Marcy Barge

Johannes Kellendonk
email: kellendonk@math.univ-lyon1.fr

Scott Schmieding

MAXIMAL EQUICONTINUOUS FACTORS AND COHOMOLOGY FOR TILING SPACES

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

Introduction

An effective procedure for studying the properties of a tiling, or point-pattern, T of R n is to consider the space Ω (called the hull of T ) of all tilings that, up to translation, are locally indistinguishable from T . Dynamical properties of the action of R n on Ω, by translation, correspond to combinatorial properties of T . Regularity assumptions on T guarantee that the dynamical system (Ω, R n ) is compact and minimal. There is then a maximal equicontinuous factor (Ω max , R n ), with semi-conjugacy π : Ω → Ω max ; Ω max is a compact abelian group on which R n acts by translation and every equicontinuous factor of (Ω, R n ) is a factor of (Ω max , R n ).

The relationship between the hull of a tiling and its maximal equicontinuous factor is of fundamental importance in certain aspects of tiling theory. For example, if T is a (sufficiently well-behaved) distribution of "atoms" in R n , the diffraction spectrum of T is pure point (that is, T is a perfect quasicrystal) if and only if the dynamical spectrum of (Ω, R n ) is pure discrete ( [LMS], [D]), if and only if the factor map π is a.e. one-to-one (with respect to Haar measure, [BaKe]).

In this article we study the properties of the homomorphism π * induced by the factor map π in cohomology. This is directly motivated by a recent formulation of the Pisot Substitution Conjecture ( [BG]) in terms of the homological properties of π * . More generally, cohomology has long been a primary tool for understanding the structure of Ω ( [AP], [S], [FHK]) and, at least for tilings with a non-trivial discrete component of dynamical spectrum, the pull-back of the cohomology of the maximal equicontinuous factor represents a sort of skeleton supporting the rest of the cohomology of Ω.

The maximal equicontinuous factor of a tiling dynamical system is always a torus or solenoid so its cohomology (as a ring) is determined by its degree one cohomology. Consequently, our focus will be on π * in degree one (this is also the important degree for deformation theory ( [CS], [START_REF] Kellendonk | Pattern Equivariant Functions, Deformations and Equivalence of Tiling Spaces[END_REF]) and the Pisot Substitution Conjecture), though we will have something to say in higher degree for projection patterns, in which cohomology is tied to complexity. A main result is that π * is injective in degree one with torsion-free cokernel. We will show by example, however, that the first cohomology of the maximal equicontinuous factor is not necessarily a direct summand of the first cohomology of Ω.

Let us say a few words about our methods. Given a continuous map f : Ω → T of the hull to the unit circle, and a vector v ∈ R n , there is a Schwartzman winding number τ (f )(v) of f with respect to the R-action T ′ → T ′ -tv on Ω in direction v ( [Sch]). This defines a functional, v → τ (f )(v), which depends only on the homotopy class of f . As the group of homotopy classes of maps of Ω to T is naturally isomorphic with the first integer cohomology H 1 (Ω) of Ω, τ provides a homomorphism from H 1 (Ω) to R n * . We will see that the degree one cohomology of the maximal equicontinuous factor can be identified with the group E of continuous eigenvalues of the R n -action on Ω. Each eigenvalue, in turn, determines a functional on R n . With these identifications, τ • π * is the identity, establishing that π * is injective in degree one.

The homomorphism τ described above is the degree one part of the Ruelle-Sullivan map ( [KP]). In the top degree, n, τ has an interpretation as the homomorphism that assigns to each finite patch of a tiling T its frequency of occurrence in T . The range of τ is then the frequency module freq(Ω) of Ω and its kernel is the group Inf(Ω) of infinitessimals with respect to a natural order on the top degree cohomology. In the special case of one-dimensional tilings, we have two related short exact sequences with H 1 (Ω) in the middle:

0 → E π * -→ H 1 (Ω) → coker π * → 0 and 0 → Inf(Ω) → H 1 (Ω) τ -→ freq(Ω) → 0.
This situation is considered, in the context of symbolic substitutions, in [AR]. We consider tilings that arise from substitutions, as well as tilings that arise from projection methods. For almost canonical projection tilings, both of these sequences split. We will give conditions under which this is true for substitution tilings, as well as examples in which one, or both, don't split.

In the next section, we briefly review the basics of substitution tilings, projection methods, tiling cohomology, and the construction of the maximal equicontinuous factor. In Section 3 we consider the map induced in cohomology by the maximal equicontinuous factor map, and in Sections 4 and 5 we restrict consideration to almost canonical projection patterns and substitution tilings, resp.

Preliminaries

2.1. Tilings and their properties. We will use the formulations and terminology of [BaKe] and just recall here what is necessary to set up the notation.

An n-dimensional tiling is an infinite collection of tiles which cover R n and have pairwise disjoint interiors. Here a tile is a compact subset of R n which is the closure of its interior. A tile may carry a mark in case a distinction between geometrically congruent tiles is necessary. A (finite) patch is a finite collection of tiles with pairwise disjoint interiors. Its diameter is the diameter of the set covered by its tiles.

The translation group R n acts on tiles, patches and tilings as on all geometric objects of R n and we write this action by t • O or O -t with t ∈ R n and O the geometric object. A collection Ω of tilings of R n has (translationally) finite local complexity (FLC) if it is the case that for each R there are only finitely many translational equivalence classes of patches P ⊂ T ∈ Ω with diameter smaller than R. A single tiling T has FLC if {T } has FLC. Finite local complexity of tilings will be a standing assumption in this article and we won't repeat it.

We say that a collection Ω of tilings of R n constitutes an n-dimensional tiling space if Ω has FLC, is closed under translation, and is compact in the tiling metric d. In this metric two tilings are close if a small translate of one agrees with the other in a large neighborhood of the origin. The main example of a tiling space is the hull of an FLC tiling T Ω T = {T ′ : T ′ is a tiling of R n and every patch of T ′ is a translate of a patch of T }.

If the translation action on Ω is free (i.e., T -v = T ⇒ v = 0), Ω is said to be non-periodic.

Of particular interest for us are repetitive tilings which have the property that for each finite patch P of T the set of occurrences of translates of P in T is relatively dense. If Ω is repetitive, then the action of R n on Ω by translation is minimal.

Another property which we will require occasionally is the existence of frequencies of patches in a tiling. The frequency of a patch P (up to translation) in T is the density of the set of occurrences of translates of P in T , and being able to define this properly, independent of the limiting procedure, is equivalent to the unique ergodicity of the dynamical system (Ω, R n ). We denote, then, the unique ergodic measure by µ.

Let p be a puncture map; that is, p assigns to each tile τ a point p(τ ) ∈ τ so that p(τ + v) = p(τ ). If a tiling T has FLC then the set of its punctures p(T ) = {p(τ ) : τ ∈ T } is a Delone set, i.e., a subset of R n which is uniformly discrete and relatively dense. The puncture map p defines a discrete hull Ξ = {T ′ ∈ Ω T : 0 ∈ p(T ′ )}. Ξ is also refered to as the canonical transversal as it is transversal in Ω T to the R n -action reducing it to the so-called tiling groupoid

G = {(ω, t) ∈ Ξ × R n : ω -t ∈ Ξ} with multiplication (ω, t)(ω ′ , t ′ ) = (ω, t + t ′ ) provided ω ′ = ω -t.
The definitions we have made for tilings all have analogs for Delone sets and whether we deal with tilings or Delone sets is mainly a matter of convenience. One could, for instance, represent a tiling T by the Delone set of its punctures, or a Delone set by its Voronoi tiling and the topological dynamical systems (Ω, R n ) are unchanged. Whereas substitutions are usually and more intuitively presented by tilings, the projection method produces Delone sets which are often referred to as projection patterns (or, under more general circumstances, model sets).

Substitution tilings.

Suppose that A = {ρ 1 , . . . , ρ k } is a set of translationally inequivalent tiles (called prototiles ) in R n and Λ is an expanding linear isomorphism of R n , that is, all eigenvalues of Λ have modulus strictly greater than 1. A substitution on A with expansion Λ is a function Φ : A → {P : P is a patch in R n } with the properties that, for each i ∈ {1, . . . , k}, every tile in Φ(ρ i ) is a translate of an element of A, and Φ(ρ i ) covers the same set as Λ(ρ i ). Such a substitution naturally extends to patches whose elements are translates of prototiles by Φ({ρ i(j) + v j : j ∈ J}) := ∪ j∈J (Φ(ρ i(j) ) + Λv j ). A patch P is allowed for Φ if there is an m ≥ 1, an i ∈ {1, . . . , k}, and a v ∈ R n , with P ⊂ Φ m (ρ i ) -v. The substitution tiling space associated with Φ is the collection Ω Φ := {T : T is a tiling of R n and every finite patch in T is allowed for Φ}. Clearly, translation preserves allowed patches, so R n acts on Ω Φ by translation. We say that a substitution Φ is FLC or non-periodic if its corresponding tiling space Ω Φ is FLC or non-periodic.

The substitution Φ is primitive if for each pair ρ i , ρ j of prototiles there is a k ∈ N so that a translate of ρ i occurs in Φ k (ρ j ). If Φ is primitive then Ω Φ is repetitive.

If Φ is primitive, non-periodic and FLC then Ω Φ is compact in the tiling metric, Φ : Ω Φ → Ω Φ is a homeomorphism, and the translation action on Ω Φ is minimal and uniquely ergodic [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF][START_REF] Solomyak | Dynamics of self-similar tilings[END_REF]. In particular, Ω Φ = Ω T for any T ∈ Ω Φ . All substitutions will be assumed to be primitive, non-periodic and FLC.

2.3. Almost canonical projection patterns. We describe here almost canonical projection patterns without going into details which the reader may find in [FHK].

Consider a regular lattice Γ ⊂ R n × R n ⊥ such that R n is in irrational position w.r.t. Γ, and a window K which is a compact polyhedron. Let π : R n × R n ⊥ → R n be the projection onto the first factor and π ⊥ : R n × R n ⊥ → R n ⊥ be the projection onto the second factor. Define the set S of singular points in R n ⊥ by

S := γ∈Γ ∂K -π ⊥ (γ)
where ∂K denotes the boundary of K. We assume that

• the restrictions of π and π ⊥ to Γ are one to one,

• the restrictions of π and π ⊥ to Γ have dense image,

• there exists a finite set of affine hyperplanes {W i } i∈I of codimension 1 in R n ⊥ such that S may be alternatively described as

S = i∈I γ∈Γ W i -π ⊥ (γ).
We call the hyperplanes W i -π ⊥ (γ), i ∈ I, γ ∈ Γ cut-planes. By the second assumption S is a dense subset of R n ⊥ but of zero Lebesgue measure. The last assumption means that, given a face f of K, the union of all π ⊥ (Γ)-translates of f contains the affine hyperplane spanned by f ; in particular the faces of K have rational orientation w.r.t π ⊥ (Γ) and the stabilizer {γ ∈ Γ :

W i -π ⊥ (γ) = W i } of an affine hyperplane W i must have at least rank n ⊥ -1.
We also assume (for simplicity) that 0 is not a singular point. Then the set

P K := {π (γ) : γ ∈ Γ, π ⊥ (γ) ∈ K}
is a repetitive Delone set, called the projection pattern with window K. With the above rather restrictive assumptions made on the window K the projection pattern is called almost canonical. There are standard ways to turn P K into a tiling which is mutually locally derivable from P K , for instance the dual of the Voronoi tiling defined by P K will do it.

2.4. Tiling cohomology and the order structure on the top degree. We are interested in the cohomology of a tiling (or pattern) T . This cohomology can be defined in various equivalent ways, for instance as the Cech cohomology H(Ω) of the hull Ω of T or as (continuous cocycle) cohomology H(G) of the tiling groupoid G (after [Re]).

The equivalence between the two formulations of tiling cohomology can either be seen by realizing that Ω is a classifying space for the groupoid, or by a further reduction: From the work of Sadun-Williams [SW] we know that we can deform the tiling into a tiling by decorated cubes without changing the topological structure of the hull (the hull of the tiling by cubes is homeomorphic to the original one). It follows then that the tiling groupoid of the tiling by cubes is a transformation groupoid Ξ ′ × Z n which is continuously similar to G [Re, FHK]. Here Ξ ′ is the canonical transversal of the tiling by cubes. Like Ξ it is a compact totally disconnected space. It follows then quickly that

H(G) ∼ = H(Ξ ′ × Z n ) and, by definition of the groupoid cohomology, H(Ξ ′ × Z n ) is the dynamical cohomology H(Z n , C(Ξ ′ , Z))
which is the cohomology of the group Z n with coefficients in the integer valued continuous (and hence locally constant) functions. Now what the construction of [SW] actually does on the level of spaces is to realise Ω as a fiber bundle over an n-torus whose typical fibre is Ξ ′ such that the above Z n action corresponds to the holonomy action induced by the fundamental group of the torus. Another way of saying this is that Ω is the mapping torus of that Z n action. It follows (as is seen for instance from the Serre spectral sequence) that

H(Z n , C(Ξ ′ , Z)) is isomorphic to H(Ω).
In the highest non-vanishing degree, namely in degree n, H n (G) is the group of co-invariants,

H n (G) ∼ = C(Ξ, Z)/B
where B is the subgroup generated by differences of indicator functions of the form

1 U -1 U -t , U ⊂ Ξ a clopen subset and t ∈ R n such that U -t ⊂ Ξ.
The group of co-invariants carries a natural order: an element is positive whenever it is represented by a positive function in C(Ξ, Z). Moreover, the order structure is preserved under groupoid isomorphism, and hence the ordered group of co-invariants is a topological invariant for the tiling system.

Let us assume that the tiling system is strictly ergodic. Hence the R n action on Ω as well as the groupoid action on Ξ are minimal and uniquely ergodic. Let ν be the unique ergodic measure on Ξ. Then 1 U → ν(U) factors through C(Ξ, Z)/B and hence, combined with the isomorphism

H n (Ω) ∼ = C(Ξ, Z)/B, defines a group homomorphism τ : H n (Ω) → R.
Now the order can be described by saying that x ∈ H n (Ω) is positive whenever τ (x) ≥ 0. We say that an element x is infinitesimal if it is neither strictly positive nor strictly negative, which is hence the case if and only if τ (x) = 0. We denote the infinitesimal elements by Inf(Ω).

It is well known that a basis of the topology of Ξ is given by the acceptance domains on patches, that is, by subsets containing all tilings which have a given patch at the origin. It follows from this (and the unique ergodicity) that ν(U P ) is the frequency of occurrence of the patch P in T where U P is the acceptance domain of P . Let us denote by freq(Ω) the subgroup of R generated by the frequencies of finite patches in T . We thus have an exact sequence

(1) 0 → Inf(Ω) → H n (Ω) τ → freq(Ω) → 0
which splits if freq(Ω) is finitely generated.

2.5. The maximal equicontinuous factor and eigenvalues. Let (X, G) be a minimal dynamical system with compact Hausdorff space X and abelian group G action.

There is a maximal equicontinuous factor (X max , G) of that system -unique up to conjugacy -and this factor can be obtained from the continuous eigenvalues of the action. In fact, a continuous eigenfunction of a dynamical system (X, G) is a non-zero function f ∈ C(X) for which there exists a (continuous) character χ ∈ Ĝ such that

f (t • x) = χ(t)f (x).
χ is called the eigenvalue of f . To stress that this eigenvalue is an eigenvalue to a continuous function (as opposed to an L 2 -function) one also calls it a continuous eigenvalue. But we will here consider only eigenvalues to continuous eigenfunctions and so drop that adjective.

The set of all eigenvalues E forms a subgroup of the Pontryagin dual Ĝ of G. We consider E with discrete topology. Then the Pontryagin dual Ê of E is a compact abelian group and the maximal equicontinuous factor can be identified with it, X max ∼ = Ê. The factor map π : X → Ê is then given by x → j x , where j x : E → T 1 is defined by j x (χ) = f χ (x), and the G-action on ϕ ∈ Ê is given by (t • ϕ)(χ) = χ(t)ϕ(χ). Here f χ is the eigenfunction to eigenvalue χ normalized in such a way that f χ (x 0 ) = 1 where x 0 ∈ X is some chosen point used to normalize all eigenfunctions.

The factor map and cohomology

We are interested in the map in cohomology induced by the factor map π:

π * : H k ( Ê) → H k (X).
(If nothing else is said this means integer valued Čech cohomology.) In particular, we consider the kernel and cokernel of π * . The situation is extremely simple in degree 0: X and X max are connected and so their cohomology in degree 0 is Z and π * an isomorphism in that degree. The situation is very complicated in degrees larger than one, and we will only be able to say something for almost canonical projection patterns. This will be done in the next section. In this section we will concentrate on degree 1 which is important for deformation theory [START_REF] Sadun | Tiling Spaces are Cantor Set Fiber Bundles[END_REF][START_REF] Clark | When Shape Matters: Deformations of Tiling Spaces[END_REF][START_REF] Kellendonk | Pattern Equivariant Functions, Deformations and Equivalence of Tiling Spaces[END_REF]Bo] and for the homological version of the Pisot conjecture [BBJS, BG].

3.1. The cohomology of the maximal equicontinuous factor. Note that the group E of eigenvalues is at most countable. This follows from the fact that L 2 (X, µ) is separable (for any ergodic invariant probability measure µ) and eigenfunctions to distinct eigenvalues are orthogonal in that Hilbert space.

We suppose that E is torsion free which is certainly the case if Ĝ is torsion free, in particular thus if G = R n . As an abelian group E is a Z-module and we may consider the exterior algebra ΛE, which is a graded ring.

As is well-known, H1 (S 1 ) is a free abelian group of rank one. We pick a generator γ ∈ H 1 (S 1 ) (which amounts to choosing an orientation). Given an element of χ ∈ E, which we may view as a character on Ê, χ : Ê → S 1 , χ * (γ) defines an element in H 1 ( Ê) and thus a group homomorphism

 : E → H 1 ( Ê), (χ) = χ * (γ). Theorem 1. Λ : ΛE → H( Ê) is a graded ring isomorphism.
Proof. As E is countable and torsion free we can write it as E = lim -→ (E n , i n+1 n ) where E n is free abelian of finite rank and i n+1 n : E n → E n+1 an injective group homomorphism 1 . We denote i n : E n → E the corresponding group inclusion. It follows that Ê = lim ← -( Ên , în+1 n ). Now Ên is a torus whose dimension equals the rank of E n and so its cohomology is generated as a ring by its degree 1 elements which, in turn, are the elements of the form (χ), χ ∈ E n . This shows that H( Ên ) ∼ = ΛE n with ring-isomorphism given by Λ :

ΛE n → H( Ên ). Hence H( Ê) = lim -→ H(E n , i n+1 n * ) ∼ = lim -→ (ΛE n , Λi n+1 n ) = ΛE.
3.2. Injectivity of π * in degree one. Let [X, S 1 ] denote the set of homotopy classes of maps from X to the circle S 1 := {z ∈ C : |z| = 1}. This is an abelian group (known as the Bruschlinsky group of X) under the operation

[f ] + [g] := [f g] and the map [f ] → f * (γ) is a natural isomorphism between [X, S 1
] and H 1 (X) (see, for example, [PT]). If we take X = Ê, the maximal equicontinuous factor, then

H 1 ( Ê) = E and the isomorphism E ∼ = [ Ê, S 1 ] is given by χ → [χ],
as is easily seen. The naturality of the isomorphism implies that

H 1 ( Ê) π * -→ H 1 (X) [ Ê, S 1 ] π * -→ [X, S 1 ] commutes.
We now suppose that G = R n and X = Ω is a tiling space. Then Rn is isomorphic to R n * , the dual of R n as a vector space, the map R n * ∋ β → e 2πıβ ∈ Rn providing a group isomorphism. We denote E = {β : e 2πıβ ∈ E}, calling it also the group of eigenvalues, let ı : E ֒→ R n * be the inclusion, and θ : E → [Ω, S 1 ] be the composition θ(β) = π * ([e 2πıβ ]). If f β is an eigenfunction to eigenvalue β, normalized so that its modulus is everywhere 1, then θ(β) = [f β ]. Indeed, by minimality any two eigenfunctions differ by a multiplicative constant and hence are homotopic.

The Lie-algebra of

G = R n is R n . Let H k (R n , C ∞ (Ω, R)) be the Lie-algebra coho- mology of R n = Lie(G) with values in C ∞ (Ω,
R) which are continuous functions that are smooth w.r.t. the derivative d defined by the Lie-algebra action. Since any continuous function on Ω can be approximated in the sup norm by a smooth function we can define a group homomorphism

ψ : [Ω, S 1 ] → H 1 (R n , C ∞ (Ω, R)), ψ([f ]) = 1 2πı f -1 df
using a smooth representative. Given an ergodic invariant probability measure µ on Ω we can define the homomorphism

C µ : H 1 (R n , C ∞ (Ω, R)) → R n * , C µ (α) = Ω α(ω)dµ(ω).
The composition τ = C µ • ψ is the degree 1 part of the Ruelle-Sullivan map of [KP].

τ ([f ])(v)
is also known as the Schwartzman winding number of f with respect to the R-action

T → T -tv on Ω. Lemma 2. τ • θ : E → R n * is given by τ • θ = ı. Proof. Since f -1 β df β = 2πıβ, a constant function, we have C µ (Ψ(θ(β))) = β. Corollary 3. π * is injective in degree 1.
Proof. τ • θ is injective and factors through the degree one part of π * .

Remark: Lemma 2 is actually the degree one part of a more general result which can be obtained with the help of the full Ruelle-Sullivan map τ :

H(Ω) → ΛR n * . The composition τ • π * • Λ : ΛE → ΛR n * is Λı. In particular imτ • π * = Λı(E)
. The proof of this statements is a slight generalization of the one given in [KP][Thm. 13].

In degree n we may identify Λ n R n * ∼ = R and obtain the same map as above (justifying this way the double use of τ in the notation).

3.3. The cokernel of π * in degree one. We start with the remark that H 1 (Ω) is torsion free, as follows from the universal coefficient theorem. In higher degrees, H k (Ω) may contain torsion.

Lemma 4. (Krasinkiewicz [Kr]) Suppose that f : Ω → S 1 is continuous and suppose that there are 0 = k ∈ Z and a continuous g

: Ω → S 1 such that [f ] = k[g] ∈ [Ω, S 1 ].
Then there is a continuous f : Ω → S 1 so that f = f k .

Proof. Let p k : S 1 → S 1 be the k-fold covering map p k (z) := z k . By assumption we have

[f ] = [p k • g]. Let H : Ω × I → S 1
be a homotopy from h := p k • g to f . Then h := g is a lift of h. Being a covering map, p k has the homotopy lifting property so there is a homotopy H : Ω × I → S 1 from h to some function f such that H(•, 0) = h and

p k • H = H. It follows that p k • f = f . Theorem 5. The cokernel of the homomorphism π * : H 1 ( Ê) → H 1 (Ω) is torsion free.
Proof. The statement of the theorem is equivalent to saying that the cokernel of θ :

E → H 1 (Ω) is torsion free. Let [f ] ∈ [Ω, S 1 ] ∼ = H 1 (Ω), k ∈ N, and β ∈ E be such that k[f ] = θ(β) = [f β ]. By Lemma 4 there is an fβ : Ω → S 1 so that f β = p k • fβ . Then k([ fβ ] -[f ]) = 0. Since H 1 (Ω) is torsion free, [ fβ ] = [f ].
We claim that fβ is an eigenfunction with eigenvalue β/k. By continuity of fβ it is enough to verify the equation fβ (T 0 -x) = exp 2πı β(x) k fβ (T 0 ) for some T 0 . We have

f β (T 0 -x) = exp(2πıβ(x))f β (T 0 ) for all x ∈ R n , thus f k β (T 0 -x) = exp(2πıβ(x)) f k β (T 0 ).
Taking the kth root we obtain fβ (T 0 -x) = u(x) exp(2πı β(x) k ) fβ (T 0 ) where u(x) is a kth root of unity. Continuity of fβ requires that u(x) = 1. Hence fβ is an eigenfunction with eigenvalue β/k. Hence θ(β/k) = [ fβ ] = [f ] and coker θ is torsion free.

Corollary 6. If coker π * is finitely generated then H 1 (Ω) is isomorphic to the direct sum of E with coker π * .

Proof. Under the assumption coker π * ∼ = Z l for some finite l, as it is torsion free.

If H 1 (Ω) is not finitely generated then it is not always the direct sum of E with coker π * as Example 5.3.1 shows.

Almost canonical projection patterns

To obtain a Cantor fiber bundle for almost canonical projection tilings we do not actually use the approach of Sadun & Williams via a deformation of the tiling, but rather consider a variant of the "rope dynamical system" of [START_REF] Kellendonk | The local structure of Tilings and their Integer group of coinvariants[END_REF], see [FHK]. This way we obtain a different Cantor fibre bundle, whose fibre we shall denote by C. The conclusion that tiling cohomology can be formulated as group cohomology of a Z naction on C remains valid and we have the benefit that the structure of C allows for a calculation of the cohomology groups. C can be obtained from the set of singular points S by disconnecting R n ⊥ along the cut-planes W i -π ⊥ (γ) and moding out the action of a subgroup of π ⊥ (Γ) of rank n ⊥ . This subgroup Z n ⊥ should be a direct summand, i.e. π ⊥ (Γ) = Z n ⊥ ⊕ Z n , and it should span R n ⊥ , but it can otherwise be chosen arbitrarily. So C = F c /Z n ⊥ , where F c is the so-called cut-up space obtained by disconnecting R n ⊥ , and the other summand Z n yields the action on C. We refer the reader to [FHK,[START_REF] Gähler | Integer cohomology for rational projection patterns[END_REF] for the precise definition of the disconnecting procedure mentioning here only that it can be obtained via an inverse limit: For any finite collection of cut-planes, disconnecting R n ⊥ along these cut-planes means taking out the cut-planes so that the remaining part of R n ⊥ falls into several connected components and then completing separately these connected components to obtain a closed space. The inverse limit is just geared to make that work for infinitely many cut-planes.

To do the actual computation it is more convenient to work with homology. Using Poincaré duality for group (co-) homology and the fact that Z n ⊥ acts freely on F c one obtains

H k (Z n , C(C, Z)) ∼ = H n-k (Z n , C(C, Z)) ∼ = H n-k (Γ, C n ⊥ )
where C n ⊥ is the Z-module generated by indicator functions on polyhedra whose faces belong to cut-planes and γ ∈ Γ acts on such a function by pull back of the translation with π ⊥ (γ). Intersections of cut-planes are affine subspaces of smaller dimension and we call such an affine space a singular space. On each singular subspace L, say of dimension k, we have a similar structure as on R n ⊥ : The intersections of the cut-planes with L are affine subspaces of co-dimension 1 in L. We let C k be the module generated by indicator functions on k-dimensional polyhedra in a k-dimensional singular space whose faces belong to cut-planes.

The polyhedral structure and the fact that R n ⊥ is contractible give rise to an acyclic complex

C n ⊥ → C n ⊥ -1 → • • • → C 0 of Γ-
modules whose differential is reminiscent of the boundary map in polyhedral complexes. As a result the homology H * (Γ, C n ⊥ ) may be computed by breaking the complex into n ⊥ short exact sequences

0 → C 0 k → C k → C 0 k-1 → 0, 0 ≤ k < n ⊥ with C 0 k equal to the image of the boundary map δ : C k+1 → C k which is, of course, the same as the kernel of δ : C k → C k-1 (and thus C n ⊥ = C 0 n ⊥ -1
) and C 0 -1 = Z. Each such short exact sequence gives rise to a long exact sequence in homology and in particular to a connecting homomorphism γ k :

H p (Γ, C 0 k-1 ) → H p-1 (Γ, C 0 k ).
We now recall:

Theorem 7 ( [BaKe]). The maximal equicontinuous factor Ê of a projection pattern is naturally isomorphic to the torus R n × R n ⊥ /Γ.

Hence, upon identifying R n × R n ⊥ with its dual we have E = Γ and so we may identify H p ( Ê) = Λ p Γ = H p (Γ, Z) ∼ = H n+n ⊥ -p (Γ, Z), the last identification by Poincaré duality.

Theorem 8 ([GHK2]). Under the above identifications H

p (Ω) ∼ = H n-p (Γ, C n ⊥ ) and H p ( Ê) ∼ = H n+n ⊥ -p (Γ, Z) the map π * : H p ( Ê) → H p (Ω) gets identified with the compo- sition of connecting maps γ n ⊥ -1 • • • • • γ 0 : H n+n ⊥ -p (Γ, Z) → H n-p (Γ, C n ⊥ ).
4.1. Injectivity of π * . We have now the tools at hand to find out in which degrees π * is injective. In fact, the long exact sequence in homology corresponding to the above exact sequence is

→ H p (Γ, C k ) δ ′ * → H p (Γ, C 0 k-1 ) γ k → H p-1 (Γ, C 0 k )→
where δ ′ is the boundary map with target space restricted to its image. Thus γ k is injective whenever δ

′ * = 0. Now the module C k decomposes, C k = θ∈I k C θ k ⊗ Z[Γ/Γ θ ]
where I k indexes the set of Γ-orbits of singular spaces of dimension k and Γ θ ⊂ Γ is the subgroup stabilizing the singular space of orbit type θ. It follows that H

p (Γ, C k ) = θ∈I k H p (Γ θ , C θ k ).
Since the singular spaces which make up

C θ k are k-dimensional H p (Γ θ , C θ k ) = 0 if p > k. But we have also H p (Γ θ , C θ k ) = 0 if p > rkΓ θ -dim RΓ θ , because Γ θ
contains a subgroup of rank dim RΓ θ which acts freely on C θ k . To summarize

H p (Γ, C k ) = 0 if p > min{k, r k } where r k = max θ∈I k (rkΓ θ -dim RΓ θ ). In particular, γ n ⊥ -1 : H p+1 (Γ, C 0 n ⊥ -2 ) → H p (Γ, C n ⊥ ) ∼ = H n-p (Ω) is injective if p ≥ min{n ⊥ -1, r n ⊥ -1 }.
We now consider first the case in which the ranks of the stabilizers are minimal. Since the stabilizer of W i must have at least rank n ⊥ -1 the minimal case is r n ⊥ -1 = 0 which then implies that r k = 0 for all k. This is in fact the generic case and it corresponds to the pattern having maximal complexity among almost canonical projection patterns; that is, the growth exponent for the complexity function is n ⊥ n [Ju]. We see from the above that H p (Γ,

C θ k ) = 0 if p > 0 and therefore π * is injective in all degrees. But H k (Ω) is infinitely generated except if n ⊥ = 1.
The situation is different if we require that the cohomology is finitely generated. By the results of [FHK] and [Ju] this is precisely the case if ν := n+n ⊥ n ⊥ is an integer and the rank of the stabilizer of a singular plane is ν times its dimension, i.e. r k = (ν -1)k ≥ k. We find it interesting to note that this case corresponds to the case of minimal complexity, that is, the number of patches of size R grows polynomially with exponent n [Ju]. This yields the bound that π * :

H k ( Ê) → H k (Ω) is injective if k ≤ n -n ⊥ + 1 = (ν -2)n + 1.
Furthermore, the calculations done in [FHK] (for codimension 3 patterns) show that this is the best possible bound: π * is never injective in degree k > (ν -2)n + 1. In particular, for the standard tilings like the Penrose, Amman-Beenker, Socolar, and the icosahedral tilings, ν = 2, and hence π * is injective only in degree 0 and 1. 4.2. On the cokernel of π * . We consider here only the case of finitely generated cohomology. It comes not as a surprise that then coker π * is also finitely generated, see [FHK]. It can, however, have torsion in higher degrees [START_REF] Gähler | Integer Cech cohomology for icosahedral projection tilings[END_REF][START_REF] Gähler | Integer cohomology for rational projection patterns[END_REF]: the Tübingen Triangle Tiling is an example of a 2-dimensional tiling which has torsion in its second cohomology and all icosahedral tilings considered in [START_REF] Gähler | Integer cohomology for rational projection patterns[END_REF] have torsion in degree 3 (the Ammann-Kramer and the dual canonical D 6 tiling also have torsion in degree 2). A more subtle question is whether π * is always onto a direct summand; that is, whether the exact sequence

(2) 0 → H k ( Ê) π * -→ H k (Ω) → coker π * → 0.
splits and hence the torsion in the tiling cohomology agrees with the torsion of coker π * . While this is generally true if n ⊥ ≤ 2 it could be answered affirmatively in higher codimension n ⊥ only for rational projection patterns [START_REF] Gähler | Integer cohomology for rational projection patterns[END_REF]. Rational projection patterns are not only almost canonical but satisfy an additional rationality assumption, roughly that the cut-planes are projections onto R n ⊥ of lattice planes in QΓ. This assumption allows for the construction of a torus arrangement in Ê. Such torus arrangements were was first proposed in [Ka].

4.3. The frequency module. In this section we will prove that the frequency module, freq(Ω), of an almost canonical projection pattern is always finitely generated and hence the sequence (2) splits.

We start with some known background material. It is known that the factor map π is almost everywhere one-to-one and the measure on Ω the push forward of the (normalized) Haar-measure on Ê. This implies that the frequency module is generated by the volumes of all polyhedra in R n ⊥ whose faces lie in S. Here the volume of a polyhedron is measured with the help of the Lebesgue measure normalized so that the window K has volume 1.

Theorem 9. The frequency module freq(Ω) of an almost canonical projection pattern is finitely generated.

Proof. We call a point x ∈ R n ⊥ a cut-point if it is the unique point in the intersection of n ⊥ cut-planes. Clearly, any polyhedron whose faces lie in S has vertices which are cut-points. Given that the cut-points are dense we may subdivide any such polyhedron into simplices of dimension n ⊥ such that all vertices of the simplices are cut-points (we do not care whether the newly introduced faces lie in S). The theorem thus follows if we can show that the Z-module of volumes of all n ⊥ -simplices whose vertices are cut-points is finitely generated.

Denote by P the cut points. We claim that P -P is contained in a finitely generated Z-module. Suppose first that x, y ∈ P lie in a common singular space L of dimension 1 and both on the intersection with the same class of cut-plane, i.e. {x} = L ∩ (W iπ ⊥ (γ x )) and {y} = L ∩ (W i -π ⊥ (γ y )). Then x -y belongs to π L W i (Γ), the projection along W i onto L of π ⊥ (Γ). This group is, of course, finitely generated and since there are only finitely many W i we see that (P ∩ L) -(P ∩ L) is contained in a finitely generated Z-module. Now we can go from any cut point x to any other cut point y along singular lines and since there are only finitely many directions of singular lines x -y lies in a finitely generated Z-module M, say.

The volume of an n ⊥ -simplex with vertices (x 0 , • • • , x n ⊥ ) is one half of the determinant of the n ⊥ vectors x i -x 0 , i = 1, • • • , n ⊥ which all lie in M. Hence the determinant also lies in a finitely generated Z-module.

Corollary 10. For almost canonical projection patterns the sequence (2) splits.

Substitution tilings

We recall quickly how to calculate the cohomology of substitution tiling space Ω = Ω Φ referring the reader to [AP] for more details.

The collared Anderson-Putnam complex Y is an n-dimensional CW-complex whose n-cells are collared prototiles. Two of these cells are glued along (n -1)-faces if translates of the corresponding collared prototiles meet along a translate of that face in some tiling in Ω. There is a natural map p : Ω → Y assigning to a tiling the point in Y which corresponds to the position of the origin 0 in the collared tile of the tiling that contains 0. Furthermore, the substitution Φ induces a continuous surjection F : Y → Y with p • Φ = F • p. By the universality property of the inverse limit, p induces a map p : Ω → lim ← -(Y, F ) where lim ← -(Y, F ) denotes the inverse limit of the stationary system

• • • Y F → Y F → Y .
It is shown in [AP] that p is a homeomorphism that conjugates Φ with the shift F on lim ← -(Y, F ) and hence p :

H k (lim ← - (Y, F )) → H k (Ω) is an isomorphism. By the continuity property of Čech cohomology, H k (lim ← - (Y, F )) is naturally isomorphic with lim -→ (H k (Y ), F * )
. This direct limit is computable and we are interested in k = 1.

H 1 (Y ) is a free abelian group of finite rank, and so the stationary system H 1 (Y )

F * → H 1 (Y ) F * → H 1 (Y ) • • • is of the form Z N A → Z N A → Z N • • • for some N and N × N integer matrix A. Let ER(A) = n A n Q N be the eventual range of A. ER(A) = Q N if A has non-vanishing determinant, but always ER(A) = A N Q N . Then lim -→ (Z N , A) = {v ∈ ER(A) : ∃nA n v ∈ Z N } = n Ã-n Σ,
where Σ = ER(A) ∩ Z N and à is the restriction of A to ER(A).

If the substitution forces its border then the above construction works already if one considers non-collared tiles [AP] for the construction of the Anderson-Putnam complex Y . In the one dimensional context, that is for a substitution which can be symbolically defined on an alphabet of N letters, and for a substitution which forces its border in the sense that all substituted tiles start with the same tile and all end with the same tile (that is, the symbolic substitution has a common prefix and a common suffix), one may replace the Anderson-Putnam complex Y simply by a bouquet, X, of circles, one circle for each letter. In this case, H 1 (X) ∼ = Z N and the matrix A representing

H 1 (X) F * → H 1 (X)
in the basis provided by the cohomology classes of the circles is the transpose of the incidence matrix for the substitution. It turns out that, at least for determining the cohomology, we may also work with the bouquet X as long as the symbolic substitution has either a common prefix or a common suffix [START_REF] Andress | The Cech cohomology and the spectrum for 1-dimensional tiling systems[END_REF][START_REF] Barge | Cohomology in one-dimensional substitution tiling spaces[END_REF]. 5.1. One dimensional irreducible substitutions. For one-dimensional tilings the first cohomology group arrises in both exact sequences, the degree 1 version of the sequence (2), namely

(3) 0 → E θ -→ H 1 (Ω) → coker θ → 0
and, assuming unique ergodicity, sequence (1)

0 → Inf(Ω) → H 1 (Ω) τ -→ freq(Ω) → 0.
Some of the main results of [AR] give complete information about the structure of the above sequences in the context of one-dimensional primitive, irreducible substitutions of FLC. The work in [AR] starts with symbolic substitutions which are then realized geometrically by assigning a length to each symbol so as to realize it as an interval. Our case is slightly more restrictive, and can be compared if the lengths of the symbols are obtained from the left Perron Frobenius vector of the substitution matrix. The results of [AR] require generally that the substitution forces its border on one side in the sense that the symbolic substitution has a common prefix. A further assumption made on the substitution is that the characteristic polynomial of its substitution matrix is irreducible or, what amounts to the same, the dilation factor λ is an algebraic integer of degree equal to the number of prototiles (letters). One simply says that the substitution is irreducible in that case.

Theorem 11 ( [AR]). Consider a one-dimensional substitution tiling with common prefix. Assume furthermore that the substitution is irreducible. Then Inf(Ω) = 0. Moreover, if its dilation factor is a Pisot number then coker θ = 0.

Note that combined with Solomyak's result ( [S3]) on the existence of eigenfunctions, this yields a dichotomy: Either λ is a Pisot number and then coker θ = 0 or λ is not a Pisot number and then E = 0.

Hence we see that in one dimension, under the assumptions of common prefix and of irreducibility, the two exact sequences (1) and ( 3) are completely degenerate. We will see below in the examples that the situation is not at all like this if we look at non irreducible substitutions. Also, the results on projection patterns indicate that this behavior is restricted to one-dimensional tilings. 5.2. On the splitting of exact sequence (3). For substitution tilings, there is a simple criterion guaranteeing that the sequence (3) splits. Clearly, if F * is an isomorphism in degree 1 then H 1 (Ω) = H 1 (Y ) and since the latter is finitely generated we obtain from Cor. 6 that the sequence (3) splits. But we can do better.

Recall that the homeomorphism Φ : Ω → Ω defined by a substitution on the substitution tiling space satisfies

Φ(T -v) = Φ(T ) -Λ(v).
Let f be an eigenfunction with eigenvalue β ∈ E. Then

f (Φ(T -v)) = e 2πıβ(Λ(v)) f (Φ(T )), showing that f • Φ is an eigenfunction with eigenvalue Λ T β. It follows that Φ * θ(β) = [f • Φ] = θ(Λ T β), i.e.
θ intertwines the action of Λ T on E with that of Φ * on H 1 (Ω). Φ * thus induces a homomorphism Φ * on coker θ. We now work with rational coefficients, i.e. rational cohomology. Then, since H 1 (Y ; Q) is a finite dimensional vector space, also H 1 (Ω; Q) and thus coker Q θ = H 1 (Ω; Q)/θ(E ⊗ Z Q) are finite dimensional and so we can view Φ * as a finite matrix with rational coefficients which we denote by Ā. Of course, this matrix depends on a choice of basis, but not its determinant.

Proposition 12. Suppose that Ā is as above and that det( Ā) = ±1. Then the sequence (3) splits with

H 1 (Ω) ∼ = E ⊕ Z l , l = dim coker Q θ.
Proof. Recall from above that we may identify H 1 (Y ) = Z N and F * = A so that H 1 (Ω) ∼ = {v ∈ ER(A) : ∃nA n v ∈ Σ} where Σ = Z N ∩ ER(A). We now denote ER Z := {v ∈ ER(A) : ∃nA n v ∈ Σ} and by V Z ⊂ ER Z the subgroup corresponding to θ(E) under the above isomorphism. Then (3) can be identified with

(4) 0 → V Z ֒→ ER Z → ER Z /V Z → 0
and our aim is to show that there is a splitting map s : ER Z → V Z .

Note that H 1 (Ω; Q) ∼ = ER(A) and let denote V the subspace corresponding to θ(E ⊗ Z Q) under this isomorphism. V is the rational span of V Z . Then Ā can be seen as the linear map induced on W = ER(A)/V by A. Let π W denote the natural projection of ER(A) onto W and let Γ := π W (Σ). Then Σ and Γ are forward invariant under A and Ā, resp., and it follows from det( Ā) = ±1 that Ā restricts to an isomorphism of Γ. Working with rational vector spaces, the corresponding exact sequence 0 → V → ER(A) → W → 0 splits and there is a linear map s ′ : W → ER(A) such that π W • s ′ = id. Let π V : ER(A) → V be the projection onto V with kernel s ′ (W ). We claim that π V (ER Z ) = V Z which then shows that the restriction of π V to ER Z is a splitting map s for the sequence (4).

By Theorem 5, ER Z /V Z is torsion free and hence V Z = V ∩ ER Z . Indeed, if x ∈ V ∩ ER Z , there is p such that px ∈ V Z and thus, if x / ∈ V Z , it would map to a p-torsion element in the quotient ER Z /V Z . So we need only to show that π V (ER Z ) ⊂ ER Z .

Let x ∈ ER Z , i.e. x ∈ ER(A) and there is n such that A n x ∈ Σ. We have

A n x = v+γ with v = π V (A n x) ∈ V and γ = s ′ •π W (A n x) ∈ s ′ (Γ). Hence Ã-n A n x = Ã-n v+ Ã-n γ ∈ V + s ′ (Γ)
as Ã, the restriction of A to ER(A), is an isomorphism of ER(A) preserving V and inducing an isomorphism Ā on Γ. So we may write x = v ′ + γ ′ with v ′ ∈ V and γ ′ ∈ s ′ (Γ). Then v ′ = π V (x) and so we get

A n π V (x) = A n (x) -A n γ ′ ∈ Σ.
By a result of [KS] (see also [BG]), the collection of eigenvalues of the linear transformation A| V equals the collection of all algebraic conjugates of the eigenvalues of the linear inflation Λ. Moreover, the multiplicity of any λ as an eigenvalue of Λ is no larger than the multiplicity of λ as an eigenvalue of A. A non-unit determinant of Ā implies the existence of a non-unit (and nonzero) eigenvalue of A that has multiplicity greater than its multiplicity as an eigenvalue of Λ (see example 5.3.1).

Corollary 13. Suppose that every eigenvalue of A that is not an algebraic unit has the same multiplicity as an eigenvalue of Λ as it does as an eigenvalue of A. Then the sequence (3) splits, as in Proposition 12.

5.3.

Examples of non-irreducible substitutions. We present one-dimensional substitution tilings for which the sequences (1) and/or (3) do not split.

Note that by Lemma 2 the group of infinitesimal elements Inf(Ω) necessarily has trivial intersection with the image of θ and so the sum Inf(Ω) + imθ is direct. 5.3.1. Non-splitting example. We consider the following substitution a → abb, b → aaa, whose letters have equal length when realized as tiles of a one-dimensional tiling. We fix this length to be 1 and so this fixes the action of R on the continuous hull Ω. The expansion matrix is Λ = (3). By Host's characterization of the eigenvalues, β ∈ E if and only if β3 n mod 1 n→∞ -→ 0 which is clearly only possible if β3 n ∈ Z for some n and hence E = Z[1/3]. It is not difficult to write down an eigenfunction f 1 for β = 1: take any T 0 ∈ Ω and define f 1 (T 0 -t) = exp(2πıt). Given that all tiles have length 1 the value of f 1 (T 0 -t) depends only on the relative position of the tile on 0 (f is strongly pattern equivariant) and hence f extends by continuity to Ω. But the substitution is non-periodic and primitive and hence recognizable. This means that one can recognize the three-letter words in any T ∈ Ω which arrise from a substitution of a letter and hence also f 3 -n (T 0 -t) = exp(2πı3 -n t) is strongly pattern equivariant and so extends. In particular, and as it should be, [f 3

-n ] = [f 1 • Φ n ].
Note that the substitution has a common prefix. As explained above, the first cohomology group of Ω is therefore given by the direct limit defined by the transpose of the incidence matrix of the substitution, which is A = 1 2 3 0 .

Since this matrix is invertible over the rationals we obtain

H 1 (Ω) ∼ = n∈N A -n Z 2 = n∈N 1 5 1 -2 1 3 3 -n 0 0 (-2) -n 3 2 -1 1 Z 2 = Z[1/3] 1 1 + Z[1/2] -2 3 + 4 k=1 k 0 .
Taking into account the isomorphism between [Ω, S 1 ] and H 1 (Ω) it is easily seen that [f 1 ] corresponds to the element 1 1 and consequently [f 3 -n ] is represented as

A -n 1 1 = 1 3 n 1 1 . Thus θ(E) = Z[1/3] 1 1 .
We now consider the map τ which, according to the general theory [AR] is given by the pairing τ (x) = ν, x with the left-Perron Frobenius eigenvector ν of A normalized to ν 1 + ν 2 = 1. This is ν = 1 5 (3, 2). So

τ (A -n v) = 1 5 3 -n (3v 1 + 2v 2 ). It follows that freq(Ω) = imτ = 1 5 Z[1/3] and Inf(Ω) = ker τ = Z[1/2] -2 3 .
In particular, imθ and Inf(Ω) are subgroups of H 1 (Ω) with trivial intersection but they generate only a sub-group of index 5 in

H 1 (Ω). Likewise, τ • θ(E) = Z[1/3] is a subgroup of index 5 in imτ .
Proposition 14. Neither of the exact sequences (1) or (3) splits.

Since the substitution has a common prefix we get

H 1 (Ω) ∼ = n∈N A -n Z 2 = n∈N 1 3 1 -1 1 2 2 -n 0 0 (-1) -n 2 1 -1 1 Z 2 = Z[1/2] 1 1 + Z -1 2 + 2 k=0 k 0 .
As above one sees that [f 2 -n ] is represented as

A -n 1 1 = 1 2 n 1 1 . Thus θ(E) = Z[1/2] 1 1 .
We now consider the map τ . The left-Perron Frobenius eigenvector of A normalized to

ν 1 + ν 2 = 1 is ν = 1 3 (2, 1). So τ (A -n v) = 1 3 2 -n (2v 1 + 1v 2 ). It follows that freq(Ω) = 1 3 Z[1/2] and Inf(Ω) = Z -1 2 . Hence imθ + Inf(Ω) is a subgroup of index 3 in H 1 (Ω).
Proposition 15. The exact sequence (3) splits and hence H 1 (Ω) ∼ = Z[1/2] ⊕ Z. This splitting does not respect the order as the infinitesimal elements form an index 3 subgroup of the second summand.

Proof. coker θ is given by the quotient n∈N A -n Z 2 / ∼ where x ∼ y if there exist k, n

such that x -y = k 2 n 1 1 . It follows that 3 0 ∼ -1 2 , A -1 1 0 = 0 1 ∼ - 1 0 , and A -2 1 0 = 1 2 1 -1 ∼ 1 0
. Thus coker θ is thus generated by the equivalence class of the element 1 0 . In particular it is finitely generated and thus the sequence splits. Now it is clear that the infinitesimal elements form an index 3 subgroup of the second summand. The only possible orderings on Z are the trivial order, in which case all elements are infinitesimal, or the standard order, in which only 0 is infinitesimal.

The above splitting of H 1 (Ω) is thus not an order preserving splitting into a direct sum of ordered groups.

Proposition 16. The exact sequence (1) does not split.

Proof. The proof is like the above: Suppose the sequence 0

→ Z ψ → n∈N A -n Z 2 → 1 3 Z[1/2] → 0 splits, where ψ(1) = 1 -2
. There is then a Z-module homomorphism

s : n∈N A -n Z 2 → Z with s • ψ = id. This s extends to a Q-linear map s : Q 2 ∼ = n∈N A -n Z 2 ⊗ Z Q → Q ∼ = Z ⊗ Z Q. There is then a w ∈ Q 2 so that s(v) = w, v for all v ∈ Q 2 . Since each element of Z[1/2] 1 1 is infinitely divisible by 2 in n∈N A -n Z 2 also s(v) = w, v is infinitely divisible by 2 in Z for each v ∈ Z[1/2] 1 1 . This means that w is orthogonal to 1 1 , say w = t 1 -1 with t ∈ Q. Then s( 1 0 ) = t 1 -1 , 1 0 = t must lie in Z; that is, t ∈ Z. Now the restriction of s to ψ(Z) = Z 1 -2 is surjective so there is an x ∈ Z such that s(x 1 -2 ) = t 1 -1 , x 1 -2 = tx3 = 1. But this
implies that 1/3 is an integer.

5.4. Thue-Morse. We finally consider the Thue-Morse substitution 1 → 1 1, 1 → 11, whose letters have equal length when realized as tiles of a one-dimensional tiling. We fix this length to be 1 and so this fixes the action of R on the continuous hull Ω. The expansion matrix is Λ = (2). The substitution does not force its border and so we use the technique of collared tiles (the bracketed tile is the actual tile, the other two are the collar): a := 1( 1)1, b := 1( 1)1, c := 1( 1) 1, ā := 1(1) 1, b := 1(1) 1, c := 1(1)1.

The Anderson-Putnam complex Γ has 6 edges, namely the collared tiles which we orient to the right in the tiling, and 4 vertices v, v, w, w. Indeed w is a vertex at the end of c and the beginning of b (and w is a vertex at the end of c and the beginning of b), and v is a vertex at the end of b, ā and the beginning of a, c (and v is a vertex at the end of b, a and the beginning of ā, c). Its cohomology is thus that of the complex

0 → Z 4 δ T → Z 6 → 0 where 2 δ =     1 1 0 -1 0 -1 -1 0 -1 1 1 0 0 -1 1 0 0 0 0 0 0 0 -1 1     .
In particular, H 1 (Γ) ∼ = Z 3 . We now have to determine the matrix A corresponding to the endomorphism induced by the substitution on H 1 (Γ). The latter reads on collared tiles as follows: 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0

      
.

We find that the left-eigenvectors of σ which are not in imδ T are, 1 1 1 1 1 1 to eigenvalue 2, 1 1 1 -1 -1 -1 to eigenvalue 0, and 1 -1 0 1 -1 0 to eigenvalue -1. It follows that the eventual range of the endomorphsim A induced by σ T on Z 6 /imδ T has dimension 2 and that the restriction à of A to its essential range is obtained by row-reducing the above left-eigenvectors of σ to eigenvalues 2 and -1 w.r.t. the rows spanning imδ T + ker σ T . The result is 2 4 0 0 0 0 , the left-eigenvector of σ modulo imδ T , ker σ T to eigenvalue 2, and 2 -2 0 0 0 0 the one to eigenvalue -1. It follows that

à = 1 3 1 1 2 -1 2 0 0 -1 1 1 2 -1 = 0 1 2 1 .
Hence

H 1 (Ω) ∼ = n Ã-n Z 2 = Z[1/2] 1 2 + Z 1 -1 + 2 k=1 k 0 .
As above one sees that E = Z[1/2] and that f 2 -n (T 0 -t) = exp(2πı2 -n t) is strongly pattern equivariant and so extends to an eigenfunction to eigenvalue 2 -n . Furthermore the eigenfunction f 1 represents the class in H 1 (Ω) given by the class of (1, 1, 1, 1, 1, 1) T modulo imδ T , ker σ T which corresponds to (2, 4) T in n Ã-n Z 2 . Hence

θ(E) = Z[1/2] 1 2 .
More generally, since τ (v) = 0 for any left-eigenvector of σ to an eigenvalue different to 2 the map τ : n Ã-n Z 2 → R is given by τ ((2, 4) T ) = 1 and τ ((1, -1) T ) = 0. It follows that

τ (A -n v) = 1 6 2 -n (v 1 + v 2 ), freq(Ω) = 1 3 Z[1/2],
and Inf(Ω) = Z 1 -1 . Now the same calculation as for the period doubling sequence yields:

Proposition 17. The exact sequence (3) splits and hence H 1 (Ω) ∼ = Z[1/2] ⊕ Z. This splitting does not respect the order as infinitesimal elements form an index 3 subgroup of the second summand. The exact sequence (1) does not split.

5.5. The action of the substitution on first cohomology. As a final observation, let us consider complex valued cohomology looking at possibly complex eigenvalues of the action of Φ * on the first cohomology. It is shown in [BG] that all eigenvalues of Λ are also eigenvalues of Φ * .

Theorem 18. The image of τ is invariant under the action of Λ T . Moreover, suppose that Φ * x = λx for some cohomology element x ∈ H 1 (Ω, C). If τ (x) = 0 then λ must be an eigenvalue of Λ.

Proof. Note that, by unique ergodicity, where I k = [-k, k] n is the cube of side length 2k centered at 0 and ν is the Lebesgue measure. This holds for any T 0 . We have Φ * (f -1 df )(T 0 -x) = Λ T f -1 (Φ(T 0 ) -Λx)df (Φ(T 0 ) -Λx). Hence C µ (Φ * (f -1 df )) = Λ T C µ (f -1 df ) implying that τ ([f • Φ]) = Λ T τ ([f ]). So if x is an eigenvector of Φ * to eigenvalue λ and τ (x) = 0 then τ (x) is an eigenvector of Λ T to λ.

Thus, if x is in a generalized eigenspace of Φ * corresponding to an eigenvalue that is not also an eigenvalue of Λ, then x ∈ ker(τ ). In the case n = 1, such x ∈ [Ω, S 1 ] must be infinitesimal (see Subsection 2.4 and example 5.3.2).

a

  → bc, b → ac, c → bā, ā → bc, b → āc, c → ba which has incidence matrix σ ij (equal to the number of tiles of type i in the supertile of type j)

  k ) I k α(T 0 -x)dν(x)

E = {g n , n ∈ N} and we may take E n to be the group generated by {g 1 , • • • , g n }.

we use the bases: a, b, c, ā, b, c, and v, v, w, w.

Proof. Suppose the sequence (3) splits. There is then a Z-module homomorphism s : n∈N A -n Z 2 → E with s • θ = id. This s extends to a Q-linear map s :

The argument for the sequence ( 1) is completely similar, one only has to interchange the role of the eigenvectors of A. 5.3.2. Period doubling. We consider the following substitution a → ab, b → aa, whose letters have equal length when realized as tiles of a one-dimensional tiling. We fix this length to be 1 and so this fixes the action of R on the continuous hull Ω. The expansion matrix is Λ = (2). As above one sees that E = Z[1/2] and that f 2 -n (T 0 -t) = exp(2πı2 -n t) is strongly pattern equivariant and so extends to an eigenfunction to eigenvalue 2 -n .

The transpose of the incidence matrix of the substitution is


Kestutis Jarasiunas

Philippe Delaye

Jean-Claude Launay

Gérald Roosen

TIME RESOLVED BUILD-UP AND DECAY OF PHOTOREFRACTIVE AND FREE CARRIER GRATINGS IN CdTe:V

Laser induced picosecond transient gratings are used to study carrier transport via free carrier and photorefractive nonlinearities. Feed-back effect of light induced space charge electric field is found responsible for diffracted beam intensity oscillations during the grating recording and its decay.

. We present time resolved and excitation energy dependent analysis of carrier transport and of space charge field mechanisms in photorefractive CdTe:V. This kind of studies may help to optimize crystal parameters for different applications as well as to improve nonlinear optical techniques for determining electrical parameters of semiconductors.

). A YAG laser with pulse duration τ L = 28 ps FWHM and output energy up to 10 mJ.cm -2 was

used. Two s-polarized beams of equal intensities were used for recording a grating with period Λ = 1.8 µm. The semi-insulating crystal (ρ =5.10 8 Ω.cm) was grown by using the modified Bridgman technique and vanadium doped to 10 19 cm -3 to produce deep donor level at ≈E c -0.75 eV [8]. The absorption coefficient of the d= 5 mm thick sample is measured equal to 2 cm -1 at λ=1.06 µm. The sample was cut with faces along crystallographic directions (110), (-110), and (001). In our experiments the orientation of the grating vector Kg was along (110) or (001) directions. Probe and diffracted beam polarization states were monitored to separate coexisting refractive index gratings from different origins. The cases when pure photorefractive (PR) grating (i.e. p-diffracted component of s-polarized probe beam for Kg along (110)) or pure free carrier (FC) grating (p-diffracted component of p-polarized probe beam for Kg along (110) or (001)) have been analyzed experimentally. A Glan-polarizer oriented to reflect ppolarization was used for separation of Bragg-diffracted beam counterpropagating to one of the recording beams. The separated beam was directed to a fast silicon photodiode through the next p-transmitting polarizer. Such a set-up allows us to detect diffracted signals as low as 10 -6 -10 -7 . Diffracted beam intensity ‹I 1 › or grating efficiency η = ‹I 1 ›/ ‹I T › (here ‹I 1 › and ‹I T › are time-integrated values of diffracted and transmitted probe beams) as a function of excitation level E and of probe beam delay time ∆t are studied. For qualitative analysis of processes involved in carrier and space charge field dynamics we solved the set of six material equations [1,5] assuming that carrier generation from and via deep traps dominates.

Results and discussion. In Fig. 2 the exposure characteristics (EC) for FC and PR gratings (i.e dependences of the diffracted beam intensity ‹I 1 › versus excitation energy I ) are presented. EC of a semiconductor contains the main information about carrier generation It usually follows a power-law dependence at low excitations energies, ‹I 1 › =I γ with γ -value corresponding to carrier generation rate [9]. Assuming that carrier concentration ΔN increases with excitation by power law as ΔN=I β , we have the following relationship between slopes γ and β in the region of linear transmittance T of the sample (i.e. low excitations)

‹I 1 › = ‹I T › (π∆n d/λ) 2 = I T (πn eh ΔNd/λ) 2 = AI 2β+1 = AI γ (1
) here ∆n = n eh ΔN is the refractive index modulation of the grating, with ΔN the spatially modulated free carrier concentration and n eh is the refractive index change due to one electron hole pair.

In a general case, band-to-band linear carrier generation will lead to value β =1 and corresponding γ =3; for two photon light absorption process β =2 and γ =5. In our CdTe:V sample, γ -values reveal three excitation regions corresponding to different light absorption mechanisms. For FC gratings, we observe carrier generation from deep levels altered by two-step transitions via impurity states (γ=4), two-step and twophoton absorption of light (γ =5), and, finally, the saturation of diffracted beam intensity at high excitations. The defocussing of transmitted and diffracted beams at high excitations was observed, and this may be the reason for FC grating profile distorsions leading to ‹I 1 › saturation. To avoid this, we mainly carried out our measurements at E ≤ 5 mJ.cm -2 . Photorefractive nonlinearity, because its lower sensitivity, reveals two-step and two-photon carrier generation (γ =5) at lower excitations with the following decrease of EC slope from value γ = 5 to γ = 3 at higher excitations (linear increase with energy). This linear increase of the space charge field with excitation while the carrier concentration increases more rapidely in a nonlinear way points out that space charge field dynamics is more complicated than FC processes. Indeed, in this regime, both electrons and holes are generated and can compete giving opposite contributions to the space charge field.

Measurements of sample transmittivity vs. excitation energy (Fig. 3) have shown a tendency of absorption bleaching at low fluences and revealed the increasing role of nonlinear absorption with energy. In this way, transmittivity measurements confirmed the observed processes of carrier generation .

Carrier transport peculiarities have been observed in the dynamics of both FC and PR gratings.(Fig. 4 and5) We found that the FC grating decay time τ g varies with excitation energy and with time. The fastest average grating decay time τ g1 = 80-100 ps was observed at the lowest excitations used here (≈1 mJ.cm -2) . With increasing the incident power, the decay speed was slowering untill it reached the valueτ g2 ≈ 230 ps.

At high excitation energies the decay rate slowers also with time: at ∆t ≥ 600 ps we found τ g3 ≈ 365-410 ps. FC gratings decay finally with time constant of 1.7 ns which we attributed to carrier recombination time.

The processes of carrier diffusion and drift in light induced space charge (SC) field must be taken into account to explain the observed behaviour. One must consider that the value of created electric field E sc is dependent on carrier redistribution and thus varies with time. Electron diffusion during the action of laser pulse will create a SC field between the mobile charges and ionized deep traps. Then hole related processes may be observed: (i) the hole drift to grating minima will reduce the negative charge in these positions due to the previous diffusion of electrons and (ii) the build-up of a pure hole grating (that is π-shifted from its former position) will take place after the holes will reach the minima of light interference field (here E sc =0). Such a behaviour of free carrier grating was predicted by computer simulation of carrier dynamics in photorefractive crystals [5]. Thus, the role of holes is more and more increasing with time but in addition it increases also with excitation. In a general case, the average speed of FC grating decay will be governed by carrier bipolar diffusion related to electron-hole concentration at the given moment. The value of the bipolar diffusion coefficient D a are given by eq. ( 2):

D a = kTµ a / e = (kT/ e)(!N + !P) / (!N / µ h + !P / µ e ) (2)
with µ a the bipolar mobility, and ΔN (ΔP) the spatially modulated concentration of free electrons (holes). Thus the grating decay constant at the lowest excitations τ g1 corresponds to µ =330-410 cm 2 .V -1 .s -1 » µ h , showing that the electron generation is much more efficient than the hole one. At higher excitations decay slowers as hole concentration approaches electron one, and valueτ g2 ≈ 230 ps corresponds to carrier ambipolar mobility for the case ΔN = ΔP with the value µ a =135-140 cm 2 .V -1 .s -1 following from eq.2. The value of hole mobility µ h = 80-90 cm 2 .V -1 .s -1 , a typical value one for CdTe [10], was obtained from decay time τ g3 .

Photorefractive gratings (PR) have shown two decay times at high excitations (Fig. 5), when η ≥ 5.10 -6 . The fast decay has a time constant τ 1 = 220-230 ps that corresponds to an ambipolar carrier diffusion coefficient equal to 3.56-3.73 cm 2 .s -1 . Thus the origin of this refractive index modulation mechanism is a transient Dember field between mobile carriers. At longer delays or at lower excitation levels a slow component of grating decay with τ 2 ≈ 700 ps dominates. We attribute it to the relaxation of the ionic grating created in deep traps. Rough estimation of the SC field relaxation time τ SC was made for these conditions by using formula for the dielectric relaxation time modified by a grating factor [START_REF] Huignard | Nonlinear Optics: Materials and Devices[END_REF] and experimental values from FC grating measurements: carrier concentration N ≈ 3.10 14 cm -3 , their lifetime τ R = 1.7 ns and electron mobility µ e = 520 cm 2 .V -1 .s -1 . A good agreement between the calculated value τ sc ≈ 600 ps and the experimental one was found. In fact, the relaxation time is dependent on the ionized donor concentration and will vary in time scale. What we observed is probably the fastest component of τ sc after the Dember field has decayed.

We performed computer simulations of carrier and E sc dynamics for two cases: high excitation case when N=P and low excitation case when N>>P (Fig. 6). The fast decay of the grating by electron diffusion creates a SC field which in turn slowers the FC grating decay and prevents its from a total decay. At high excitations ambipolar diffusion governs the grating decay and the Dember field dominates over E sc in deep traps. This kind of PR and FC grating behaviour was found experimentally too. The simultaneous existence of two refractive index modulation mechanisms and their interaction lead to peculiarities observed in time-resolved grating build-up and decay. We have observed a step-like increase of the diffracted beam intensity during the action of laser pulse as well as saw-like oscillations (i.e. a fine structure of very fast decay and recovery of the grating on an essentially slower slope of diffusive grating decay) (see Fig. 4,5). We attribute this to a feed-back effect of the created space-charge field on the grating modulation, i.e. to the redistribution of the electron concentration due to the SC field. In addition to impeding their diffusion, the field will inforce carriers to move back to grating peaks from previous nearly homogeneous distribution. This will lead to electron grating reconstruction and to an increase of the grating efficiency. During the action of laser pulse, the competition of carrier generation, their diffusion and drift may create a continuing process of variation of τ g which will lead to a step-like shape of the grating build up. A similar shape of grating formation was observed in type-II quantum well heterostructures [START_REF] Weinert | [END_REF] where carrier confinement to the different layers leads to induced electric field.

This fine structure of grating decay and build-up exists on both FC and PR grating decay curves what shows that grating modulation and E sc temporal value do not equilibrate independently. A time and space varying feed-back strength may create trapezoidal carrier distribution with steep edge at position l= Λ/4. As discussed in [1], at low excitations the drift current evolves as the square of the input fluence while the diffusion current remains linear. The modelisation of carrier dynamics in nonhomogeneous electric fields is a very complicated task indeed. The linear modelisation we have given above has not shown evidences of temporal oscillations. However, experiments of light diffraction on FC gratings in external strong microwave fields [13] (with electric vector oriented along the grating vector as in our experiments) have also shown similar peculiarities of diffracted signal: an increase of diffraction efficiency that was dependent on grating period and five-fold stronger in GaAs than in Si, i.e. related to electron mobilities.

Further analysis of the obtained data allows us to calculate the electron mobility in CdTe:V. Using the average value of hole mobility µ h = 85 cm 2 .V -1 .s -1 (from FC decay) and the ambipolar one, µ a = 146 cm 2 .V -1 .s -1 (obtained from Dember field decay of PR grating, i.e. for the case ∆N= ∆P), we calculate µ e = 520 cm 2 .V -1 .s -1 . This value is twice smaller than in undoped CdTe [10]. The reason is probably a more efficient carrier scattering by neutral and ionized impurities. The mobility-lifetime product µ e τ R = 8.8 10 -7 cm 2 .V -1 follows from our measurements and is in good agreement with the one obtained for electrons by CW photorefractive gain measurements in this crystal (µτ = 8.10 -7 cm 2 .V -1 [8]). In addition, this transient grating technique permits detail studies of electron and hole generation rates. Directly measured grating decay times will give values of µ a and thus the corresponding N/P ratios at different excitation fluences will be obtained.

Conclusion.

We have demonstrated that the simultaneous investigations of coexisting optical nonlinearities in photorefractive media allow direct studies of crystal properties as monopolar carrier mobilities and carrier concentrations. This kind of knowledge is very important for the understanding of deep level behaviour with excitation energy, wavelength, doping, etc... Diffracted beam intensity oscillations during grating build-up and its decay were found and explained qualitatively by light induced space-charge field feed-back effect on carrier transport . Fig. 4. Free carrier (FC) grating dynamics at high excitation energies: E = 8.5 ± 0.5 mJ.cm -2 (1), 6.5 ± 0.3 mJ.cm -2 (2), 4.5 ± 0.25 mJ.cm -2 (3), and 3.75± 0.25 mJ.cm -2 (4); sample orientation, Kg along (110). Fig. 5. Photorefractive (PR) grating build-up and decay at different excitation levels: E= 6.5 ± 0.5 mJ.cm -2 (1), 4.75 ± 0.25 mJ.cm -2 (2), 2.88 ± 0.12 mJ.cm -2 (3), and 1.88 ± 0.12 mJ.cm -2 (4); sample orientation, Kg along (110). Error bars are presented on curve 2 showing the reality of the oscillations. Fig. 6. Computer simulations of carrier and space charge field dynamics for two cases: low excitation case (N>>P, on left side) and high excitation case (N=P, right side). Abbreviations correspond to refractive index modulation (Δn), electric space sharge field (E sc ), electron and hole grating modulation depths (N 1 and P 1 ); here ∆n=N 1 /m e + P 1 /m h .We used the following parameters for calculations: deep trap density -4.10 16 cm -3 , ionization cross-sections for electrons and holes S n =S p = 5.10 -17 cm 2 , carrier recombination coefficients γ n = 10γ p = 1.2 10 -7 cm 3 . s -1 , µ e = 520 cm 2 .V - 1 .s -1 , µ h = 80 cm 2 .V -1 .s -1 . Note: the delay in E sc build-up at low excitations is an important feature pointing out diffusive origin of E sc. This was observed experimentally only at low excitations (Fig. 5).

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig.1. Experimental set-up: GP -Glan polarizer, λ/2 -half-wavelength plate, Ddetector. Fig.2. Dependence of diffracted probe beam intensity ‹I 1 › and efficiency η vs. recording energy I for free carrier (FC) and photorefractive (PR) gratings (probe beam delay time corresponds to the end of the laser pulse; sample orientation Kg along (110)). Fig.3. Transmittivity of the sample at different excitation energies for p-polarized probe beam (filled circles) and for s-polarized one (open circles).Fig.4. Free carrier (FC) grating dynamics at high excitation energies: E = 8.5 ± 0.5 mJ.cm -2 (1), 6.5 ± 0.3 mJ.cm -2 (2), 4.5 ± 0.25 mJ.cm -2 (3), and 3.75± 0.25 mJ.cm -2 (4); sample orientation, Kg along (110). Fig.5. Photorefractive (PR) grating build-up and decay at different excitation levels:

Figure 3

 3 Figure 3

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Figure 6

 6 Figure 6

Acknowledgment. K.Jarasiunas acknowledges the support by M.R.T-France.


Kestutis Jarasiunas

Philippe Delaye

Gérald Roosen

OPTICAL NONLINEARITIES AND CARRIER TRANSPORT IN GaAs:EL2 AT HIGH EXCITATION LEVELS

Laser induced picosecond transient gratings are used to study carrier transport via free carrier and photorefractive nonlinearities in semi-insulating undoped GaAs bulk crystals. Carrier lifetime (τ R =1.5 ns), hole and ambipolar mobilities (µ h =410 cm 2 V -1 s -1 , µ a =760 cm 2 V -1 s -1 ) are measured directly and electron mobility equal to 5200 cm 2 V -1 s -1 is calculated. The optical properties of highly excited GaAs reveal deep donor EL2 transformation and intracenter absorption under picosecond excitation. Rotation of diffracted beam polarization is observed for the first time in orientation of grating vector along crystallographic axis (001) and attributed to a strain field at dislocations in LEC-grown GaAs wafers.

Résumé. La diffraction de la lumière sur les réseaux de porteurs libres et photoréfractif engendrés par des impulsions picoseconde permet d'étudier les non-linéarités optiques ainsi que le transport des porteurs libres dans le GaAs semi-isolant. Le temps de vie (τ R =1.5 ns), la mobilité des trous et la mobilité ambipolaire (µ h =410 cm 2 V -1 s -1 , µ a =760 cm 2 V -1 s -1 )) sont mesurés directement ; la valeur de la mobilité des électrons en est déduite (µ n =5200 cm 2 V -1 s -1 ).

Les propriétés optiques du GaAs sans fortes excitations révèlent les transformations du niveau profond EL2 et l'absorption intracentre sans excitation picoseconde. Une rotation de la polarisation du faisceau diffracté est observée pour la première fois avec un vecteur réseau orienté suivant l'axe cristallographique (001). Ce signal est attribué à un champ de contrainte à longue distance autour des dislocations dans le GaAs fabriqué par la technique LEC.

Introduction.

Numerous experiments on nonlinear optical phenomena in semi-conductors have been performed using transient grating techniques /1-3/. Most of previous studies used band-toband bipolar carrier plasma generation to create free carrier in Si, Ge, A 2 B 6 , A 3 B 5 compounds by one or two-photon absorption of light. Time resolved grating decay measurements of selfdiffracted beam intensity characteristics revealed carrier diffusion and recombination processes in pure crystals, carrier generation and transport peculiarities due to doping by shallow impurities, ion irradiation, external electric fields /4/. Possible applications of this technique in the fields of material research have been shown /5/.

In recent years, studies of optical nonlinearities in photorefractive crystals have been carried out in picosecond time scale /6-12/. In this kind of crystals photorefractive and non-photorefractive nonlinearities may take place simultaneously. Here a large variety of carrier generation mechanisms exists (absorption from deep levels, both single two-step processes from deep levels and two photon absorption) which finally will determine the magnitude and dynamics of these nonlinearities. Experiments in undoped CdTe crystals have revealed the role of Dember field in the enhancement of the photorefractive effect /8,9/ and the contribution of this nonlinearity to thin grating decay kinetics /10/. In vanadium-doped CdTe the separation of free carrier (FC) and photorefractive (PR) contributions allowed to measure monopolar and bipolar carrier mobilities and to observe a space-charge field feed-back effect on carrier transport /11,12/.

In order to explain peculiarities of optical nonlinearities described in these works, further theoretical and experimental investigations of carrier transport and of spacecharge field dynamics in photorefractive semiconductors are necessary. We thus continue this kind of experimental studies on picosecond time scale in highly excited GaAs crystals.

The sample used was a nominally undoped semi-insulating GaAs crystal, grown by the liquid encapsulated Czochralski (LEC) technique /13/. The crystal was cut with the faces along the crystallographic directions (110), (-110), and (001). The absorption coefficient of 1.5 mm thick sample was found equal to 1 cm -1 at 1.06 µm. It is well known that the dominant defect in undoped material is an intrinsic deep donor level EL2 and it solely determines the absorption at 1.06 µm /14, 15/. In this way we estimated deep donor concentration in our sample as NEL2=1.5 10 16 cm -3 , being a typical value for LEC-grown GaAs crystals /16/.

The configuration of the experimental set-up was a degenerate four-wave mixing scheme (figure 1). Using the Yag-laser with pulse duration τ L =28 ps FWHM and output energy up to 10 mJ.cm -2 , gratings with period Λ=1.8 µm were recorded by two s-polarized beams. The polarization states of transmitted probe beam and of diffracted one were monitored at different orientations of grating vector Kg with respect to sample crystallographic axis. In this way a contribution of a PR nonlinearity was separated from a coexisting and much stronger FC non linearity /8,17/. The cases when pure PR grating (i.e. p-diffracted component of s-polarized probe beam for Kg //(110)) or pure FC grating (i.e. p-diffracted component of p-polarized probe beam at Kg //(110)) were analyzed experimentally. A λ/2 plate was used for selecting the probe beam polarization (p or s) (figure 1). A Glan polarizer oriented to reflect ppolarization was used for separation of Bragg-diffracted beam counterpropagating to one of recording s-polarized beams. Then the diffracted beam was directed to a fast silicon photodiode through the next p-transmitting polarizer. Such a set-up allowed us to realize a noise-free interferometer able to detect diffracted signals as low as 10 -6 -10 -7 . Diffracted beam intensity I 1 or grating efficiency ! =< I 1 > / < I T > (here <I 1 > and <I T > are time-integrated values of diffracted and transmitted probe beams) as a function of excitation level I and of probe beam delay time Δt were studied. Sample transmittivity kinetics <I T > =f (Δt) were also measured at different excitation levels or probe beam polarizations in order to analyze light absorption mechanisms and their peculiarities.

Exposure characteristics.

The exposure characteristics (EC) of a semiconductor (i.e. dependence of the diffracted beam intensity <I 1 > versus excitation energy I) contains the main information about carrier generation processes. If light induced changes of optical properties are small enough (diffraction efficiency η < 0.01), EC follows a power law dependence, <I 1 > = I γ /18/. Assuming that carrier concentration increases with excitation as N = I β , the following relationship between slopes γ and β in log-log scale results /18,19/.

< I 1 > =< I T > !"nd/# ( ) 2 = I p T !n eh "Nd /# ( ) 2 = A I 2$ +1 = AI % (1)
here ΔN is the modulated FC concentration, neh is the refractive index change due to one electron-hole pair according to Drude model /1/, d is the thickness of the sample, and I p is the probe beam intensity. In this way the carrier generation rate parameter β can be easily obtained /18, 19/.

In a general case, a linear carrier generation with β= 1 will lead to γ = 3 and, correspondingly, two-step and two-photon light absorption (β=2) will increase the slope of EC to γ=5. In our semi-insulating undoped sample we observed increase of EC slope with value γ 1 =3.9 at excitation levels below 4 mJ.cm -2 both for FC and PR gratings (Fig. 2,3). From the experimental value of γ 1 we estimated parameter of carrier generation rate β 1 ~1.5 according to relationship given by equ. ( 1). This value of β 1 is determined by linear electron generation from deep EL2 traps which is altered due to nonlinear mechanisms of bipolar carrier plasma creation (i.e. by two-step transitions via ionized EL2 traps and by two-photon absorption). Diffraction efficiencies η = I 1 /I T = 4.10 -3 and 4.10 -5 and for FC and PR gratings respectively were reached at excitation level 4 mJ.cm -2 , and the following values of light induced refractive index modulation were thus estimated : Δn FC =2.10 -5 and Δn PR =2.10 -6 (at the end of excitation pulse, when Δt = τ L ). Assuming that the refractive index changes are due to FC concentration ΔN and space charge field E sc , the following absolute values of ΔN = 6.10 15 cm -3 and E sc =2000 V.cm -1

were calculated and used in the further analysis of results.

At excitation levels above 4 mJ.cm -2 a strong decrease of EC slope was observed (Fig. 2, γ 2 = 1.7 for FC grating), and corresponding value of sublinear carrier generation rate β 2 =0.5 was estimated. At these excitation levels PR grating also exhibits a slowering in space charge field creation with excitation (Fig. 3). In addition, exposure characteristics of sample transmittivity (Fig. 2, 3) revealed an increasing light induced absorption with excitation. These Time resolved measurements of induced absorption have been carried out to investigate its origin. We observed time-resolved structure of sample transmittivity : a bleaching on the front edge of pulse and a subsequent darkening with at least two components of the latter process (Fig. 4). The first component of darkening evolves as an integral of excitation pulse, reaches the value Δα=3-4 cm -2 and relaxes in 100 ps. Then the next competing component is arising, which reaches Δα=1-1.5 cm -1 and relaxes with time constant of 5-10 ns. This behaviour of transmitted beams was observed both for p-and s-polarized probe beams but the timeresolved structure was more evident in the latter case. For the second process (ii), transformation of EL2+ to EL2++ states may be assumed as a new channel of light absorption which lowers the part of fluence used for efficient electron generation from neutral donor states. Again, if transformation to EL2++ and its recovery to EL2+ take place during the action of the laser pulse, this will result in more efficient generation of holes because the optical cross section for transfer from EL2++ to EL2+ (σ p2 ~ 6.10 -17 cm 2 /28-29/) is larger than one from EL2+ to ground state (σ p1 ~2.10 -17 cm 2 /21/). As a result, one would also observe an increasing absorption and decreasing diffraction efficiency.

The observed processes of induced absorption due to intracenter transitions and EL2 transformation under picosecond excitation also might explain the variation in measured ratio of the hole cross section to electron cross section for the EL2 defect in undoped semi-insulating GaAs /24/. At low excitation levels this ratio was found varying from 0.1 to 0.25 (see /24/ and Ref. therein), but at picosecond excitations it increased to 0.8-1 /24/. The latter increase may result from activation of more efficient channels of hole generation via highly excited EL2 system.

In summary, we conclude that the simultaneous studies of light diffraction on FC and PR gratings as well as of dynamics of transmittivity have revealed different light absorption mechanisms, and, thus, may serve as a tool for further investigation of EL2 defect transformation features with excitation. The appearence of new states will cause modification of carrier generation rates, and of the built-in electric field ; in this way, creation and dynamics of PR and FC gratings will be affected. These features will be analyzed below from the transient grating decay measurements at different excitation levels.

3. Time-resolved grating decay.

Free carrier gratings (Kg//(110))

At low excitation levels, when 0.8 < I < 2.5 mJ.cm -2 , we observed two components in grating decay, τ e1 and τ e2 , both varying with the excitation in the intervals 80 ps < τ e1 < 160 ps and 30 ps < τ e2 < 60 ps (grating decay time corresponds to time interval Δt = τ e when diffracted signal decreases to level e -2 ). Decay times slower with excitation, and the first component shows tendency to saturation (fig. 5). At fluences I > 3 mJ/cm 2 grating decay starts with the time constant τ e2 =50-60 ps, which with the further increase of excitation approaches its shortest value observed in our experiments, namely, 42-43 ps (Fig. 6a). Moreover, this decay is followed by a slower part with a new time constant τ e3 =110-120 ps, and finally long lasting component of grating decay with τ e4 =1.5 ns appears (Fig. 6b).

The carrier dynamics must be analyzed taking into account light created space charge (SC) field and its feed-back effect to carrier transport /12/. At low excitations, electron diffusion will decrease the grating modulation depth with time constant of 6 ps, in this way creating SC field. The latter will slower or even stop carrier diffusion in this way preventing grating from its total decay /11/. The diffusion of holes during the action of laser pulse can be neglected due to their low mobility. After the pulse is over and electron grating has been partly erased, hole diffusion and drift become the dominant processes of grating decay. The diffusion will follow carrier bipolar mobility according to the well known relation :

µ a = N + P ( ) / P/ µ e + N / µ p ( ) (2)
which in our conditions (N >> P, thus excess holes diffuse in a n-type crystal) leads to µ a =µ p .

The carrier will drift with an effective mobility µ* :

µ * = N ! P ( )/ P /µ e + N /µ p ( ) (3)
which also has a limit value µ*=µ p under above conditions (N >> P at low excitation). The dependences of diffusion and drift related grating decay times (with Λ = 1.8 µm) calculated according to eq. 2 and eq. 3 are shown in Fig. 7. Thus, the first component of FC grating decay observed at the lowest excitations, τ 1e =80 ps, corresponds to hole drift in SC field and gives the value µ*=410 cm 2 .V -1 .s -1 from the expression of grating decay time due to drift : τ DRIFT =1/K g E sc µ*=τ 1e . At higher excitations, holes are generated more efficiently, and hole drift slowers : the observed τ 1e ≈ 100-160 ps corresponds to an N/P ratio varying from 6 to 2 with excitation (see eq. 3). Finally the drift component is saturated when N = P.

Hole drift with respect to the ionic grating will cause a decrease of SC field, and the diffusive current increases. Thus the second component of FC grating decay is a diffusion of a remaining electron grating (at low excitations) or bipolar electron-hole plasma diffusion (at high excitations) ; for the latter case, when N=P, we derive µ a = 760 cm 2 V -1 s -1 and correspondingly τ e2 = 43 ps. This qualitative model explains quite well the observed averaged time constants of grating decay, wich depend on the N/P ratio at a given excitation. In fact, the processes of diffusion and drift do not equilibrate independently, and the dynamics of grating modulation due to SC field feed-back effect is more complicated (especially at lower fluences, when electron-like grating is created).

The time required for holes to drift a distance x = Λ/2 is estimated as 250 ps. In this way a π-shifted from the former position hole grating will be created /21/ and its relaxation may be observed. Experimentally we indeed found the appearance of a new grating decay component

τ e3 .
Grating decay in this time domain is nonmonotonous and modulated by a faster one :

possibly, there is a competition of a simultaneously existing π-shifted hole grating and an electron grating reconstructed by the SC field. Even at a small modulation depth of the latter grating, its influence may be significant due to a much more efficient refractive index modulation by the electron concentration than by the hole one (the refractive index modulation coefficient by electrons is much larger than by holes : according to Drude model, this ratio is proportionnal to the inverse ratio of carrier effective masses, i.e. approximately to 10).

We attribute the last component of the grating decay with τ e4 ≈ 1.5 ns (Fig. 6) to the relaxation of an amplitude grating which is formed in EL2 traps due to their spatial photoionization. This trap modulation will be maintained until recombination of free carriers will reconstruct the initial neutral trap density. Thus, carrier lifetime is deduced directly from this relaxation, τ R =τ e4 . In addition, we estimated the value of the absorption coefficient modulation from the amplitude of the grating efficiency ! = "#d/ 2cos $ ( ) 2 % 2.10 &5 at !t " 300ps . This experimental value !" exp # 0.09cm $1 is in good agreement with the calculated one Δα=0.1 cm -1 deduced from the product of initial trap density N EL2 and their effective cross-section at high excitations ! n ! p / ! n + ! p the darkening induced at grating maxima becomes rapidly homogeneous along the grating vector This is possible if this darkening is due to mobile species which distribute homogeneously in some hundreds of picoseconds, and that these species are not creating significant refractive index modulation. Thus, generation of free holes might explain negligible influence of amplitude grating to diffraction efficiency. Computer simulation of processes involved in absorption of light and modulation of optical properties will give more detail insight into observed peculiarities.

3.2. Photorefractive gratings (K g //(110)).

We observed two components of pure PR grating decay, varying with excitation in a similar way as we have found for FC gratings.

At low fluences, SC electric field between positive charge at grating maxima (ionized donors and holes) and negative charge in grating minima (electrons transfered there by diffusion from grating maxima) is established very fast. Hole drift in this SC field is expected to be the next step in charge and field dynamics. The estimated value of PR grating decay at N >> P, according to eq. 3 and fig. 7, equals to 80 ps. This value correlates well with the experimental one τ e1 =80ps, the first component of FC grating decay at the lowest excitations. There is clear tendency of a grating decay slowering with excitation, and the values of 120 ps < τ e1 < 240 ps have been measured (Fig. 8a). This is consistent with the decrease of drift mobility with increasing N/P ratio (eq. 3), and the corresponding ratio N/P varying from 3.5 to 1.5 was estimated from the measured decay times.

At delay times larger than 60 ps, the speed of decay increased and was below our temporal resolution, being limited by duration of the laser pulse, i.e. τ e2 < τ L . This observation and the temporary oscillations of diffracted beam intensity at low excitations (Fig. 8a) point out that very fast processes are involved in SC field modulation. Similar perculiarities were found for picosecond PR gratings in CdTe and GaAs and attributed to carrier transport in nonhomogeneous electric fields or to hot carrier enhanced diffusion /9, 12/.

At higher excitation levels (I ~5 mJ.cm -1 ) grating decay has a smoother profile : the time domain, where τ e1 dominates, is compressed, and decay starts with the second components τ e2 , reaching values of 44-48 ps (Fig. 8b). This time constant at given grating period corresponds to ambipolar diffusivity of carrier plasma with N/P ratio varying from 1 to 1.3. This time constant indicates that Dember field between mobile species dominates and governs grating decay.

Again, when diffraction efficiency drops to level of 10 -5 , the decay is faster and corresponds to the case τ e2 <τ L .

Direct determination of monopolar mobilities is possible from the experimental data of PR and FC grating decay times measured at different fluences. First, hole diffusion coefficient Fig. 5 Free carrier grating dynamics at low excitation levels: E= 0.88±0.12 mJ.cm -2 (1), 1.12 ±0.12 mJ.cm -2 (2), 1.62 ± 0.12 mJ.cm -2 (3). Fig. 7. Grating decay times due to carrier diffusion (1) and drift (2) calculated by using the parameters of experiment : carrier mobilities in undoped GaAs (µ e ≈5200 cm 2 .

V -1 .s -1 , µ h ≈400 cm 2 .V -1 .s -1 ) and grating period Λ =1.8 µm.

  features alltogether point out that a new mechanism of light absorption at 1.06 µm in picosecond time scale dominates what leads to quenching of FC and PR gratings diffraction efficiency /20/.

  /. In this way the origin of amplitude modulation is confirmed.The strong induced absorption observed in transmittivity should lead to creation of an amplitude grating with Δαd = 0.2 -0.4 in time interval Δt = 100 -300 ps. Experiments have not revealed such a grating neither with corresponding amplitude nor with relaxation time correlating with recovery of transmittivity. This behaviour may be explained only in one way :

Fig. 1

 1 Fig. 1 Experimental set-up : G.P. : Glan Polarizer, λ/2 : half wave plate, D : Detector.

Fig. 2

 2 Fig. 2 Exposure characteristics of light diffraction on free carrier gratings (1a, 1b) and of sample transmittivity (2) . The p-polarized probe beam delay time is Δt= 26 ps (1a and 2) and Δt=170 ps (1b); for crystal orientation Kg along (110).

Fig. 3

 3 Fig. 3 Exposure characteristics of light diffraction on photorefractive gratings (1) and of sample transmittivity (2) . The s-polarized probe beam delay time is Δt=26 ps and the sample orientation corresponds toKg along (110).

Fig. 4 .

 4 Fig. 4. Dynamics of light induced change in transmittivity of the sample for p-polarized (1)and for s-polarized (2) probe beam at excitation fluence of 8 mJ.cm -2 .

Fig. 6

 6 Fig. 6 Free carrier grating dynamics at high excitation levels: (a) E= 3.25 ±0.25 mJ.cm -2 (1), 5.5 ±0.5 mJ.cm -2 (2) (b) 8.5 ± 0.5 mJ.cm -2 .

Fig. 8

 8 Fig. 8 Photorefractive grating dynamics at different excitation levels : (a) 2.25 ± 0.25 mJ.cm -2 (1) and 2.75 ± 0.25 mJ.cm -2 (2) ; (b) 6.75 ± 0.25 mJ.cm -2 (1) and 8.5± 0.5 mJ.cm -2 (2).

Figure 1 Figure 2 Figure 3

 123 Figure 1

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Figure 6

 6 Figure 6

Figure 7

 7 Figure 7

Figure 8

 8 Figure 8

  We ascribe this complicated dynamics of transmittivity and of diffraction quenching to deep EL2 trap related transitions under powerful picosecond excitation. Two kind of processes must be considered at excitation wavelength of 1.06 µm : (i) intracenter absorption in deep EL2 traps which does not affect the concentration of free electrons /26, 27/ and/or (ii) transformation of EL2+ to doubly ionized state EL2++ /28,29/. Our experiments provide some arguments that both processes may take place. First, the energy range for intracenter transition with its maxima at 1.18 eV /26/ corresponds to the laser wavelength used in our experiment ; next, this process is supposed to be non-photoactive /27/ what is identified from the quenching of diffraction efficiency. Moreover, photon capture cross section for this transition is quite large /26/ (about 10 16 cm 2 ). This transition to a non-photoactive state without charge generation is assumed to be the first step in a chain of processes leading to metastable state creation /30/. From ref.24, the rate of desexcitation of this transient non-photoactive state is supposed to be very fast at room temperature /24/ and is followed by hole generation. Thus the mechanism of intracenter transitions might explain diffraction quenching and a fast relaxation of induced absorption but not the full value of it.

	The origin of small bleaching at low excitation is due to partial deep level depletion what
	has been observed in previous works /6, 21, 22/. But the peak of induced absorption can not be
	explained neither by free carrier intraband absorption /23, 24/ nor by transitions between
	intervalence bands /25/. Peculiarities of two-photon absorption in transient energy transfer

and in transmittivity were carefully examined /6/ but this mechanism is not able to explain the features of our experiment : delayed position of transmittivity maximum, its fine structure, and long tail of relaxation.

Acknowledgments. Kestutis Jarasiunas thanks the Ministère de la Recherche (France) for his financial support and is very grateful to the Institut d'Optique where he spent his research stay.

FC =80 ps at Λ=1.8 µm. Next, the shortest limit of ambipolar decay τ 3e PR =43-44 ps leads to the values of ambipolar diffusion coefficient D a =18.6 -19 cm 2 .s -1 as well as of ambipolar mobility µ a ≈ 760 cm 2 .V -1 .s -1 . Thus, the electron mobility may be estimated from our measurements, and the value µ e ≈ 5200 cm 2 .V -1 .s -1 was obtained.

3.3. Gratings at K g //(001).

The photorefractive signal in the case Kg//(110) is usually separated from the nonphotorefractive one by using the effect of polarization rotation of linearly polarized probe beam to the orthogonal one /12, 17, 31/. Another usual test procedure to confirm the photorefractive origin of the refractive index modulation is the crystal rotation by 90° (so that crystallographic (001) axis is parallel to the grating wave vector Kg) and subsequent measurement of p-polarized component of the diffracted beam using s-polarized probe beam. No signal is to be observed in the latter case, and this was reported in number of experiments on semi-insulating CdTe and GaAs samples /8, 9, 12/.

In contrary, to this expectation, we have here observed a strong p-polarized diffracted signal from a grating oriented with K g //(001) and probed by a s-polarized beam. The diffracted signal reached values of η up to 10 -4 at the end of the excitation pulse, being of the same intensity as for the studied above photorefractive geometry. To test leakage of separationdetection system, we inserted a p-Si sample (ρ ≈ 10 Ω cm) instead of the GaAs, and found no signal. Moreover, using the same set-up, no diffracted signal was observed in photorefractive CdTe:V sample (or η value was below 10 -7 in the case K g //(001)) while for photorefractive orientation K g //(110), the value of η PR = 7.10 -5 has been measured /12/.

The preliminary studies of the origin of the observed effect have been carried out. The measurements of the grating decay at different fluences has revealed three components : the first one, related to hole drift and its saturation with excitation, the second one, relaxing with the time of ambipolar diffusivity, and the last component of decay with τ e ≤ τ L . These data indicate that the behaviour of the optical nonlinearity in K g //(001) is similar to the one for a typical photorefractive case studied above (i.e. in orientation K g //(110)) and, thus, give strong arguments towards the identification of a photorefractive origin of the grating in orientation K g //(001).

One of possible mechanisms that leads to photorefractive modulation mechanism is bulk photovoltaic effect in media containing macroscopic inhomogeneities which act as local space charges /32/. Indeed, LEC-grown semi-insulating GaAs crystals exhibit specific macroscopic distribution of dislocations /13, 16/ which are able to getter intrinsic defects. The concentration of midgap EL2 centers duplicates almost exactly the characteristic dislocation distribution pattern /5, 13/. The local region around dislocations (or order 30-50 µm) is perturbated : the inhomogeneous stress field and electric field accompany the dislocations /33/, which in semiinsulating GaAs are charged by electrons from EL2 centers /33, 34/. These fields may break the crystal symmetry, and non-zero components of the electrooptic tensor will appear. From the other hand, the existence of dislocation bands and bound states /33, 34/ may cause carrier confinement, anomalies in carrier transport, and new photovoltaic effects /32/. These proposed models will be worked out by further studies of optical and electrical activity of defects in LECgrown GaAs crystals with different dislocation densities.

Conclusion.

We have demonstrated novel possibilities of dynamic grating techniques to study carrier transport peculiarities via free carrier and photorefractive nonlinearities. Carrier generation and transport have been found strongly dependent on excitation level, and grating decay measurements at different fluences allowed direct determination of monopolar and bipolar mobilities, lifetime. Strong induced absorption and its non-photoactive features have been found at high excitation levels and discussed in terms of EL2 level transformation in picosecond time domain. Light diffraction with photorefractive features has been found in orientation K g //(001). This previously not observed phenomenon was attributed to long-range internal strain and electric fields at dislocations in LEC-grown bulk GaAs crystals.