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Another point of view on Kusuoka's measure

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, because its potential is discontinuous, while the standard theory of Gibbs measures requires continuous (actuallly, H\"older) potentials. In this paper, we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar Dirichlet forms on fractals. Moreover, we shall see that Kusuoka's measure can be recovered in a simple way from the matrix-valued Gibbs measure.


Introduction
First of all, let us briefly explain what we mean by a fractal G on R d ; our definition is less general than the one in [12]. We consider n contractions ψ 1 , . . . , ψ n ∈ C 1,ν0 (R d , R d ) (1) with ν 0 ∈ (0, 1]; it is standard ( [9]) that there is a unique compact set G ⊂ R d such that We shall also suppose that the maps ψ i are the "branches of the inverse" of a Borel map F : G → G. Since the maps {ψ i } n i=1 are contractions, their inverse F is expanding; in Dynamical Systems, expanding maps have been studied extensively (see for instance [13] or [19]); as we shall see, many results on expanding maps carry over to Dirichlet forms on fractals (see [9] or [17] for an introduction to this theory).
Applying ergodic theory to the study of Dirichlet forms on fractals is not new: in section 1.4 of [7] the idea of applying the Ruelle operator to the study of Kusuoka's measure is attributed to Strichartz. Actually, in this paper we try to understand the results of [7] and [12] by looking at them from a slightly different perspective.
In order to explain the connection between expanding maps and Dirichlet forms, we begin recalling the scalar Gibbs measure; though it will play no rôle in our paper, it will guide us in the construction of the matrix-valued one.
For ν 0 ∈ (0, 1] we take V ∈ C ν0 (G, R) and define the scalar Ruelle operator as Using the Perron-Frobenius theorem one can prove ( [18]) that there is β > 0 and a continuous function h > 0 such that L sc h = βh. Since L sc is a continuous operator from the space of continuous functions into itself, its adjoint L * sc brings the space of Borel measures into itself; it can be shown that there is a measure µ, called the Gibbs measure, such that hµ is probability and L * sc µ = βµ. One of the properties of µ is the following: for all u, v ∈ C(G, R) we have that We point out a few consequences of (3). First of all, we can write the adjoint L * sc explicitly, at least for measures absolutely continuous with respect to µ.
Moreover, (3) tells us that 1 β L sc v is the density of F ♯ (vµ), where F ♯ ν denotes the push-forward of the measure ν by F ; the push forward is defined by for all f ∈ C(G, R). Since L sc h = βh, this implies that F ♯ (hµ) = hµ, i. e. that hµ is an invariant measure.
Kusuoka's measure κ is defined by a similar scaling property; when the maps ψ i of (1) are affine, i. e.
Dψ i is a constant matrix, we have where Q,Q are two suitable symmetric d × d matrices, β > 0, t Q denotes the transpose of Q and tr is the trace. As explained in [7] (see [3] for the proof and further details), κ is a Gibbs measure, but for a discontinuous potential V , and the standard theory does not apply to it.
Let us come to the Dirichlet form. On many fractals, the space L 2 (S, κ) admits a "heat semigroup" P s which is induced, as in R d , by a Brownian motion ( [1], [2], [6], [11]). The semigroup P s has a generator A Af = lim Conversely ( [5]), given a Dirichlet form E it is relatively easy to check whether it is induced by a Brownian motion; since Dirichlet forms are easier to study than the Brownian motion itself, they immediately attracted attention ( [14], [15]; a counterexample to the existence of Dirichlet forms is in [16]).
This brings us to the matrix-valued Gibbs measure: under suitable hypotheses on the fractal G ⊂ R d , the "natural" Dirichlet form E on L 2 (G, κ) can be written in the following way: if u, v ∈ C 1 (R d , R), then where (·, ·) denotes the standard inner product in R d , T x is a Borel field of symmetric matrices (in many cases, projections) and κ is Kusuoka's measure.
The matrix-valued measure τ : = T x κ appears in a natural way in the formula above; the aim of this paper is to show that τ is a Gibbs measure as well.
More precisely, we denote by M d the space of symmetric d × d matrices; we can define a Ruelle operator The dual space of C(G, M d ) is the space M(G, M d ) of M d -valued measures on G, and the adjoint L * G of L G brings M(G, M d ) into itself. As we shall see, L G and L * G have each a positive-definite eigenvector, which we shall call Q G and τ G respectively. In lemma 4.8 below we shall prove the following version of (3): if g ∈ C(G, R) and A ∈ C(G, M d ), defining the integral as in section 2 below we have The formula above implies that the scalar measure (Q G , τ G ) HS (again, see section 2 for the definition) is invariant.

Theorem 1.
Let the maps ψ 1 , . . . , ψ n and F satisfy hypotheses (F1)-(F4) of section 1 below and (ND) with constant b > 0 at the beginning of section 4. Let G be the fractal associated with ψ 1 , . . . , ψ n and let M d denote the space of d × d symmetric matrices. Let the operator L G be defined as in (4) and let ||Dψ i || ν0 be smaller than a positive constant that depends on b > 0 (which is always true if (ND) holds and the maps ψ i are affine); then the following holds.
1) There are Q G ∈ C(G, M d ) and β > 0 such that The map Q G belongs to C ν0 (G, M d ) and is unique up to multiplication by a constant; again up to multiplication by a constant, Q G (x) is positive-definite for all x ∈ G.
2) Let L * G denote the adjoint of L G ; then, there is a Borel measure τ G on G which takes values in M d and such that The measure τ G is unique up to multiplication by a constant; again up to multiplication by a constant, τ G takes values in semi-positive definite matrices.
3) The measures ||τ G || and κ G : = (Q G , τ G ) HS are mutually absolutely continuous. Moreover, κ G is ergodic for the map F .

4)
We define the form E: → R in the following way (the notation for the integral is in section 2 below): Then, E is self-similar, i.e.
for all f, g ∈ C 1 (R d ).

5)
Let the maps ψ i be affine. Then, the measure τ G has the Gibbs property, i. e., for all x 0 , . . . , x l ∈ (1, . . . , n) we have that We haven't written explicitly the point where we calculate D(ψ x0 • . . . • ψ x l ) since these maps are affine and their derivative is constant.
Note that in point 4) we do not assert that E is closable: actually, we don't know any criteria for closability other than the ones in [9] and [12].
The paper is organised as follows. In section 1 we recall the notation and the basic facts about the Perron-Frobenius theorem, fractal sets and Dirichlet forms. In section 2 we define the convex cones to which we are going to apply the Perron-Frobenius theorem. In section 3 we define the Ruelle operator L G on matrices and show that the fixed points of its adjoint L * G induce a self-similar quadratic form E on C 1 (R d ). In section 4, we apply the Perron-Frobenius theorem to find the maximal eigenvector of L G and the matrix-valued Gibbs measure τ G . In section 5, we show that τ G has the Gibbs property. §1

Preliminaries and notation
The Perron-Frobenius theorem. We follow [19] (see also [4] for the original treatment).
Let X be a real vector space; we say that C ⊂ X \ {0} is a cone if v ∈ C and t > 0 implies that tv ∈ C.
Let C ⊂ X be a convex cone; we say that w ∈C if there are v ∈ C and t n ց 0 such that w + t n v ∈ C for all n ≥ 1. In what follows, we shall suppose that C is a convex cone such that Since θ(v, λv) = 0 for all λ > 0, we identify the points of a ray; namely, we say that for some t > 0; we shall denote by C ≃ the set of equivalence classes. We have that θ(v 1 , v 2 ) ∈ [0, +∞] for all v 1 , v 2 ∈ C; if θ never assumes the value +∞, then θ is a distance on C ≃ . The following proposition from [19] allows us to use the contraction principle.
Proposition 1.1. 1) Let L: X → X be a linear operator such that L(C) ⊂ C and let us define Then, if D < +∞, L is a contraction on ( C ≃ , θ), namely Moreover, if w ∈ C, then (1.5) Fractal sets. We make the following hypotheses on the fractal set.
(F1) There is ν 0 ∈ (0, 1] and diffeomorphisms By theorem 1.1.7 of [9], this implies that there is a unique non empty compact set G ⊂ R d such that In the following, we shall always rescale the norm of R d in such a way that If (F1) holds, then the dynamics of F on G can be coded. Indeed, we define Σ as the space of sequences with the product topology. This is a metric space; for instance, if γ ∈ (0, 1), we can define the metric with the convention that the inf of the empty set is +∞.
We define the shift σ as If x 0 , . . . , x l ∈ (1, . . . , n), we define the cylinder We also set From (1.7), (1.9) and (1.10) we get that Let {x i } i≥0 ⊂ Σ; by the last two formulas and the finite intersection property we have that is a single point, which we call Φ({x i } i≥0 ); formula (1.12) implies in a standard way that the map Φ: Σ → G is continuous. It is not hard to prove, using (1.8), that Φ is surjective. We shall calld the distance on G induced by the Euclidean distance on R d and, from now on, in our choice of the metric on Σ we take γ ∈ (η, 1); this implies by the definition of d γ and (1.12) that Φ is 1-Lipschitz.
(F3) We ask that G is post-critically finite, which means the following: if we set A = Φ −1 (F ), then the set We define a map F : since the maps ψ i are diffeos), this implies the first equality below.
We call a i the unique fixed point of ψ i ; note that, by (1.8), a i ∈ G. If x ∈ F , we define F (x) = a j for some arbitrary a j . This defines F as a Borel map on all of G, which satisfies (1.13).
We point out a consequence of (F4): the sets O i and ψ j (G) do not intersect unless i = j. By the definition of the coding this implies that, if z = Φ({x j } j≥0 ) ∈ O i , then x 0 = i. Since ψ x0 is a diffeo, also the sets ψ x0 (O j ) and ψ x0 • ψ l (G) do not intersect unless l = j. Thus, if in addition z ∈ n j=1 ψ x0 (O j ), z belongs to a unique ψ x0 (ψ x1 (G)) and also x 1 is uniquely determined. Going on, we see that z has a unique coding unless it belongs to the countable set n≥1 i0,...,in ψ i0...in (F ).
By (1.10) and the definition of Φ we easily get that [x 0 . . .
is not injective is countable, we have that The definition of F implies the first equality below; if we suppose that Φ(x) ∈ O x0 and recall that F = ψ −1 x0 on O x0 we get the middle one while the last equality comes from the formula above.
In other words, the first equality below holds save when Φ(x) ∈ F . The second equality below follows for all In other words, up to a change of coordinates, shifting the coding one place to the left is the same as applying F . Iterating the first one of (1.15) we get that, for all l ≥ 1, Note that the union on the right is a countable set, since Φ −1 (F ) is finite by (F3).
A particular case we have in mind is the harmonic Sierpinski gasket on R 2 ([8], [10]). We set Referring to the figure below, ψ 1 brings the triangle ABC into Abc; ψ 2 brings ABC into Bac and ψ 3 brings as three disjoint open sets which contain, respectively, the triangle Abc minus b, c, Bca minus c, a and Cba minus a, b.
We define the map F as and we extend it as in (F4) on {a, b, c}.
It is easy to check that the fractal G generated by ψ 1 , ψ 2 , ψ 3 satisfies hypotheses (F1)-(F4) above; it is easy to check that it also satisfies (ND) of section 4 below.
The invariant Dirichlet form. Let (X,d, ν) be a metric measure space; we suppose for simplicity that (X,d) is compact and ν is probability.
A Dirichlet form is a symmetric bilinear form such that the two conditions below hold.
(D1) D(E) is closed under the graph norm; in other words, D(E) is a Hilbert space for the norm We list some additional properties a Dirichlet form can have.
(D3) E is regular; this means that D(E) ∩ C(X, R) is dense in C(X, R) for the uniform topology and in D(E) for the graph norm (1.17).
whenever f, g ∈ D(E) and f is constant in a neighbourhood of the support of g.
It can be proven ( [5]) that, if E satisfies (D1)-(D4), then it is the Dirichlet form of a Brownian motion.
As we stated in the introduction, we shall be able to build a local and self-similar form on but not to prove its closability. §2

Function spaces and cones
We begin listing three equivalent ways to define the norm of a matrix in M d .
The first one is the sup norm We denote by t A the adjoint of the matrix A and by tr(A) its trace; on the space of all matrices we can define the inner product which induces the Hilbert-Schmidt norm If A is symmetric we can set where we have denoted by (·, ·) the inner product of R d .
Clearly, if A is symmetric, |||A||| is the modulus of the largest eigenvalue of A, while ||A|| HS is the quadratic mean of the eigenvalues; it is standard that there is D 1 > 0 such that For the converse, we note that, since A and B are symmetric and semi positive definite, their square roots are symmetric and real, which implies the first and third equalities below; for the second one, we use the fact that the trace of a product is invariant under cyclic permutations. Let now (E,d) be a compact metric space; in the following, (E,d) will be either one of (G,d) or (Σ, d γ ).
We define C(E, M d ) as the space of continuous functions from E to M d ; for A ∈ C(E, M d ) we define Putting on M d the Hilbert-Schmidt norm, we can define in the usual way the total variation ||τ || of a measure τ ∈ M(E, M d ); clearly, ||τ || is a scalar-valued, non-negative, finite measure on the Borel sets of E.
Several other products are possible; for instance, if u, v: E → R d are Borel vector fields such that we can define the real number If Q ∈ C(E, M d ) and τ ∈ M(E, M d ), we define the scalar measure (Q, τ ) HS in the following way: if B ⊂ E is Borel, then In other words, (Q, τ ) HS = (Q x , T x ) HS ||τ ||.
By Lusin's theorem we get in the usual way that, if B ⊂ E is a Borel set, then Remark. In the discussion above, we should have distinguished between M d and its dual (M d ) * ; strictly . In order to have a simpler notation, we identify M d and , we shall deal with its gradient ∇f and not with its differential df .
We shall say that In turn, by (2.3) this is equivalent to . Then, there is D 3 > 0 (depending on the constant D 2 of (2.7)) such that for all τ ∈ M + (E, M d ) and all Borel sets B ⊂ E we have where || · || denotes total variation. As a consequence, Proof. By the definition of total variation, we must find D 3 > 0 with the following property: for all Borel By (2.1), this follows if we show that, for the constant D 2 of (2.7), By the definition of |||τ (B i )||| before (2.1), we can find unit vectors v i such that Let now v ∈ R d ; the inequality below follows in a standard way from the fact that τ (B i ) is symmetric and non-negative-definite; the equality comes from the definition of the Hilbert-Schmidt product.
Since v i has unit length, the last two formulas imply the first inequality below; the first equality follows since This is (2.9) and we are done.
In order to prove the last assertion, we note that, by (2.6), clesed for the weak * topology; as a consequence, also P Q (E, M d ) is a closed convex set, while (2.8) implies that it is relatively compact for the weak * topology.

\\\
Definitions. Let (E,d) be a compact metric space with diam(E) ≤ 1. We define C + as the set of all the For a > 0 and ν ∈ (0, 1] we define C + (E, a, ν) as the set of all the A ∈ C + such that We also define C ν (E, M d ) as the set of all ν-Hölder maps from E to M d , with the seminorm As lemma 2.3 below shows, the last two formulas are two different ways to look at the same seminorm, but we shall need both.

Lemma 2.2.
Let ǫ > 0 and let A, B ∈ M d such that For fixed ǫ, the function D 3 (ǫ, B) is bounded when B is bounded.
As a converse, there is D 5 > 0 such that the following holds. Let A, B ∈ M d be semi-positive definite and let us suppose that there is D 4 > 0 such that (2.14) Then, Proof. We begin with the direct part. Let C ∈ M d ; it is easy to see (for instance, choosing a base in Since B satisfies (2.12), this implies the first inequality below.
This yields the inequality on the right of (2.13). We prove the inequality on the left; the first inequality below is (2.16) with the names changed, the second one follows from (2.12).
Again by (2.12) this implies that The left hand side of (2.13) now follows from two facts: the first one is that, for D 3 large enough, The second one is the formula below. The first inequality comes taking ||B − A|| HS ≥ ǫ, the second one taking γ > 0 so small that γB ≤ Id; the third one taking D 3 so large that We prove the converse. By (2.14) we have that By the definition of ||| · ||| and the fact that e D4 − 1 ≥ 1 − e −D4 this implies that Now (2.15) follows from (2.1).
2) There is D 6 > 0 such that for all A ∈ C + (E, a, ν) we have that Proof. We don't dwell on the proof of point 1), since it follows immediately from the definitions of C + and C + (a, ν).
Since we are supposing that the diameter of E is 1 (for E = G this is (1.9)), we get (2.17).
Note that the term on the right is finite since, by lemma 2.2, D 3 (ǫ, ·) is bounded on bounded sets and ||A|| ∞ is finite.

\\\ §3
The Ruelle operator From now on, we suppose that (F1)-(F4) hold; we let (G,d) be the fractal defined by (1.8) with the distanced induced by the immersion in R d .
We define the Ruelle operator L G on G as We also define a Ruelle operator Then, we define Note also that : x → Dψ i | Φ(x) is ν 0 -Hölder, since Dψ i is ν 0 -Hölder by (F1) and we saw in section 1 that Φ is Lipschitz; if A =Ã • Φ for someÃ ∈ C(G, M d ), we get by the second formula of (1.15) that The next lemma shows that the fixed points of the adjoint of L G , which we call L * G , induce a self-similar form on C 1 (R d ).
Let f, g ∈ C 1 (R d , R) and let us define, with the notation of section 2 for the integral, Then, Proof. By the polarisation identity, it suffices to show (3.3) when f = g; we shall take advantage of the fact that ∇f ⊗ ∇f ∈ C(G, M d ), i. e. is in the domain of L G . We recall that, if a ∈ R d and A ∈ M d , then (a, Aa) = (a ⊗ a, A) HS where by a ⊗ a we denote the tensor product of the column vector a with itself: a ⊗ a = a · t a.
As we shall see in the formula below, this explains the position of the transpose sign in (3.1).
The definition of E τ and the formula above imply the first equality below; the second one comes from the chain rule (recall that ∇(f • ψ i ) = t Dψ i · ∇f ) and the definition of L G ; this third one follows since L * G τ = βτ and the last one is again the definition of E τ .

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Remark. We can read lemma 3.1 as a statement about the push-forward of the measure τ , the positive eigenvector of L * . Indeed, let f, g ∈ C 1 (R d ); by (F3), f • F and g • F are not defined at the points of F , which are a finite set. As we shall see in lemma 5.3 below, the measure τ G of point 2) of theorem 1 is non-atomic; in particular, the points where f • F and g • F are not defined have measure zero. Together with (1.13), this implies the first equality below; the second one follows from lemma 3.1; the last one follows from the chain rule.
In other words, denoting by F * ω the pull-back by F of the two-tensor ω, we have that where the "push-forward" (F ♯ ) ♯ τ is defined as in the last term on the right in the formula above. Though this is not the standard push-forward operator, it is natural if we regard τ not as a measure, but as a linear operator on 2-tensors. §4

Fixed points of the Ruelle operator
We shall suppose that the maps {ψ i } n i=1 satisfy the following nondegeneracy condition; it is stronger than the one in [12], but it allows us to use the Perron-Frobenius theorem without modifications.
(ND) We suppose that, for all v ∈ R d \ {0} and all x ∈ G, the set { t Dψ i (x)v} n i=1 generates R d . Actually, we ask for a quantitative version of this, i. e. that there is b > 0 such that the following holds. Let v, v 0 ∈ R d and let x ∈ G; then, there isī ∈ (1, . . . , n), depending on x, v and v 0 , such that If we denote by P the orthogonal projection on v 0 , the formula above implies the inequality below.
It is easy to verify that the harmonic Sierpinski gasket of section 1 satisfies (ND); as we shall see in the next lemma, (ND) implies a bound from below on LA. Then, for all a > 0 there is D 1 = D 1 (a, b) > 0 such that the following happens. Let ν ∈ (0, ν 0 ] and let A ∈ C + (E, a, ν); then, for all x ∈ E, Proof. We prove the left hand side of (4.3); for the right hand side it suffices to note that L is continuous from the || · || ∞ topology to itself.
By compactness, there is x max ∈ E such that By the definition of |||A xmax ||| we can find v max ∈ R d with ||v max || = 1 such that By (2.11) and the fact that diam(E) = 1 we have that Let v ∈ R d and letī ∈ (1, . . . , n); the formula above implies the first inequality below. Since v max is an eigenvector of the symmetric matrix A xmax , we have that A xmax preserves the space generated by v max and its orthogonal complement. Thus, if we denote by P the orthogonal projection on v max , we get the second inequality below. Next, we chooseī in such a way that (4.1) holds with v 0 = v max ; the choice ofī depends on x and v. By (4.2) we get the third inequality below; the last equality comes from (4.5) and the last inequality from (2.1) and (4.4).
We chooseī as above; the definition of L implies the first inequality below; the second one comes from the formula above.

Lemma 4.2.
Let the maps {ψ i } n i=1 satisfy (F1)-(F4) and (ND) for some b > 0. Then, for all a > 0 there is ω(a, b) > 0 such that, if A ∈ C + (G, a, ν 0 ) and then for all x, y ∈ G we have that Note that, if the maps ψ i are affine and satisfy (ND), (4.6) is always verified.
Since A ∈ C + (a, ν) we can apply lemma 4.1 and get that there is D 6 = D 6 (a, b) such that We take and we recall that, if x ∈ [0, 1], 1 − 1 4 x ≥ e −x and 1 + 1 4 x ≤ e x .
From the last three formulas, (4.6) and the fact that diam(E) = 1 we get that which is (4.7) G .
We shall prove the lemma on G, since the proof on Σ is analogous. Let x, y ∈ G; the first equality below is the definition of L G in (3.1); the first inequality is the left hand side of (4.7) G and holds if ||Dψ i || ν0 ≤ ω(a, b).
The second one follows from two facts: the map : A → t BAB is order-preserving and A ∈ C + (a, ν 0 ), i. e.
it satisfies (2.11). The third inequality comes from the fact that η is the common Lipschitz constant of the maps ψ i , i. e. formula (1.7).
Since η ∈ (0, 1) we can choose a 0 so large that, for a ≥ a 0 , From the last two formulas we get that The opposite inequality follows similarly, implying (4.9).

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Definitions. Let λ 1 , ǫ > 0; we denote by C ǫ + (E, λ 1 a, ν 0 ) the subset of the A ∈ C + (E, λ 1 a, ν 0 ) such that, for all x ∈ E, (4.10) Moreover, we shall call θ (a,ν0) the hyperbolic distance on C + (E, a, ν 0 ) and θ + the hyperbolic distance on C + ; we recall that the hyperbolic distance on a cone has been defined at the beginning of section 1.

Lemma 4.4.
Let E be either one of G or Σ; let a > 0. Then, the following holds.
Since C + (E, a, ν 0 ) ⊂ C + , the definition of the hyperbolic distance implies that θ + ≤ θ (a,ν0) ; thus, {A n } n≥1 is Cauchy also for the θ + distance; the definition of θ + in (1.4) implies that By the definition of α + and β + in (1.2) and (1.3) respectively, we get that, for all x ∈ E, Let δ > 0; by (4.12) we have that, for n and m large enough and all x ∈ E, By the converse part of lemma 2.2 this implies that, for n and m large, By (4.11) and the triangle inequality this implies that β + (A m , A n ) → 1; again by the formula above, Step 2. We show that θ (a,ν0) (A n , A) → 0 as n → +∞. It is easy to see that θ (a,ν0) is lower semicontinuous under uniform convergence; together with step 1, this yields the first inequality below. Since {A n } n≥1 is Cauchy for θ (a,ν0) , there is δ n → 0 such that also the second inequality below holds.
End of the proof of point 1. It only remains to prove that A ∈ C + (E, a, ν 0 ). First of all, A satisfies (2.11), since this condition is closed under uniform convergence. We have to show that A x is positive-definite for all x ∈ E. We recall that A n,x is positive-definite for all x ∈ E, since A n ∈ C + (E, a, ν 0 ); since θ + (A n , A) < +∞, we have that α + (A n , A) > 0; since by (1.2) A ≥ α + (A n , A)A n and A n satisfies (2.10) for some ǫ > 0 (which depends on A n ), we get that A x is positive-definite for all x ∈ E.
Proof of point 2). The proof is in two steps: first, we show that and then that diam θ (a,ν 0 ) C ǫ + (E, λ 1 a, ν) ≤ diam θ + C ǫ + (E, λ 1 a, ν 0 ) + D 5 (λ 1 ). (4.14) Step 3. We prove (4.13); this follows if we show that C ǫ + (E, λ 1 a, ν 0 ) is compact in (C + , θ + ). Thus, let {A n } n≥1 ⊂ C ǫ + (E, λ 1 a, ν 0 ); we can suppose that {A n } n≥1 is normalised, i. e. that (4.11) holds. Now, point 2) of lemma 2.3 implies that the Hölder seminorm of A n is bounded. Thus, by Ascoli-Arzelà there is a subsequence {A n h } h≥1 which converges uniformly to A ∈ C(E, M d ); we see as in point 1) that A satisfies (2.11); it also satisfies (4.10) because this formula is stable under uniform convergence. Thus, λ 1 a, ν 0 ). Since A n h and A satisfy (4.10) and A n h → A uniformly, we can apply the direct part of lemma 2.2 and get that θ + (A n h , A) → 0, ending the proof of compactness.
Step 4. We prove (4.14) on the general space E with distanced. For starters, let us see how α + and α (a,ν0) are related. Let A 1 , A 2 ∈ C + (E, a, ν 0 ); by the definition of α (a,ν0) (A 1 , A 2 ) we have that which by (2.11) implies that, for all x, y ∈ E, Rearranging the terms of the inequality on the left, we get that for all x, y ∈ E. Since A 1 , A 2 satisfy (2.11) for λ 1 a, the formula above implies that We set and by a function study we see that D 6 ∈ (0, 1). By (4.15) we get that which by the definition of α + implies that Analogously, we can set A function study shows that D 7 > 1 and the same argument that yielded (4.16) yields Using this, (4.16) and the definition of θ + in (1.4) we get that which ends the proof of (2.18).
1) There is a simple, positive eigenvalue β of Denoting by Q the eigenfunction of β, we have that Q ∈ C + (a, ν 0 ). In particular, Q x is positive-definite for all x ∈ E. If the maps ψ i are affine, then there isQ ∈ M d such that Q x =Q for all x ∈ E.
2) Recall that after formula (2.7) we defined P Q (E, M d ); we assert that there is τ ∈ P Q (E, M d ) such that uniformly on E. Note that this implies that the measure τ of the previous point is unique and that β is simple eigenvalue of L. Moreover, if B ∈ C + (E, a, ν) with ν ∈ (0, ν 0 ], the convergence above is exponentially fast. Proof. Step 1. Let sup i∈(1,...,n) ||Dψ i || ν0 be so small that lemma 4.3 hold. Since L: is continuous, the left hand side of (4.3) implies that, possibly increasing the constant D 1 (a, b), Together with lemma 4.3 this implies that L(C + (E, a, ν 0 )) ⊂ C ǫ + (E, λ 1 a, ν 0 ) for some λ 1 ∈ (0, 1) and ǫ equal to the constant 1 D1(a,b) of the formula above; by point 2) of lemma 4.4 this implies that diam θ (a,ν) L(C + (E, a, ν 0 )) < +∞.
By point 1) of proposition 1.1, we get that L is a contraction of C + (E, a, ν) into itself; since Since Q is unique up to multiplication by a scalar, we can normalise it in such a way that ||Q|| ∞ = 1.
Step 2. If the maps Dψ i are constant, we see that, if A is a constant matrix, then LA is constant too.
Applying the Perron-Frobenius theorem to the positive cone of M d , we can find a constant, positive-definite matrixQ and β ′ > 0 such that LQ = β ′Q . By the uniqueness of step 1, we have that Q ≡Q.
Step 3. We prove point 2). We saw in lemma 2.1 that P Q (E, M d ) is a convex, compact set of M(E, M d ); thus, by Schauder's fixed point theorem, it suffices to show that 1 β L * brings P Q (E, M d ) into itself. Let τ ∈ P Q (E, M d ); we skip the proof that 1 β L * τ is non-negative definite (it follows easily by (2.3) and the definition of the adjoint), but we show that its integral against Q is 1. The first equality below is the definition of the adjoint, the second one is point 1) and the last follows sinceτ ∈ P Q (E, M d ).
Step 4. We prove point 3). Since L is linear and B = B + − B − with B + , B − ≥ 0, it suffices to prove (4.17) when B ∈ C(E, M d ) and B ≥ 0; in other words, when B ∈C + .
We begin to show that, if B ∈ C + , then It is clear that (4.19) follows from the three points below.
b) If a is large enough, θ (a,ν0) (L lB , Q) → 0. This follows sinceB ∈ C + (E, a, ν 0 ) and L is a contraction on C + (E, a, ν 0 ) by step 2. By (1.5), this convergence is exponentially fast. If we apply this argument tô B ∈ C + (E, a, ν 0 ) we get the last assertion of the thesis.
c) Since C + (E, a, ν 0 ) ⊂ C + , the definition of hyperbolic distance in section 1 immediately implies that θ + ≤ θ (a,ν0) ; by the triangle inequality, this implies the first inequality below; the second one comes from the fact that, since L(C + ) ⊂ C + , then Lip θ+ (L) ≤ 1.
Now the first term on the right is arbitrarily small by point a) and the second one tends to zero by point b).
We show how (4.19) implies (4.17) when B ∈ C + . We begin to note that, since L * τ = βτ by point 2) of the thesis, we have for all l ≥ 1 The right hand side in the formula above is 1 since τ ∈ P Q (G, M d ); this implies that Note that the numerator is 1 since τ ∈ P Q (G, M d ); the denominator is different from zero since B ∈ C + , (2.4) holds and τ ∈ P Q (E, M d ).
Recall that (4.21) and the last formula imply that uniformly. Now (4.17) follows from the last two formulas.
The last case is when B ∈C + \ {0}. In this case, we consider B + δId for δ > 0; since B + δId ∈ C + we have just shown that uniformly as n → +∞. We saw above that, since Id ∈ C + , uniformly. Now the thesis follows subtracting the last two formulas.

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Definition. By point 1) of proposition 4.5, the operator L G on C(G, M k ) has a couple eigenvalue-eigenvector which we call (β G , Q G ); the operator L Σ on C(Σ, M k ) has a couple eigenvalue-eigenvector which we call (β Σ , Q Σ ). By point 2) of proposition 4.5 there is a Gibbs measure on G, which we call τ G , and one on Σ, which we call τ Σ . We shall say that κ G : = (Q G , τ G ) HS is Kusuoka's measure on G and that κ Σ : = (Q Σ , τ Σ ) HS is Kusuoka's measure on Σ. Since τ G ∈ P QG and τ Σ ∈ P QΣ , κ G and κ Σ are both probability measures.
The next lemma shows that there is a natural relationship between these objects.
We have that β G = β Σ ; we shall call β their common value. Up to multiplying one of them by a positive constant, we have that Proof. The first equality below comes from the formula after (3.2) and the second one from the definition of L G in (3.1).
Since Q G is a fixed point of L G on C+(G,a,ν0) ≃ , the formula above implies that Q G • Φ is a fixed point of L Σ on C+(Σ,a,ν0) ≃ . By the uniqueness of proposition 4.5 we get that, up to multiplying one of them by a positive We prove the relation between the Gibbs measures. Let A ∈ C(G, M k ); the first equality below is the definition of the adjoint, the second one follows from the definition of push-forward; the third one comes from (4.23) while the fourth one comes from the fact that τ Σ is an eigenvector of L * Σ ; the last one comes again by the definition of push-forward.
Moreover, the fact that Q Σ = Q G • Φ easily implies that Φ ♯ τ Σ ∈ P QG ; since lemma 4.5 implies the uniqueness of the eigenvector of L * G in P QG , the last formula implies that τ G = Φ ♯ τ Σ . We leave to the reader the easy verification that κ G = Φ ♯ κ Σ .

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In order to prove that Kusuoka's measure (Q, τ ) HS is ergodic, we need a lemma. Proof. Let us define the measure t as if we are on G; on Σ we set By Lusin's theorem there is a sequence A k ∈ C(G, M d ) such that A k → A t-a. e. on G; moreover, ||A k || ∞ is bounded. By dominated convergence, this implies that Since A k is continuous, point 2) of proposition 4.5 implies that The thesis follows from the last three formulas.

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The next lemma recalls some properties of Kusuoka's measure. 2) The scalar measures κ G and κ Σ defined above are ergodic.
Proof. We begin with point 1) on Σ; we follow [PP].
First of all, we note that, if h: Σ → R is a bounded Borel function, then h • σ(ix) = h(x) for all i ∈ (1, . . . , n) and all x ∈ Σ. (4.25) By (3.2) this implies that, if A ∈ C(Σ, M d ), Integrating against τ Σ , we get the first equality below; the second equality comes from lemma 4.7.
In particular, if A = Q Σ where Q Σ is the eigenfuction of proposition 3.2, we have that Iterating (4.26) for h = g ∈ C(Σ, R) we get that Next, we show (4.24) when E = G. Anticipating on lemma 5.3 below, the measure τ Σ is non-atomic.
In particular, the countable set N ⊂ Σ on which Φ is not injective is a null set for κ Σ . Let g ∈ C(G, R) and let A ∈ C(G, M k ); the first equality below comes from lemma 4.6; the second one is the definition of push-forward; the third one comes from (1.16), which holds save on a null-set; the limit is (4.24) on Σ, which we have just proven. The last equality follows again from lemma 4.6.
This is (4.24) for G, ending the proof of point 1).
We prove point 2). First of all, since (4.27) holds for all h ∈ C(Σ, R) we get that σ ♯ κ Σ = κ Σ , i. e. that κ Σ is σ-invariant. With the same argument we used for the formula above this implies that Now we work on E, with E = G or E = Σ. Setting A = f Q for a continuous function f , (4.24) implies which implies that (Q, τ ) HS is strongly mixing; in particular, it is ergodic.

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At this stage it is natural to ask whether, when the maps ψ i are affine, Kusuoka's measure κ G coincides with (Q G , τ G ) HS ; in the remark at the end of section 5 we shall prove that this is the case. §5 The Gibbs property Let τ Σ be the positive eigenvector of L * as in lemma 4.5; we briefly prove that τ Σ (Σ) = 0. Let Q Σ be as in proposition 4.5; the inequality below comes from (2.4) and the fact that, for some ǫ > 0, ǫQ Σ (G) ≤ Id for all x ∈ G by compactness; the equality comes from the fact that τ Σ ∈ P Q (Σ, M d ).
We have the following analogue of proposition 3.2 of [18]. Letting A vary among the one-dimensional projections we get that To get (5.2) it suffices to multiply the formula above by 1 β · (Dψ x0 ) on the left and by t (Dψ x0 ) on the right. be as in proposition 4.5. then, τ Σ is a Gibbs measure for the constant C = log β.
Proof. Iterating the right hand side of (5.2) and using the chain rule we get the following.  Let the maps ψ i satisfy (F1)-(F4) and let (ND) hold. Then, we have the following.
1) The measure τ Σ is positive on open sets.

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Remark. In corollary (4.2) we have supposed that the maps Dψ i are constant, which is the case of Kusuoka's paper [12]. We prove that, up to multiplication by a positive constant, (Q G , τ G ) HS coincides with Kusuoka's measure κ.
When Dψ i is constant, point 1) of proposition 4.5 implies that Q G is constant too and solves By point 2) of lemma 3.1 we have that , dτ t Dψ i · ∇g| ψi(x) ).
If we choose as f and g two linear functions and we recall that Dψ i is a constant matrix, we see that the last formula implies that Dψ i · τ G (G) · t Dψ i .