On the uniqueness of an ergodic measure of full dimension for non-conformal repellers

We give a subclass L of Non-linear Lalley-Gatzouras carpets and an open set U in L such that any carpet in U has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general Sierpinski carpet has a unique ergodic measure of full dimension.


Introduction
It is well known that a C 1+α conformal repeller has a unique ergodic measure of full dimension. This is a consequence of Bowen's equation together with the classical thermodynamic formalism developed by Sinai-Ruelle-Bowen, see [14], [12], [3] and [13]. Moreover, this measure is a Gibbs state relative to some Hölder-continuous potential. Is this true for non-conformal repellers?
The simplest examples of non-conformal repellers are the general Sierpinski carpets, whose Hausdorff dimension was studied by Bedford [2] and McMullen [10]. They computed the Hausdorff dimension of these sets by establishing the variational principle for the dimension. As a consequence, these repellers have an ergodic measure of full dimension (in fact Bernoulli) and, by [11], this measure is unique.
In [6] Lalley and Gatzouras introduced a larger class of non-conformal repellers and computed their Hausdorff dimension also by establishing the variational principle for the dimension, and so these repellers have a Bernoulli measure of full dimension (see also [9] for a random version of this result). In [1] the authors give an example of a Lalley-Gatzouras carpet which has two Bernoulli measures of full dimension. So the answer to the question formulated above is negative.
In this paper, we study this problem -existence and uniqueness of an ergodic measure of full dimension -for a larger class of non-conformal repellers which we shall call Non-linear Lalley-Gatzouras carpets. As the name suggests, these repellers are the C 1+α non-linear versions of the Lalley-Gatzouras carpets. They are defined by an Iterated Function System {f ij } where f ij : [0, 1] 2 → [0, 1] 2 , i = 1, ..., m, j = 1, ..., m i have the skew-product form f ij (x, y) = (a ij (x, y), b i (y)), with the domination condition 0 < |∂ x a ij (x, y)| < |b ′ i (y)| < 1, and the corresponding attractor Λ (see Section 2 for precise definitions). The Hausdorff dimension of these repellers was, essentially, computed in [7] by establishing the variational principle for the dimension. Because of the non-linearity of the transformations f ij , the existence of an ergodic measure of full dimension turns out to be a non-trivial problem. This was proved to be true in [8] (in a more general context). Then we have the following. Theorem 1. A Non-linear Lalley-Gatzouras carpet has an ergodic measure of full dimension. Moreover, this measure is a Gibbs state for a relativized variational principle.
As we know now (by [1]), such a measure is, in general, not unique. The main purpose of this paper is to give sufficient conditions for having a unique ergodic measure of full dimension, based on an idea introduced in Remark 2 of [8].
We can introduce a natural topology on the class of Non-linear Lalley-Gatzouras carpets by saying that two of these carpets are close if the corresponding functions of the Iterated Function System are C 1+α close (with alphabet (i, j) fixed). We denote by L the subclass of Non-linear Lalley-Gatzouras carpets for which ∂ xx a ij = 0, i.e. a ij (x, y) =ã ij (y)x + u ij (y). Of course, L contains the Lalley-Gatzouras carpets. In this paper, a general Sierpinski carpet is a Lalley-Gatzouras carpet for which ∂ x a ij = a and b ′ i = b for some positive constants a and b and every (i, j) (this is a more general definition than usual). We say that such a carpet is non-trivial if a < b and the natural numbers m i ≥ 2, i = 1, ..., m are not all equal to each other.

Theorem 2.
There is an open set U in L such that: (i) U contains all non-trivial general Sierpinski carpets; (ii) every reppeller K in U has a unique ergodic measure of full dimension µ K ; (iii) the map U ∋ K → µ K is continuous.
We believe that Theorem 2 also holds in the class of Non-linear Lalley-Gatzouras carpets. The reason for restricting to the subclass L relies on the necessity of considering basic potentials in the relativized variational principle of [5], which we use, in order to have additional properties (see Remark 2). This paper is organized as follows. In Section 2 we introduce the class of Nonlinear Lalley-Gatzouras carpets and say how Theorem 1 follows from the works [7] and [8]. In Section 3, within the more general context of [8], we prove some properties of measures of maximal dimension, a relativized version of Ruelle's formulas for the derivative of the pressure, and a criterium for uniqueness of a measure of maximal dimension (Theorem 5). In Section 4 we use this criterium to prove Theorem 2.
When ∂ xx a ij = 0 we get the definition of a carpet in L. When the functions a ij and b i are linear and ∂ y a ij = 0, we get the definition of a Lalley-Gatzouras carpet, see [6] (where equality is allowed in (H4)). When, moreover, ∂ x a ij = a and b ′ i = b for some positive constants a and b and every (i, j), we get the definition of a general Sierpinski carpet, see [2] and [10] (in fact, our definition is a little more general).

2.2.
Hausdorff dimension. The Hausdorff dimension of Non-linear Lalley-Gatzouras carpets was, essentially, computed in [7] by establishing the variational principle for the dimension. In fact, the theorems in [7] are formulated in terms of a Dynamical System f instead of an Iterated Function System {f ij }, although in its proofs we mainly used the f ij approach. The relation between the two approaches is given by f ij = (f |R ij ) −1 where R ij is an element of a Markov partition for f . Beside imposing a skew-product structure for f (which translates to (H1)), we considered a C 2 perturbation of the 2-torus transformation f 0 (x, y) = (lx, my), where l > m > 1 are integers. The only reason for doing this is to inherit from the linear system a domination condition (which translates to (H4)) and a simple Markov partition (inducing a full shift) which is smooth. More precisely, the Markov partition is constructed using the invariant foliation by horizontal lines (due to the skew-product structure) and an invariant smooth vertical foliation, which exists because the vertical lines constitute a normally expanding invariant foliation for f 0 . In the present setting, all we need to show is that the sets have vertical boundaries formed by C 1 curves with uniformly bounded distortion for all n ∈ N. But, as we shall see, this is a consequence of the domination condition (H4). Let We will see that each f ij transforms vertical graphs with distortion ≤ C into vertical graphs with distortion ≤ C.
Then, starting with the vertical graphs {0} × [0, 1] and {1} × [0, 1] and using induction on n, we get the desired property for the sets Then it follows from the proof of Theorem A in [7] that, there exists A > 0 such that, for every n ∈ N, where Λ n is a Lalley-Gatzouras carpet defined using an appropriate linearization of the functions More precisely, given n ∈ N, consider the n-tuples i = (i 1 , ..., i n ) and j = ( Consider the numbers Let p n = (p n i ) be a probability vector in R nm . We define and t n (p n ) as being the unique real in [0, 1] satisfying Theorem 3 (Proof of Theorem A, [7]). Let ({f ij }, Λ) be a Non-Linear Lalley-Gatzouras carpet. There exist constants A, B > 0 such that, for every n ∈ N, Remark 1. The continuity of ({f ij }, Λ) → dim H Λ follows from the Proof of Corollary A in [7]. In fact, there we used the C 2 topology but it is clear that we can use the C 1+α topology.
As a consequence, the variational principle for dimension holds, i.e. the Hausdorff dimension of Λ is the supremum of the Hausdorff dimension of ergodic measures (with respect to {f ij }) on Λ. In [8] we prove the existence of an ergodic measure of full dimension for Λ, which is a Gibbs state for a relativized variational principle. Thus we have Theorem 1.

Properties of measures of maximal dimension
The results given in this section hold in the more general context of [8]. We consider (X, T ) and (Y, S) mixing subshifts of finite type such that (Y, S) is a factor of (X, T ) with factor map π : X → Y . Assume that each fibre π −1 (y) has at least two points.

Characterization of measures of maximal dimension.
We use the following notation: M(T ) is the set of all T -invariant Borel probability measures on X; h µ (T ) is the metric entropy of T with respect to µ ∈ M(T ).
Let ϕ : X → R and ψ : Y → R be positive Hölder-continuous functions. We define Note that if µ is ergodic then D(µ) might be interpretated as the Hausdorff dimension of the measure µ (see Remark 5 of [8]). We say that µ is a measure of maximal dimension if D(µ) = D. In [8] we prove the existence of an ergodic measure of maximal dimension, and give a characterization of measures of maximal dimension that we shall describe now (for more details see this reference). We use the following version of the relativized variational principle by [4] and [5]. Given an Hölder-continuous function φ : X → R and ν ∈ M(S), there exists a positive Hölder-continuous function A φ : Y → R (not depending on ν) such that Moreover, there is a unique measure µ for which the supremum in (1) is attained which we call the relative equilibrium state with respect to φ and ν, and µ is ergodic if ν is ergodic. Given ν ∈ M(S), there is a unique real t(ν) ≥ 0 such that Y log A −t(ν)ϕ dν = 0.
Then it easy to see that Throughout this paper we assume D and t are uniformly bounded (with respect to ψ and ϕ), since in applications these numbers have dimension interpretations. We assume the following technical condition: (H) the supremum in (2) is not attained at an ergodic measure ν with t(ν) = t or t.
Let P (·) denote the classical Pressure function with respect to (Y, S), and let ν g denote the corresponding Gibbs state with respect to the Hölder-continuous potential g : Y → R. Given t ∈ (t, t), let where β(t) is the unique real satisfying log A −tϕ dν Φt = 0 (see [8] for details). Finally, let µ Φt be the relative equilibrium state with respect to −tϕ and ν Φt . The following result follows from the proof of Theorem A and Remark 3 in [8].
Let φ t : Z → R be a one-parameter family of continuous functions. We say that t → φ t is differentiable if its partial derivative in t exists, let us call itφ t or d dt φ t , and it is a one-parameter family of continuous functions.
Then the following result follows from [13].
where Q φt (·, ·) : H C,θ (Y ) × H C,θ (Y ) → R is given by There exists a constant B > 0 (depending only on C and θ) such that Also, for each h 1 , h 2 ∈ H C,θ (Y ), is a continuous function.
Now we recall some definitions from [4] and [5] that are used to define A φ , for φ ∈ H C,θ (X). Given y ∈ Y , let C y denote the space of bounded continuous functions f : π −1 (y) → R. For each y ∈ Y and n ∈ N, define the operators G (n) y and P (n) y : C y → C y by .

Then (see Proposition 2.5 of [5]),
, uniformly in y ∈ Y , x ∈ π −1 (y). Moreover (see Corollary 4.14, Remark 4.16 and Proposition 5.5 of [4]), the rate of convergence is exponential depending only in C and θ. Also, for any y ∈ Y , the operators P (n) y converge to a conditional expectation operator P y which gives a probability measure µ y in π −1 (y), in the sense that (P y f )(x) = f dµ y , for any x ∈ π −1 (y).
The system {µ y : y ∈ Y } is called a Gibbs family for φ.
We will use the following property of A φ . Given y ∈ Y , consider the operators V y : C y → C S(y) and U y : C S(y) → C y given by and (U y f )(x) := f (T (x)).
(Note that the operators G (n) y , P (n) y , P y and V y depend on the potential φ.) We say that φ ∈ H C,θ (X) is a basic potential (see Definition 4.1 of [5]), if for y ∈ Y and x ∈ π −1 (S(y)) we have (7) A φ (y) = (V y 1)(x), i.e., for each y ∈ Y , the function V y 1 is constant. In this case we have the following.
for each y ∈ Y ; (b) the relative equilibrium state for (1) with respect to φ and ν is given by µ = µ y × ν; Now we are ready to prove the following.
Then t → A −tϕ is differentiable and Proof. The differentiability of t → A −tϕ is an immediate consequence of (7), and (where V t,y is V y with the potential φ = −tϕ). In particulary, (V t,y ϕ)(x) does not depend on x ∈ π −1 (S(y)). Then applying (6) to ϕ we get A −tϕ (y)P t,y ϕ = V t,y ϕ, which together with (10) gives (8). The Hölder-continuity of d dt log A −tϕ follows from Theorem 2.10 of [4].
Recall the definition of Φ t from (3).

3.3.
Criterium for uniqueness of measure of maximal dimension. Now we give sufficient conditions for having d 2 P (Φt) dt 2 < 0 which, by Theorem 4, implies the existence of a unique measure of maximal dimension (the existence follows from Theorem A in [8]), as already noticed in Remark 3 in [8].
Therefore, ifψ,φ are as described in statement of Theorem 5, there is a unique measure of full dimension µψ ,φ , and we can infer about its continuity.
Then, Theorem 2 follows from applying Theorem 5 to carpets in L which are C 1+α close to a non-trivial general Sierpinski carpet.