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On invariant measures of finite affine type tilings

In this paper, we consider tilings of the hyperbolic 2-space, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T, we associate a space of tilings: the continuous hull Omega(T) on which the affine group acts. This space Omega(T) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First we prove the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally we give examples with an arbitrary number of ergodic invariant probability measures.


Introduction
Let N be either the hyperbolic 2-space H 2 , identified with the upper half complex plane: {z ∈ C |Im(z) > 0} with the metric ds 2 = dx 2 +dy 2 y 2 , or the Euclidean plane R 2 .
A tiling T = {t 1 , . . . , t n , . . .} of N , is a collection of convex compact polygons t i with geodesic borders, called tiles, such that their union is the whole space N , their interiors are pairwise disjoint and they meet full edge to full edge. Let G denote a Lie group of isometries of N preserving the orientation. A tiling is said of G-finite type if there exists a finite number of polygons {p 1 , . . . , p n } called prototiles such that each t i is the image of one of these polygons by an element of G. For instance, when F is a fundamental domain of a discrete cocompact group G of isometries of N , then {γ(F ), γ ∈ G} is a tiling of N . However the set of finite type tilings is much richer than the one given by discrete cocompact groups. When N = R 2 , R. Penrose [15] gave an example whose set of prototiles is made with teen rhombi: the Penrose's tiling. When N = H 2 , Penrose also constructed a finite type tiling made with a single prototile which is not stable for any Fuchsian group. This example is the typical example of tilings studied in this paper. The construction goes as follows. Let P be the convex polygon with vertices A p with affix (p − 1)/2 + i for 1 ≤ p ≤ 3 and A 4 : 2i + 1 and A 5 : 2i (see figure 1): P is a polygon with 5 geodesic edges. Consider the two maps: R : z → 2z and S : z → z + 1.
The hyperbolic Penrose's tiling is defined by T = {R k • S n P |n, k ∈ Z} (see figure 2). This tiling is an example of P-finite type tiling where P denote the group of affine maps i.e. isometries of H 2 of the kind z → az + b with a, b reals and a > 0. The argument of Penrose is a homological one: he associates with the edge A 4 A 5 a positive charge and two negative charges with edges A 1 A 2 , A 2 A 3 . If T was stable for a Fuchsian Figure 1: The prototile P group, then P would tile a compact surface. Since the edge A 4 A 5 can meet only the edges A 1 A 2 or A 2 A 3 , the surface has a neutral charge. This is in contradiction with the fact P is negatively charged. G. Margulis and S. Mozes [12] have generalized this construction to build a family of prototiles which cannot be used to tile a compact surface. Notice the group of isometries which preserves T is generated by the transformation R. In order to break this symmetry, it is possible to decorate prototiles to get a new finite type tiling which is not stable for any non trivial isometry (we say in this case that the tiling is aperiodic). Using the same procedure, C. Goodmann-Strauss [10] construct a set of polygons which can tile H 2 only in an aperiodic way.
To understand the combinatorial properties of a tiling, it is useful to associate with this tiling, a set of tilings that we can study both from a geometric and dynamical point of view. The image of a G finite type tiling T by an element of G is again a G finite type tiling. We consider a compact metric space Ω(T ), which is the completion of the set of tilings image of T by elements of G, for a natural metrizable topology defined in section 2. The space Ω(T ) is called the continuous hull of T . The group G acts continuously on this space. In this paper we are mainly interested in the situation when the G-action on the hull is free (without fixed point). This is the case for the P-action on the hulls of examples in [10] as well as for the translation group action on the hull of the Euclidean Penrose's tiling. In this case, the G-action induces a specific laminated structure on the hull: a G-solenoid structure, where leaves are orbits for the group G-action (see section 2). The combinatorics properties of the tiling T are related to geometrical properties of Ω(T )and dynamical properties of (Ω(T ), G). In particular, the distribution of tiles of the tiling, which is our main interest for this paper, can be described by the statistical properties of the leaves of the solenoid. On one hand, these properties can be grasped from a dynamical point of view. When the group G is amenable, the G-action possesses finite invariant measures. R. Benedetti, J.-M. Gambaudo [2] and L. Sadun [17], show that a G-solenoid can be seen as a projective limit lim ← (B n , π n ) of branched manifold B n . Furthermore, when the group G is unimodular (for example when N = R 2 and G is the translation group), authors of [2] prove that the notions of transverse invariant measure, foliated cycle and finite G invariant measure, are equivalent. Thanks to this, they characterize finite G-invariant measures as the elements of a projective limit of cones in the dim G-homology groups of the branched manifolds B n . When the group G is amenable and not unimodular (this is the case when G is the affine group P), their results do not apply. Actually, we prove that on P-solenoid there is no transverse invariant measure (Proposition 3.1). On the other hand, statistical properties of the leaves can be studied through a geometric point of view. Following the work of L. Garnett [7] on foliations, we can consider harmonic currents on the hull (such currents always exist on laminations). A riemannian metric one the leaves yields a correspondence between harmonic currents and finite harmonic measures and these measures give statistical properties of random path in a leaf of Brownian motions. More particulary, harmonic measures enable to define the average time of a generic path crossing an open subset of the hull. We prove that, for a P-solenoid, both geometrical and dynamical approaches are related: Theorem 1.1 A finite measure on a P-solenoid is harmonic if and only if it is invariant for the affine group action.
By using the structure of projective limit lim ← (B n , π n ) of a P-solenoid, we give a characterization of harmonic measures of a P-solenoid: Theorem 1.2 There exists a sequence of linear morphisms A n such that the set of harmonic measures is isomorphic to the projective limit of cones in 2 chains spaces of branched manifold B n , lim ← (C 2 (B n , R) + , A n ).
The linear morphisms A n will be defined in section 4. We deduce from Theorem 1.2 that the number of ergodic invariant probability measures on the solenoid is bounded from above by the maximal number of faces of the branched manifolds. Finally we prove, by giving explicit examples:

Background on tiling spaces
We recall here different useful notions defined in [11] and [2] 2.1 Action on the hull Let G be the subgroup of isometries acting transitively, freely and preserving the orientation of the surface N , thus G is a Lie group homeomorphic to N . The metric on N gives a left multiplicative invariant metric on G. We fix a point O in N that we call origin. For a tiling T of G finite type and an isometry p in G, the image of T by p −1 is again a tiling of N of finite affine type. We denote by T.G the set of tilings which are image of T by isometries in G. The affine group G acts on this set by the right action: We equip T.G with a metrizable topology, finer as one induced by the standard hyperbolic metric. A base of neighborhoods is defined as follows: two tilings are close one of the other if they agree, on a big ball of N centered at the origin, up to an isometry in G close to the identity. This topology can be generated by the metric δ on T.G defined by : for T and T ′ be two tilings of T.G, let The continuous hull of the tiling T , is the metric completion of T.G for the metric δ. We denote it by Ω(T ). Actually this space is a set of tilings of N of G-finite type. A patch of a tiling T is a finite set of tiles of T . It is straightforward to check that patches of tilings in Ω(T ) are copies of patches of T . The set Ω(T ) is then a compact metric set and the action of G can be extended to a continuous right action on this space. The dynamical system (Ω(T ), G) has a dense orbit (the orbit of T ). We fix in each prototile prot of T , a marked point x prot in its interior. Consequently, each tile t of a tiling T ′ ∈ Ω(T ) admits a distinguished point x t . Let Ω 0 (T ) denote the set of tilings of Ω(T ) such that one x t coincides with the origin O. With the induced topology, Ω 0 (T ) is compact and completely disconnected.
Definition 2.1 A tiling T satisfies the repetitivity condition if for all patch P , there exists a real R(P ) such that every ball of N with radius R(P ) intersected with the tiling T contains a copy of the patch P .
This definition can be interpreted from a dynamical point of view (see for instance [11]). We call a tiling non-periodic if the action of G on Ω(T ) is free: for all p = Id of G and all tilings T ′ of Ω(T ) we have T ′ .p = T ′ . It is straightforward to show that when the stabilizer of T is reduced to the identity (T is aperiodic) and T is repetitive then T is non periodic. In this case the space Ω 0 (T ) is a Cantor set. For example when N = R 2 and G is the translation group, the Euclidean Penrose's tiling is a non-periodic repetitive tiling of R 2 finite type. When N = H 2 and G is the affine group P, we saw that the hyperbolic Penrose's tiling is not aperiodic, however we shall construct in the last section examples of repetitive and aperiodic affine finite type tilings (with specific ergodic properties).

Structure of G-solenoid
where f i,j .x means the multiplication of x ∈ V j with an element f i,j of G, independent of x and c ∈ C j ; and g i,j is a continuous map from C j to C i independent of x. Two atlases are equivalent if their union is again an atlas. We will call a compact metric space M with an equivalence class of atlas, a G-solenoid.
The transition maps structure provides the following important notions: 1. slices and leaves: a slice is a set of the kind h −1 i (V i × {c}). The leaves are the union of the slices which intersect. The global space M is laminated by these leaves. Leaves are differentiable manifolds of dimension 2. A G-solenoid M is called minimal if every leaf of M is dense in M .

2.
Vertical germs: it is a set of the kind h −1 i ({x} × C i ). Transition maps map vertical germs onto vertical germs, and thus this notion is well defined (independently of the charts).
These transition maps enable to define right multiplication by an element of G close to the identity. We suppose furthermore that each leaf is diffeomorphic to N and that this local G right action on a leaf extends to a free G right action on M . Leaves correspond to orbits of the action of G by right multiplication. This action is minimal if and only if the G-solenoid is minimal.
Furthermore this action has locally constant return times: if an orbit (or a leaf) intersects two verticals V and V ′ at points v and v.g where g ∈ G, then for any point w of V close enough to v, w.g belongs to V ′ .
It turns out that the hull of a tiling has a laminated structure (see for instanceÉ. Ghys [8]). More precisely, in [2] authors prove that the hull Ω(T ) of a non periodic G finite type tiling T , has a G-solenoid structure. The boxes of Ω(T ) are homeomorphic to spaces V i × C i where V i is an open subset of G ≃ N and C i is a closed and open subset of Ω 0 (T ). The charts are the inverse of the maps f i : The action of the group G on the solenoid coincides with the action of this group on the hull. This G-action is expansive: there exists a positive real ǫ such that for every points T 1 and T 2 in the same vertical in Ω(T ), if δ(T 1 .g, T 2 .g) < ǫ for every g ∈ G, then T 1 = T 2 .
If furthermore T verifies the repetitivity condition, the hull Ω(T ) is minimal, and the transversal in any point in any box is homeomorphic to a Cantor set.

Branched manifolds and projective limits
A box decomposition of a solenoid M is a finite collection of charts B 1 , . . . , B n such that: any two boxes are disjoint and the closure of the union of all boxes is the whole space M ; The hull of a finite affine type tiling has a natural box decomposition, where boxes are homeomorphic to the product of a prototile of the tiling times a disconnected set. Boxes are sets of tilings having the same tile on the origin. We say that this box decomposition is associated to tiles of the tiling.
Let us consider a box decomposition on M . We consider now the equivalence relation ∼ generated by the relation ≈: x ≈ y ⇔ x and y belong to the closure of the same box and are in the same vertical.
Let B be the quotient space M/ ∼ and let p be the projection of M onto B. Authors of [2] prove that the set B with the quotient topology, has a differentiable structure and is a branched manifold, a structure by R. Williams (see [22]). Actually, in the proof of Theorem 1.2 we will only use the simplex structure of B.
Example: consider a non-periodic tiling T which is a decorated hyperbolic Penrose's tiling (see section 5). The set of prototiles is a finite union of different copies of P . Let us consider now the box decomposition of Ω(T ) associated to its prototiles. The quotient space Ω(T )/ ∼ is then homeomorphic to the collapsing of prototiles along edges. Points on prototiles are identified if somewhere, on T , their copies meet (see [1]). For the Penrose's tiling T , this identification leads to a branched manifold N homeomorphic to P with edges identified as follows: edges A 1 A 2 , A 2 A 3 and A 4 A 5 are identified themselves and edge A 4 A 1 is identified with A 5 A 3 . This space is homeomorphic to the mapping torus of the application x → 2x mod 1 on the circle S 1 ≃ R /Z . There is a natural projection of Ω(T )/ ∼ onto N .
We say that the box decomposition B 2 is zoomed out of the box decomposition B 1 if: 4. if a vertical in the vertical boundary of a box in B 1 contains a point in a vertical boundary of a box in B 2 , then it contains the whole vertical.
A tower system of a solenoid M is a sequence of box decompositions (B n ) n≥1 , such that for any n ≥ 1, B n+1 is zoomed out of B n . In [2] it is proved that any P-solenoid admits a tower system(B n ) n . From above, for every n, there exists a branched manifold B n associated to the box decomposition B n and a projection p n : M → B n . By definition, the set of verticals of boxes of B n+1 is included in the set of verticals of B n , this induces a natural map π n : B n+1 → B n such that p n = π n • p n+1 . We recall that lim ← (B n , π n ) is a subspace of ΠB n defined by {(x n ) ∈ ΠB n | x n = π n (x n+1 )} and equipped with the topology induced by the product topology.

Foliated cycles and harmonic currents 3.1 Foliated cycles
The leaves of a G solenoid M carry a 2-manifold structure. Following [8], we call k-differential form the data, in any box, of a family of real k-differential forms (C ∞ ) on slices V i × {c} which depends continuously of the parameter c (in the C ∞ -topology) and such that each family is mapped onto each other by the transition maps. We denote by A k (M ) the set of k-differential forms on M . The differentiation along leaves gives an operator d : Foliated cycles, introduced by D. Sullivan [20], are a continuous linear forms A 2 (M ) → R which are positive on positive forms and vanish on exact forms. Proposition 3.1 A P-solenoid does not admit a foliated cycle.
In order to prove this result, let us introduce the following definition.
Definition 3.2 A finite transverse invariant measure on M is the data of a finite positive measure µ i on each set C i such that for any borelian set B in some C i which is contained in the definition set of the transition map g ij then The data of a transverse invariant measure for a given atlas provides another invariant transverse measure for any equivalent atlas and thus gives an invariant measure on each verticals. Thus it makes sense to consider a transverse invariant measure µ t of a P-solenoid. It turns out that finite transverse invariant measures are in one-to-one correspondence with foliated cycles (also called Ruelle-Sullivan current) and that conversely any foliated cycle implied the existence of a transverse invariant measure.
Proof of Proposition 3.1: if µ t is a finite invariant transverse measure of a P-solenoid Ω and λ is a left invariant Haar measure on borelian sets of P (for example the measure induced by the standard metric on H 2 ). We can define a global finite measure µ on Ω as follows. On a box U i × C i we consider the product measure λ ⊗ µ t , which is well defined thanks the invariance properties of considered measures. Up to multiplication by a scalar, we can suppose the measure µ is a probability measure on Ω. As P acts on Ω, any element g of P defines an homeomorphism of Ω denoted τ g . Let f be a continuous function on Ω with value in R with support included in a box B ≃ U ×C. Thanks the locally constant return times property, we can decompose B into a finite disjoint union of boxes b i ≃ U × C i where C i is a closed and open subset of C, such that b i and τ (b i ) are included in the same box D i . We consider now the probability measure τ g * µ obtained by the transport of µ by τ g . We have In each box By taking a partition of the unity associated with open sets of an atlas, it is possible to prove the equality (♯) holds true for any continuous function f : Ω → R. Thus the measure τ g * µ is the measure aµ. This is a contradiction with the fact that µ is a probability measure.
Remark 1 When the Lie group G is unimodular, a G-solenoid admits foliated cycles, which are characterized in [2].

Remark 2
The existence of a foliated cycle is a very strong hypothesis. The non existence of foliated cycle gives information on geometric behavior of leaves. Following J. Plante [16], it implies the exponential growth for every leaf of a P-solenoid. L. Garnett [7] gives the local structure of such measures. In a box U i ≃ V i × C i a harmonic measure µ disintegrates into a probability measure ν i on C i times the measure f i (z, c)dz where dz denotes the Riemannian leaf measure and f i :

Harmonic currents
defined for almost all c of C i and harmonic on all the slices V i × {c}. Thus for any Borelian B included in U i : This local decomposition is not unique. If two decompositions µ i , f i and µ ′ i , f ′ i define the same measure, then it exists a measurable application δ i : Remark 3 For a R 2 -solenoid, leaves are then homeomorphic to the plane. The harmonic function obtained is positive and defined on all the plane then it is a constant function. The harmonic measure associated with is then locally disintegrated into a measure µ i on C i times the Riemannian measure, and µ i is then a transverse invariant measure.

Harmonic measures and ergodic theorem
Let x ∈ M be a point of the hull and let Γ x be the set {γ : R + → L x continuous |γ(0) = x, γ(R + ) ⊂ L x } where L x is the leaf passing trough x. The set Γ x is the set of continuous path beginning at x and strictly include in L x . We equip this set with the topology of uniform convergence on compact sets. On the space of borelians, there exists a natural finite measure w x called the Wiener measure. This measure is defined so that the motion Γ x × R + : (γ, t) → γ(t) ∈ L x is a brownian motion. Let Γ = x∈Ω(T ) Γ x be the set of continuous paths of M strictly included in leaves, we equip again this set with the topology of uniform convergence on compact sets. If µ is a finite measure on M , then µ = w x ⊗ µ(x) is a finite measure on Γ. The semi-group R + acts on the space Γ by time translations: for τ > 0 and γ ∈ Γ we define the semi-group of transformations S τ with S τ (γ)(s) = γ(s + τ ). It is straightforward to check transformations S τ preserve µ if and only if µ is a harmonic measure. This comes the Wiener measure is built with the heat kernel. For a harmonic measure µ, we can apply the Birkhoff ergodic theorem.  Since the group P is the extension of two Abelian groups, P is amenable, and the set of invariant measures is a closed non-empty set for the weak topology. Actually, for a P-solenoid invariant measures and harmonic measures are the same (Theorem 1.1).
First let us prove that a harmonic measure of M is an invariant finite measure for the Paction. We will use the lemma: such that for each i, b i and b i .τ −1 are included in a same box D i . By taking ǫ small enough, for every element g of a neighborhood of τ , we have also that b i and b i .g −1 are included in D i . Therefore where g is the map z → az + b. We recall here for z = (x, y) in H 2 , z.g = (x + by, ay).
As shown in section 3.2, the map f i (., t) for a fixed t, can be extended to a harmonic map on the whole half plane H 2 . The map g → f i (z.g, t) is defined on P and it is straightforward to check it is a harmonic map. Thus the bounded map g ∈ H 2 → b i φdg * µ ∈ R reads  Proof : Fix a box V × C, we decompose m in this box into a transversal measure ν on C and a system of leaf measure σ c on V × {c} for each c of C. Hence we have for any measurable function f with support included in the box, We fix a point x of the box and a closed neighborhood K included in the box. Let A be the set of bounded measurable functions with support in K. If m is P-invariant for any f ∈ A and for any g ∈ P s.t. It follows that when m is invariant, for ν almost all c in C, f (x)−f (x.g)dσ c = 0. Therefore, by identifying the leaf with the Lie group P, for ν almost all c, σ c is a right invariant Haar measure.
When identifying the Lie group P with H 2 , a right invariant measure reads λ y dxdy for some constant λ > 0. Therefore an invariant measure m on M can be written in a box λ c dxdy y dν(c), where c ∈ C → λ c ∈ R + is a measurable map. Then the measure m is harmonic. This ends the proof of Theorem 1.1.
As we know, the local decomposition of an invariant measure m is not unique. If λc y dxdydν(c) and λ ′ c y dxdydν ′ (c) are two decompositions of the same measure m, the measures ν and ν ′ are in the same class, and thus there exists a positive measurable map defined almost everywhere δ : C → R + * such that ν = 1 δ(.) ν ′ and λ c = δ(c)λ ′ c . An important consequence is that the value C λ c dσ(c) is well defined. Consider f the positive function H 2 → R defined by f (x, y) = C λ c dσ(c).y, then the measure of a cylinder A × C (where A is a measurable set of V ) of the box is m(A × C) = A f (x, y) dxdy y 2 . We will use this function to characterize invariant measures. We consider first a box decomposition of the P-solenoid M . With each box B and for an invariant measure m, we have seen that we can associate a non negative number b = C λ c dσ(c) > 0. The identification of elements belonging to the same vertical of the box decomposition leads to a fibration p of M over a branched manifold B. We associate to the interior F i of a 2-face of B a box B i = p −1 (F i ) with the fibration and then we consider the 2-chain Σ i b i F i ∈ C 2 (B, R) + . Therefore the fibration p : M → B induces a linear map

Combinatorics of the invariant measures
If we consider now a tower decomposition (B n ) n , we obtain a sequence of fibration p n over branched manifolds B n and a sequence of map π n : B n+1 → B n such that p n = π n • p n+1 and M ≃ lim ← (B n , π n ). These maps induce linear maps (p n ) * : M(M ) → C 2 (B n , R) + .
The relation between (p n ) * (m) and (p n+1 ) * (m) can be described as follows. We denote by B n i ≃ F n i × C n i the boxes of B n , where the index i is an enumeration of these boxes. Let f i (x, y) be the function (x, y) → C n i λ n ic dσ n i (c).y = b n i y for a local decomposition of the measure m. The intersection of B n i and B n+1 j is either empty or a disjoint union of boxes l D l ij . In the non trivial case, there exists transition maps h l ij : z, γ ij (c)) and g l ij ∈ P. Thus for any cylinder A × C n i of B n i we have Since this is true for any A ⊂ V n i , we have the relation: Let us denote p(n) the dimension of C 2 (B n , R) and A n the p(n) × p(n + 1) matrix with positive coefficients a n i,j = l α(g l ij ) when B n i and B n+1 j intersect and 0 otherwise. We have the relation (p n ) * (m) = A n ((p n+1 ) * (m)), and thus the sequence ((p n ) * (m)) n is an element of Lim ← (C 2 (B n , R) + , A n ). This enables us to extend maps (p n ) * to a map It is obvious that p * maps the set of probability invariant measures to the set lim ← (P(B n , R), A n ).
Actually this linear map is an isomorphism whose inverse can be constructed as follows. Let (v n ) n be an element of lim ← (C 2 (B n , R) + , A n ). We consider the family of cylinder A such that there exists a box B n i ≃ V n i × C n i where A ⊂ B n i and A ≃ A n i × C n i for some measurable subset A n i of V n i . Let m(A) be the value A n i b n i dxdy y where v n = (b n 1 , . . . , b n i , . . . , b n p(n) ). Thanks to the relations between v n and v n+1 , the value m(A) is well defined and can be extended by additivity to the σ-algebra generated by cylinders A. This set is big enough so that its σ-algebra is actually the Borel tribe. It is then straightforward to check that p * (m) = (v n ) n . Furthermore, since m disintegrates locally into a transverse measure times a measure of the kind by dxdy y 2 on the slices, m is a harmonic measure, then from Theorem 1.1 m is also an invariant measure.
The above result can be summarized in the following theorem which is an explicit reformulation of Theorem 1.2 : Theorem 4.3 If M is a P solenoid and M is homeomorphic to a projective limit of branched manifolds B n , lim ← (B n , p n ).
The restriction to the set of invariant probability measure is then homeomorphic to Lim ← (P(B n , R) + , A n ).
This last theorem allows us to exhibit some criteria to bound the number of invariant probabilities. 2. If furthermore M is minimal and the linear map A n are uniformly bounded, then there is a unique invariant probability measure.
Proof : Without loose of generality, we may assume that for all n ≥ 1, dim R C 2 (B n , R) = N . Let us consider N sequences (w n j ) n ∈ Π n C 2 (B n , R) + for j ∈ {1, . . . , N } where w n j = (w n j,1 , . . . , w n ji , . . . , w n j,N ) and w n j,i = 0 if j = i and 1 otherwise. Fix an integer n, for any j in {1, . . . , N } and m > n let w nm . Up to a choice of a subsequence, we can suppose that the sequences (w nm j ) m>n converge to w j ∈ P(B 1 , R). Let us denote proj n the projection of the product Π n C 2 (B n , R) onto C 2 (B n , R), and P rob n = proj n (Lim ← (P(B n , R), A n ). The set P rob n is a convex set and if H m is the convex hull of {w nm j |j = 1, . . . , N }, we have P rob n = m>n A n • . . . • A m−1 (H m ). Therefore P rob n is the convex hull of {w j |j = 1, . . . , N }. Suppose now there is more than N ergodic invariant probabilities then for n big enough, there would be more than N extremal points in P rob n , a contradiction.
In order to prove the second statement, we show that for any n, P rob n is reduced to a point. For this we define the hyperbolic distance between two points x, y in P(B n , R).
where m is the Euclidean length of the segment [x, y] and l, r are the length of connected components of S\[x, y] where S is the largest line segment containing [x, y] in P(B n , R). It is straightforward to check positive matrices contract this distance and the minimality of the action implies the positivity of matrices. Since linear maps A n are uniformly bounded and defined on space with bounded dimension, the contraction is uniform. Therefore P rob n = m>n A n • . . . • A m−1 (P(B m , R)) is reduced to a point.

Examples and proof of Proposition 1.3
We give an example of a non periodic repetitive P finite type tiling with exactly r ergodic invariant probability measures, for any integer r > 0. The idea is to decorate the Penrose's tiling with a non periodic bi-infinite sequence. We choose a sequence such that the action of the shift on the closure X of the orbit for the action, is minimal and has r ergodic invariant probability measures.
First, consider the case r ≥ 2. Let Σ be the set {1, . . . , r}. We associate to each symbol in Σ a different color. Let P be the polygon defined in the introduction to build the Penrose's tiling. Let R and S be the affine maps defined in the introduction. For an element i of Σ, let P i be the prototile P painted in the color i. To a sequence w = (w k ) k∈Z ∈ Σ Z , we associate the decorated tiling T (w) of finite affine type, with prototiles P i for i in Σ, defined by T (w) = {R q • S n (P wq )|n, q ∈ Z}.
Its tiles are isometric to P and its stabilizer is included in < R >. To a sequence (w n ) n∈Z the shift σ associates the sequence (w ′ n ) n∈Z where w ′ n = w n+1 . Thus we have T (w).R = T (σ(w)). Therefore if the sequence w is not periodic for the action of the shift, then T (w) is not stable for any element of P. The product space Σ Z is equipped with the product topology and is a Cantor set. Let X denote the closure of the orbit of w by the action of the shift σ: X = {σ n (w), n ∈ Z}. The set X is a compact metric space stable under the action of σ. When the dynamical system (X, σ) is minimal then Ω(T (w)) is minimal.
In [23], S. Williams generalizes an example of J. C. Oxtoby ([14]) and defines a Toeplitz sequence w ∈ Σ Z for which the action of the shift is minimal and has r ergodic probability measures. We recall here the definition of this sequence. Consider the sequence of natural numbers (p i ) i∈N with p 0 = 3 and p i+1 = 3 i .p i and the sequence s i ≡ i mod r ∈ Σ for i ∈ N. Define then the sequence w = (w q ) q∈Z ∈ Σ Z by inductive steps. The first step (step 1) is to set w q = s 1 for all q ≡ 0 or − 1 mod p 1 . In general for i ∈ N, k in Z, let J(i, k) denote the set of integers q ∈ [kp i , (k + 1)p i ) for which w q has been not yet defined at the end of the step i. The step (i + 1) is to set w q = s i+1 for q ∈ J(i, k) with k ≡ −1 or 0 mod 3 i . The dynamical system (X, σ) is minimal and X is a Cantor set.
Let us define now a sequence of atlas of words for the sequence w. Let A 0 be the set of words {s i , i = 1 . . . r}. Let A 1 be the set of words {s 1 s p 1 −2 i s 1 , i = 1, . . . , r}, where for two words a and b, ab denotes the concatenation of the two words and a q denotes the concatenation of q times the word a. In the general case for any integer q ≥ 1, we denote by p q,i i ∈ {1, . . . , r} the word of A q indexed by i and for q > 1, A q is the set of words The suspension of the action of σ on X, is the quotient space X = R × X/σ where points (t, x) and (s, x ′ ) are identified if s − t ∈ Z and x = σ s−t (x ′ ). The natural R-action by time translation on the space R × X induces a R-action on the suspension. It turns out that the suspension R × X/σ is a R-solenoid ( [2]) which has exactly r invariant ergodic probability measures ( [23]). For any q ≥ 0, A q defines a box decomposition of the suspension X . Each box is identified with a unique word of A q .
We will construct a tower system for Ω(T (w)) associated to the former box decompositions of the suspension, thanks to a collection of patches for the tiling T (w). For a word b = w i 0 . . . w i 0 +l of w, let Pa(b) be the patch l j=0 {R −j • S k (P w i 0 +j ) for k = 0, . . . , j} of T (w). Now let us consider for q ≥ 0 the collection of patches Pa q = {Pa(p q,i ), for i = 1, . . . , r}. For any q, the tiling T (w) is an union of elements of Pa q , copies of patches meeting only on their borders. Remark that all the patches of Pa q have the same size and actually, the box decompositions of Ω(T (w)) associated to Pa q define a tower system of the hull. If we denote by ∼ q the relation generated by the identification of borders of patches of Pa q which meet somewhere in the tiling T (w) and B q = r i=1 Pa p q,i / ∼ q , we have applications π q such that: Now we construct a natural continuous map h from Ω(T (w)) onto X . For an element g : z → az + b of the group P, we define h(T (w).g) = [(log 2 (a), w)] ∈ X where [(t, x)] denotes the class of the element (t, x) in R × X for the relation defined by σ. The map h is then a continuous map from T (w).P to X . Remark that if the origin O lies in a copy of a patch Pa(p q,i ) for some q ≥ 1 and i ∈ Σ in the tiling T (w).g, then O lies also in a copy of the patch Pa(p q,i ) in the tiling T (σ n (w)), where n denotes the integer part of log 2 (a). Thus the origin of the sequence σ n (w) lies in the word p q,i . As h(T (w).g) = [(log 2 (a) − n, σ n (w))], we get that h(T (w).g) is in the box of the suspension defined by the word p q,i . It follows that for any q ≥ 1, the map h sends the restriction to the orbit of T (w) of the box associated to the patch Pa(p q,i ) to the box of the suspension associated to the word p q,i . Thus the map h is uniformly continuous. It follows that h can be extended to a map from Ω(T (w)) onto X also denoted h. It is straightforward to check that each fiber of the map h is stable under the action of the group N = {z → z + t, t ∈ R}. Furthermore, as P is an extension over N and the the group {z → az, a > 0}, the action of the group P preserves the set of fibers. Then the P-action on the hull Ω(T (w)) defines through the application h, a P-action on the suspension X and h is a semi-conjugacy from the hull Ω(T (w)) to X . The group N acts trivially on X . The invariant measures for the P-action on X are the invariant measures for the R-action. We claim that the map h sends the invariant measures of the hull onto the invariant measures of the suspension. To prove this, we use a Følner's base of P that we denote (A n ) n and a right multiplicative invariant Haar measure on P that we denote λ. Let µ be a ergodic invariant probability measure for the P-action on X . By the ergodic theorem, there exists a point x in the suspension such that the sequence of probability measures µ n = 1 λ(An) An δ g.x dλ(g) converges, when n grows to infinity, to the measure µ. Let y be a point in Ω(T (w)) such that h(y) = x. Then, up to the choice of a subsequence, the sequence of probability measures on Ω(T (w)) ν n = 1 λ(An) An δ g.y dλ(g) converges to a probability measure ν invariant for the P-action. As h * ν n = µ n , we get h * ν = µ. It follows that the map h sends the set of invariant measures of Ω(T (w)) onto the set of invariant measures of X . Furthermore the map h sends ergodic measures on ergodic measures. Then Ω(T (w)) has at least r independent ergodic probability measures. From Proposition 4.4, we also know that the hull Ω(T (w)) admits at most r invariant ergodic probability measures. Thus there are exactly r probability measures.
To obtain an example of a minimal P-solenoid with a single P-invariant probability measure, we use the same strategy as before. We keep the same notations as the case r = 2 but we define an other Toeplitz sequence w on which the shift action is free, minimal and uniquely ergodic ( [9]). We consider the substitution S over the alphabet Σ = {1, 2} defined by S(1) = 112, S(2) = 122. Using the extension of the substitution over the words by the concatenation, we can iterate the substitution. The sequence w is then the bi-infinite sequence defined by: where the dot . is placed between the 0 and −1 coordinate.
Let A 0 be the set {1, 2}, and for any integer q ≥ 1, let A q be the atlas of words {S q−1 (1)S q−1 (i)S q−1 (2), i = 1, 2} for the sequence w. The sequence w is a bi-infinite sequence of words of A q . Now let us consider the collection of patches Pa q = {Pa(wo), wo ∈ A q }. For any q ≥ 0, the tiling T (w) is an union of elements of Pa q and the box decompositions of Ω(T (w)) associated to Pa q define a tower system of the hull. The hull Ω(T (w)) is then homeomorphic to lim ← (B q , π q ) where B q = wo∈Aq Pa(wo)/ ∼ q . By Theorem 4.3, the space of invariant measures M(Ω(T (w))) is isomorphic to lim ← (C 2 (B n , R) + , A n ). A simple calculation shows that the linear applications A n are defined by the matrices: A n = 1 + 2 −3 n +1 1 2 −3 n 2+2 2 −3 n +1 + 2 −3 n 2+2 .
Proposition 4.4 enables us to conclude that the hull Ω(T (w)) admits only one P-invariant probability measure.