of φ is given by Replacing this expression in the equation for entropy we get Now, we use the concept introduced in the present section.
Proof. Let P u be the normalization of P φ . Then, The equality follows from the Lemma 26 From Theorem 29 and Lemma 26, it follows that there exists equilibrium states, more precisely, given a ρ-weighted system, all holonomic liftings of the normalized probability, are equilibrium states.
The Variational principle in the formulation of the present section is stronger than the formulated in the first part. The change in the definition of entropy allow us to get a characterization of the equilibrium states as holonomic liftings of the P u -invariant probabilities of the normalized transference operator. This point will become clear in the proof (of the "if, and only if," part) of the next theorem.
Proof. By Lemma 26, it follows that: ifν 0 is the holonomic lifting of the normalized probability π * ν0 , thenν 0 is an equilibrium state. The converse is also true. In fact, suppose thatν 0 is a equilibrium state, that is, h(ν 0 ) + ln(φ)dν 0 = ln(ρ) Using the normalization we get, where ([0, 1], τ i , v i ) is the 1-weighted system associated to holonomic disintegration ofν 0 . From the invariance of P v we have Finally, from the relations of normalization P u of P φ This is equivalent to it follows that u i = v i , π * ν0 − a.e. As, π * ν0 is P v -invariant, we get P * u (π * ν0 ) = π * ν0 Finally, we point out that for φ fixed one can consider a real parameter β and the problem p(φ β ) = sup ν∈H {h(ν) + β ln(φ)dν}.
For each value β, denote byν β a solution (therefore, normalized) of the above variational problem. Any subsequence (weak limit)ν βn → ν will determine a maximizing holonomic probability ν (in the sense, of maximizing supν ∈H { ln(φ)dν }) because the entropy of any holonomic probability is bounded by ln d. We refer the reader to [6] for properties on maximizing holonomic probabilities and we point out that these results apply also for the iterated setting as described above in the first two sections.