Vincent Iampietro

Formal semantics of H-VHDL

Keywords:

This document serves as a reference for the definition of the syntax and semantics of a subset of VHDL named H-VHDL. H-VHDL aims at the definition of synthesizable designs with a synchronous behavior (i.e. the execution of the designs is synchronized with a clock signal).

= (v + )
NATMAX denotes the maximum value for a natural number. The NATMAX value depends on the implementation of the VHDL language; NATMAX must at least be equal to 2 31 -1.

Definition 1 (Elaborated design). An elaborated design ∆ ∈ ElDesign is a record <G, I, O, S, P, C> where:

-G ∈ id (t × v)
is the function yielding the type and the value of generic constants.

-I ∈ id t is the function yielding the type of input ports.

-O ∈ id t is the function yielding the type of output ports.

-S ∈ id t is the function yielding the type of declared signals.

-

P ∈ id (id (t × v))
is the function associating process identifiers to their local environment.

-C ∈ id ElDesign is the function mapping component instance identifiers to their own elaborated design version.

We assume that there is no overlapping between the identifiers of the sub-environments of an elaborate design (i.e, an identifier belongs to at most one sub-environment), and also between the identifiers of the sub-environments and the identifiers of local environments. When there is no ambiguity, we write ∆(x) to denote the value returned for identifier x, where x is looked up in the appropriate field of ∆. We write x ∈ ∆ to state that identifier x is defined in the domain of one of ∆'s field. We note ∆(x) ← v the overriding of the value associated to identifier x with value v in the appropriate field of ∆, ∆ ∪ (x, v) to note the addition of the mapping from identifier x to value v in the appropriate field of ∆, that assuming x / ∈ ∆. We write x ∈ F(∆), where F is a field of ∆, when more precision is needed regarding the lookup of identifier x in the record ∆.

Definition 2 (Design state). A design state σ ∈ Σ is a couple (S, C) where:

-S ∈ id v is the signal store, i.e. the function yielding the current values of ports and declared signals.

-C ∈ id Σ is the design instance store, i.e. the function yielding the current state of design instances.

When there is no ambiguity regarding which store a given identifier belongs to, we use σ(id) as a shorthand notation for S(σ)(id) or C(σ)(id). Similarly, we write id ∈ σ as a shorthand notation for id ∈ dom(S(σ)) or id ∈ dom(C(σ)).

3 Elaboration rules

Implicit default value

The following predicate states that a semantic type is well-formed.

WFBool

WF(bool)

WFNat l ≤ u ≤ NATMAX WF(nat(l, u)) WFArr WF(t) l ≤ u ≤ NATMAX
WF(array(t, l, u))

Table 7: Well-formed semantic type

According to the VHDL LRM, at the declaration of a port, a signal or a variable, these items must receive an implicit default value depending on their types [1, p.61, 64, 173]. The dv relation determines the default value for a given type.

DefaultVBool bool dv -→ false DefaultVCNat WF(nat(l, u)) nat(l, u) dv -→ l DefaultVCArr WF(array(t, l, u)) t dv -→ v size = (u -l) + 1 array(t, l, u) dv -→ create_array(size, v)
Table 8: Type default value

The create_array(size, v) expression yields an array of size size where each element is initialized with the value v.

Typing relation

The typing relation ∈ c checks that a given value conforms to a given type.

IsBool b ∈ B b ∈ c bool IsCNat WF(nat(l, u)) n ∈ [l, u] n ∈ c nat(l, u) IsArray WF(array(t, l, u)) v i ∈ c t i = 1, . . . , n n = (u -l) + 1 (v 1 , . . . , v n ) ∈ c array(t, l, u)

Static expressions

Static expressions are either locally static or globally static; the LRM defines locally static and globally static expressions as follows.

Locally static expressions

The SE l relation, defined by the following rules, states that an expression is locally static.

LSENat n ∈ N SE l (n) LSEBool b ∈ B SE l (b) LSEUOp SE l (e)
SE l (uop(e))

LSEBOp SE l (e 1 ) SE l (e 2 ) SE l (bop(e 1 , e 2 ))

Globally static expressions

The SE g relation, defined by the following rules, checks that an expression is globally static in the context of an elaborated design ∆.

GSELocal SE l (e) ∆ SE g (e) GSEGen id ∈ G(∆) ∆ SE g (id) GSEAggregate ∆ SE g (e i ) i = 1, . . . , n ∆ SE g ((e 1 , . . . , e n ))

Type indication elaboration

The econstr relation checks that a constraint is well-formed and evaluates the constraint bounds.

EConstr ∆ SE g (e 1 ) ∆ SE g (e 2 ) ∆ e 1 e -→ n 1 ∆ e 2 e -→ n 2 n 1 ∈ c nat(0, NATMAX) n 2 ∈ c nat(0, NATMAX) n1 ≤ n2 ∆ (e 1 , e 2 ) econstr -----→ (n 1 , n 2 )
The et relation checks the well-formedness of a type indication τ , and transforms it into a semantic type t (as defined in Table 6).

ETypeBool ∆ bool et -→ bool ETypeNat ∆ (e 1 , e 2 ) econstr -----→ (n 1 , n 2 ) ∆ nat(e 1 , e 2 ) et -→ nat(n 1 , n 2 ) ETypeArray ∆ τ et -→ t ∆ (e 1 , e 2 ) econstr -----→ (n 1 , n 2 ) ∆ array(τ, e 1 , e 2 ) et -→ array(t, n 1 , n 2 )
The econstr g relation checks that a generic constraint (i.e, a constraint appearing in a type indication associated with a generic constant declaration) is well-formed and evaluates the constraint bounds.

EConstrG SE l (e 1 ) SE l (e 2 ) e 1 e -→ n 1 e 2 e -→ n 2 n 1 ∈ c nat(0, NATMAX) n 2 ∈ c nat(0, NATMAX) n1 ≤ n2 (e 1 , e 2 ) econstrg -----→ (n 1 , n 2 )
The et g relation is specially defined to check the well-formedness of a type indication associated with a generic constant declaration.

ETypeGBool bool etg --→ bool ETypeGNat (e 1 , e 2 ) econstrg -----→ (n 1 , n 2 ) nat(e 1 , e 2 ) etg --→ nat(n 1 , n 2 )

Design elaboration

Given design store D ∈ id design and a dimensioning function M g ∈ id v, the elaboration phase generates an elaborated design ∆ ∈ ElDesign along with a default state σ e ∈ Σ out of a H-VHDL design d ∈ design. The elaboration performs static type-checking over the declarative (gens, ports and sigs) and behavioral (behavior) parts of a design, but also checks the well-formedness of generic and port maps in design instantiation statements, and finally checks that there is no multiply-driven signal.

DesignElab ∆ ∅ , M g d.gens egens ----→ ∆ ∆ , σ d.sigs esigs ---→ ∆ , σ ∆, σ ∅ d.ports eports ----→ ∆ , σ D, ∆ , σ d.beh ebeh ---→ ∆ , σ NoMDInCs(∆ , d.beh) D, M g d elab --→ ∆ , σ 3.6 Generic clause elaboration GenElabDimen τ etg --→ t SE l (e) e e -→ v v ∈ c t M(id) ∈ c t M(id) = v id / ∈ ∆ ∆, M (id, τ, e) egens ----→ ∆ ∪ (id, (t, M(id))) GenElabDefault τ etg --→ t e e -→ v SE l (e) v ∈ c t id / ∈ ∆ id / ∈ M ∆, M (id, τ, e) egens ----→ ∆ ∪ (id, (t, v)) GenElabComp ∆, M (id, τ, e) egens ----→ ∆ ∆ , M gens egens ----→ ∆ ∆, M (id, τ, e), gens egens ----→ ∆ 3.7 Port clause elaboration InPortElab ∆ τ et -→ t t dv -→ v id / ∈ ∆ id / ∈ σ ∆, σ (in, id, τ ) eports ----→ I(∆) ∪ (id, t), σ ∪ (id, v) OutPortElab ∆ τ et -→ t t dv -→ v id / ∈ ∆ id / ∈ σ ∆, σ (out, id, τ ) eports ----→ O(∆) ∪ (id, t), σ ∪ (id, v) PortElabComp ∆, σ pdecl eports ----→ ∆ , σ ∆ , σ ports eports ----→ ∆ , σ ∆, σ pdecl, ports eports ----→ ∆ , σ
where pdecl ::= (in|out), id, τ .

Architecture declarative part elaboration

SigElab ∆ τ et -→ t t dv -→ v id / ∈ ∆ id / ∈ σ ∆, σ (id, τ ) esigs ---→ S(∆) ∪ (id, t), σ ∪ (id, v) SigElabComp ∆, σ (id, τ ) esigs ---→ ∆ , σ ∆ , σ sigs esigs ---→ ∆ , σ ∆, σ (id, τ ), sigs esigs ---→ ∆ , σ 3.9 Behavior elaboration CsParElab D, ∆, σ cs ebeh ---→ ∆ , σ D, ∆ , σ cs ebeh ---→ ∆ , σ D, ∆, σ cs || cs ebeh ---→ ∆ , σ CsNullElab D, ∆, σ null ebeh ---→ ∆, σ PsElab ∆, Λ ∅ vars evars ---→ Λ ∆, S, Λ valid ss (ss) id / ∈ ∆ σ = (S, C) D, ∆, σ ps(id, vars, ss) ebeh ---→ ∆ ∪ (id, Λ), σ CompElab M ∅ g emapg ----→ M D, M d elab --→ ∆ c , σ c ∆, ∆ c , S valid ipm (i) ∆, ∆ c valid opm (o) idc / ∈ ∆, idc / ∈ σ σ = (S, C) D(ide) = d D, ∆, σ comp(id c , id e , g, i, o) ebeh ---→ ∆ ∪ (id c , ∆ c ), σ ∪ (id c , σ c ) 3.9.1 Process declarative part elaboration VarElab ∆ τ et -→ t t dv -→ v id / ∈ Λ id / ∈ ∆ ∆, Λ (id, τ ) evars ---→ Λ ∪ (id, (t, v)) VarElabComp ∆, Λ (id, τ ) evars ---→ Λ ∆, Λ vars 3.9.3 Map validity Dimensioning function construction AssocGElab SE l (e) e e -→ v id / ∈ M M (id ⇒ e) emapg ----→ M ∪ (id, v) GMElab M (id ⇒ e) emapg ----→ M M g emapg ----→ M M (id ⇒ e), g emapg ----→ M Input port map validity ListIPMSimple ∆, S e e -→ v v ∈ c t id / ∈ L, id ∈ I(∆c) vi s.t. (id, vi) ∈ L ∆c(id) = t ∆, ∆ c , S, L (id ⇒ e) listipm ----→ L ∪ {id} ListIPMPartial SE l (e i ) e i e -→ v i ∆, S e e -→ v v i ∈ c nat(n, m) v ∈ c t id / ∈ L, (id, vi) / ∈ L id ∈ I(∆c) ∆c(id) = array(t, n, m) ∆, ∆ c , S, L (id(e i ) ⇒ e) listipm ----→ L ∪ {(id, v i )} ListIPMCons ∆, ∆ c , S, L (name ⇒ e) listipm ----→ L ∆, ∆ c , S, L i listipm ----→ L ∆, ∆ c , S, L (name ⇒ e), i listipm ----→ L
where L ⊆ id ∪ (id × N).

check pm (P orts, L) ≡ ∀id f ∈ dom(P orts), id f ∈ L ∨ ∃t ∈ type, n, m ∈ N, (P orts(id f ) = array(t, n, m)∧ ∀i ∈ [n, m], (id f , i) ∈ L)
where P orts ∈ id type, and

L ⊆ id ∪ (id × N). ValidIPM ∆, ∆ c , S, L ∅ i listipm ----→ L check pm (I(∆ c ), L) ∆, ∆ c , S valid ipm (i)
Output port map validity

ListOPMSimpleToSimple id f / ∈ L id f ∈ O(∆c) ida ∈ S(∆) ∪ O(∆) ∆c(id f ) = ∆(ida) = t ∆, ∆ c , L (id f ⇒ id a ) listopm -----→ L ∪ {id f } ListOPMSimpleToPartial SE l (e i ) e i e -→ v i v i ∈ c nat(n, m) id f / ∈ L id f ∈ O(∆c) ida ∈ S(∆) ∪ O(∆) ∆c(id f ) = t ∆(ida) = array(t, n, m) ∆, ∆ c , L (id f ⇒ id a (e i )) listopm -----→ L ∪ {id f } ListOPMSimpleToOpen id f / ∈ L id f ∈ O(∆c) ∆, ∆ c , L (id f ⇒ open) listopm -----→ L ∪ {id f } ListOPMPartialToSimple SE l (e i ) e i e -→ v i v i ∈ c nat(n, m) id f , (id f , vi) / ∈ L id f ∈ O(∆c) ida ∈ S(∆) ∪ O(∆) ∆c(id f ) = array(t, n, m) ∆(ida) = t ∆, ∆ c , L (id f (e i ) ⇒ id a ) listopm -----→ L ∪ {(id f , v i )} ListOPMPartialToPartial SE l (e i ) SE l (e i ) e i e -→ v i e i e -→ v i v i ∈ c nat(n , m ) v i ∈ c nat(n, m) id f , (id f , vi) / ∈ L id f ∈ O(∆c) ida ∈ S(∆) ∪ O(∆) ∆c(id f ) = array(t, n, m) ∆(ida) = array(t, n , m ) ∆, ∆ c , L (id f (e i ) ⇒ id a (e i )) listopm -----→ L ∪ {(id f , v i )} ListOPMCons ∆, ∆ c , L assoc po listopm -----→ L ∆, ∆ c , L o listopm -----→ L ∆, ∆ c , L assoc po , o listopm -----→ L
where L ⊆ id ∪ (id × N) and assoc po ::= (id ⇒ (name|open)) (id(e) ⇒ name).

ValidOPM ∆, ∆ c , L ∅ , L ids∅ o listopm -----→ L, L ids ∆, ∆ c valid opm (o)

Detection of multiply-driven signals

Assignment in sequential statement

The is_assgd_in_ss function returns true if the signal identifier id s is assigned in the sequential statement ss, i.e. if id s appears in the left part of a signal assignment statement.

S(id) = v id ∈ S(∆) ∪ I(∆) ∆, S, Λ id e -→ v Var Λ(id) = (t, v) ∆, S, Λ id e -→ v Gen G(∆)(id) = (t, v) ∆, S, Λ id e -→ v Out S(id) = v id ∈ O(∆) ∆, S id eo -→ v IdxOut e i e -→ n i a[i] = v id ∈ O(∆) ∆(id) = array(t, n, m) S(id) = a i = ni -n ∆, S id(e i ) eo -→ v IdxSig ∆, S, Λ e i e -→ n i a[i] = v id ∈ S(∆) ∪ I(∆) ∆(id) = array(t, n, m) S(id) = a i = ni -n ∆, S, Λ id(e i ) e -→ v IdxVar ∆, S, Λ e i e -→ n i a[i] = v id ∈ Λ Λ(id) = (array(t, n, m), a) i = ni -n ∆, S, Λ id(e i ) e -→ v Cst vcst(cst) = v ∆, S, Λ cst e -→ v BOp ∆, S, Λ e 1 e -→ v 1 ∆, S, Λ e 2 e -→ v 2 vbop(bop, v 1 , v 2 ) = v ∆, S, Λ bop(e 1 , e 2 ) e -→ v Not ∆, S, Λ e e -→ b ∆, S, Λ not(e) e -→ ¬b Aggreg ∆, S, Λ e i e -→ v i i = 1, . . . , n ∆, S, Λ (e 1 , . . . , e n ) e -→ (v 1 , . . . , v n ) Evaluation of constants vcst(b) = b vcst(n) = n if n ≤ NATMAX ∅ otherwise Evaluation of binary operators vbop(and, b 1 , b 2 ) = b 1 && b 2 vbop(or, b 1 , b 2 ) = b 1 || b 2 vbop(add, n 1 , n 2 ) = n 1 + n 2 if n 1 + n 2 ≤ NATMAX ∅ otherwise vbop(sub, n 1 , n 2 ) = n 1 -n 2 if n 1 ≥ n 2 ∅ otherwise vbop(eq, b 1 , b 2 ) = (b 1 = B b 2 ) vbop(eq, n 1 , n 2 ) = (n 1 = N n 2 ) vbop(eq, a 1 , a 2 ) = (a 1 = a a 2 ) vbop(neq, v 1 , v 2 ) = ¬b if vbop(eq, v 1 , v 2 ) = b ∅ otherwise vbop(gt, n 1 , n 2 ) = n 1 > n 2 vbop(ge, n 1 , n 2 ) = n 1 ≥ n 2 vbop(lt, n 1 , n 2 ) = n 1 < n 2 vbop(le, n 1 , n 2 ) = n 1 ≤ n 2
The function = a ∈ a → a → B is the equality between two arrays, where a ::= (v + ) as defined in the value set presented in Table 6. It yields true if the two compared arrays are of the same size and have all their elements positionally equal; the result is false otherwise.

Evaluation of sequential statements

SigAssign ∆, S r , Λ e e -→ v v ∈ c t id ∈ S(∆) ∪ O(∆) ∆(id) = t ∆, S r , S w , Λ id ⇐ e ss -→ S w (id) ← v, Λ IdxSigAssign ∆, S r , Λ e i e -→ n i ∆, S r , Λ e e -→ v v ∈ c t n i ∈ c nat(n, m) id ∈ S(∆) ∪ O(∆) ∆(id) = array(t, n, m) Sw(id) = a i = ni -n a[i] ← v = a ∆, S r , S w , Λ id(e i ) ⇐ e ss -→ S w (id) ← a , Λ VarAssign ∆, S r , Λ e e -→ v v ∈ c t Λ(id) = (t, val) ∆, S r , S w , Λ id := e ss -→ S w , Λ(id) ← (t, v) IdxVarAssign ∆, S r , Λ e i e -→ n i ∆, S r , Λ e e -→ v n i ∈ c nat(n, m) v ∈ c t Λ(id) = array(t, n, m), a i = ni -n a[i] ← v = a ∆, S r , S w , Λ id(e i ) := e ss -→ S w , Λ(id) ← (t, a ) IfElse ∆, S r , Λ e e -→ true ∆, S r , S w , Λ ss 1 ss -→ S w , Λ ∆, S r , S w , Λ if(e) {ss 1 } else {ss 2 } ss -→ S w , Λ IfElse⊥ ∆, S r , Λ e e -→ false ∆, S r , S w , Λ ss 2 ss -→ S w , Λ ∆, S r , S w , Λ if (e) {ss 1 } else {ss 2 } ss -→ S w , Λ Loop⊥ ∆, S r , Λ e 2 > 0 e -→ true ∆, S r , Λ e 1 e -→ v 1 v 1 ∈ c nat(0, NATMAX) ∆, S r , S w , Λ(id) ← nat(0, NATMAX), v 1 ss ss -→ S w , Λ ∆, S r , S w , Λ for (id, e 1 + 1, e 2 -1) {ss} ss -→ S w , Λ ∆, S r , S w , Λ for (id, e 1 , e 2 ) {ss} ss -→ S w , Λ Loop ∆, S r , Λ e 2 = 0 e -→ true ∆, S r , S w , Λ for (id, e 1 , e 2 ) {ss} ss -→ S w , Λ \ {id} RisingEdgeDefault f =↑ f ∈ {↓, i, c} ∆, S r , S w , Λ rising {ss} ss f --→ S w , Λ FallingEdgeDefault f =↓ f ∈ {↑, i, c} ∆, S r , S w , Λ falling {ss} ss f --→ S w , Λ RisingEdgeExec ∆, S r , S w , Λ ss ss ↑ --→ S w , Λ ∆, S r , S w , Λ rising {ss} ss ↑ --→ S w , Λ FallingEdgeExec ∆, S r , S w , Λ ss ss ↓ --→ S w , Λ ∆, S r , S w , Λ falling {ss} ss ↓ --→ S w , Λ RstDefault ∆, S r , S w , Λ ss 2 ss f --→ S w , Λ f = i f ∈ {↑, ↓, c} ∆, S r , S w , Λ rst {ss 1 } else {ss 2 } ss f --→ S w , Λ RstExec ∆, S r , S w , Λ ss 1 ssi --→ S w , Λ ∆, S r , S w , Λ rst {ss 1 } else {ss 2 } ssi --→ S w , Λ Seq
∆c(id) = t ∆, ∆ c , S, S c (id ⇒ e) mip --→ S c (id) ← v MipPartial ∆, S e e -→ v e i e -→ n i v ∈ c t n i ∈ c nat(n, m) id ∈ I(∆c) ∆c(id) = array(t, n, m) i = ni -n Sc(id) = a a[i] ← v = a ∆, ∆ c , S, S c (id(e i ) ⇒ e)
-→ v v ∈ c t id ∈ S(∆) ∪ O(∆) ∆(id) = t ∆, ∆ c , S, S c (name ⇒ id) mop ---→ S(id) ← v MopPartial e i e -→ n i ∆ c , S c name eo -→ v v ∈ c t n i ∈ c nat(n, m) id ∈ S(∆) ∪ O(∆) ∆(id) = array(t, n, m) i = ni -n S(id) = a a[i] ← v = a ∆, ∆ c , S, S c (name ⇒ id(e i )) mop ---→ S(id) ← a MopComp ∆, ∆ c , S, S c assoc po mop ---→ S ∆, ∆ c , S , S c o mop ---→ S ∆, ∆ c , S, S c assoc po , o mop ---→ S
where assoc po ::= (id ⇒ (name|open)) (id(e) ⇒ name).

Evaluation of concurrent statements

1 Definition merge(σ, σ , σ ) := 2 let σ = (S, C) in 3 let σ = (S , C ) in 4 let σ = (S , C ) in 5 let Sm(id) =      S (id) if S (id) = S(id) S (id) if S (id) = S(id) S(id) otherwise in 6 let Cm(id) =      C (id) if C (id) = C(id) C (id) if C (id) = C(id) C(id) otherwise in 7 (Sm, Cm).
Listing 1: The merge function that fuses together an origin state σ, with two states σ and σ generated by the execution of two H-VHDL concurrent statements.

The merge function is correct if the three input design states assume the same domains in their signal store and design instance store, and also if there are no multiply driven signal, i.e. a signal that would have a value that is different from the original signal store S in both signal stores S and S . In practice, the case where a design instance identifier could be associated with a different state in both design instance store C and C is not possible. Such a case could only arise if two design instances with the same identifier exist in a design's behavior, and such a design can never be elaborated.

Ps

∆, S, S, Λ ss

ss f --→ S , Λ ∆(id) = Λ σ = (S, C) D, ∆, σ ps(id, vars, ss) cs f --→ (S , C) Comp ∆, ∆ c , S, S c i mip --→ S c D, ∆ c , (S c , C c ) d.beh cs f --→ σ c ∆, ∆ c , S, S c o mop ---→ S D(ide) = d ∆(idc) = ∆c σ(idc) = σc σ = (S, C) σc = (Sc, Cc) σ c = (S c , C c ) D, ∆, σ comp(id c , id e , g, i, o) cs f --→ (S , C(id c ) ← σ c ) Par D, ∆, σ cs cs f --→ σ D, ∆, σ cs cs f --→ σ D, ∆, σ cs || cs cs f --→ merge(σ, σ , σ ) Null ∆, σ null cs f --→ σ
where f ∈ {i, ↑, ↓, c}. The i flag (resp. ↑, ↓, c) stands for the evaluation of concurrent statements during the initialization phase (resp. rising edge, falling edge and stabilization phases).

We choose to remove from the side condition of the rule Par that was stating the absence of multiply-driven signal, and of twin design instances. The two can be obtain after a successful elaboration phase.

  ps(id, process identifier vars = {(id, τ ) * }, local variable declarations body = ss) statement body comp ::= comp(id c , component instance identifier id e , instantiated design identifier g = {(id ⇒ e) * }, generic constant map i = {(name ⇒ e) * }, input port map o = {((id ⇒ (name|open))|(id(e) ⇒ name)) * }) output port map Table 4: Concurrent statements design ::= {gens = {(id, τ, e) * }, generic constants ports = {((in|out), id, τ ) * }, input and output ports sigs = {(id, τ ) * },

  No multiply-driven signal check NoMDInCs(∆, cs) ≡ ∀id s ∈ O(∆) ∪ S(∆), is_md_in_cs(cs, id s ) = false.

Sig

  ∆, S r , S w , Λ ss 1 ss -→ S w , Λ ∆, S r , S w , Λ ss 2 ss -→ S w , Λ ∆, S r , S w , Λ ss 1 ; ss 2 ss -→ S w , Λ Null ∆, S r , S w , Λ null ss -→ S w , Λ 4.3 Evaluation of input and output port maps 4.3.1 Evaluation of an input port map

  mip --→ S c (id) ← a MipComp ∆, ∆ c , S, S c (name ⇒ e) mip --→ S c ∆, ∆ c , S, S c i mip --→ S c ∆, ∆ c , S, S c (name ⇒ e), i mip --→ S c 4.3.2 Evaluation of an output port map MopOpen ∆, ∆ c , S, S c (id ⇒ open) mop ---→ S MopSimple ∆ c , S c name eo

  No multiply-driven signal nmds(S, S , S ) ≡ dom(S) = dom(S ) = dom(S) ∧ ∀id s , S(id s ) = S (id s ) ∨ S(id s ) = S (id s ) where S, S , S ∈ id v, i.e. three signal stores.State equality relationσ Σ = σ ≡ ∀id s , S(σ)(id s ) = S(σ )(id s ) ∧ ∀id c , C(σ)(id c ) Σ = C(σ )(id c ) No twin design instance ntdi(C, C , C ) ≡ dom(C) = dom(C ) = dom(C) ∧ ∀id c , C(id c ) Σ = C (id c ) ∨ C(id c ) Σ = C (id c ) where C, C , C ∈ id Σ, i.e. three design instance stores. Input port values update). Given a simulation environment E p ∈ N → (id v), let us define the function that update of the value of signals at a given design state σ ∈ Σ and clock cycle count τ ∈ N. The function is defined as follows: inj(σ, E p , τ ) = (S ← ∪ E p (τ ), C) where σ = (S, C), and for all sets X and Y , and for all partial functionf, f ∈ X Y , f ← ∪ f (x) = f (x) if x ∈ dom(f ) f (x) otherwise . Cycle D, ∆, inj(σ, E p , τ ) cs cs ↑ --→ σ ↑ D, ∆, σ ↑ cs -→ σ D, ∆, σ cs cs ↓ --→ σ ↓ D, ∆, σ ↓ cs -→ σ D, E p , ∆, τ, σ cs p , ∆, 0, σ cs → [ ] SimLoop D, E p , ∆, τ, σ cs ↑,↓ --→ σ , σ D, E p , ∆, τ -1, σ cs → θ τ > 0 D, E p , ∆,τ, σ cs → (σ :: σ :: θ) p , ∆, τ, σ 0 d.beh → θ ∀τ, dom(Ep(τ )) ⊆ dom(I(∆)) D, M g , E p , τ d f ull --→ (σ 0 :: θ)

Table 1 :

 1 Expressions τ ::= bool boolean | nat (e 1 , e 2 ) natural range e 1 to e 2 | array (τ , e 1 , e 2 ) array of τ with index range e 1 to e 2

	1 Abstract syntax of H-VHDL
	e ::= name	read a signal, a local variable
		or a generic constant value
	| cst	constant
	| bop(e 1 , e 2 )	binary operation
	| uop(e)	unary operation
	| (e + )	aggregate expression
	name ::= id	read a signal, local variable,
		or generic constant value
	| id(e)	read value of an array signal
		or local variable at index e

cst ::= n | b natural or Boolean bop ::= and | or Boolean operators | add | sub natural number arithmetic | eq | ne | gt | ge | lt | le comparisons uop ::= not Boolean negation

Table 2 :

 2 Type indication

	ss ::= name ⇐ e	assignment to a signal
	| name := e	assignment to a local variable
	| if(e){ss 1 } else {ss 2 } conditional
	| for(id, e 1 , e 2 ){ss}	range loop
	| falling{ss}	falling edge block
	| rising{ss}	rising edge block
	| rst {ss 1 } else {ss 2 }	reset conditional
	| ss 1 ;ss 2	sequence
	| null	no operation

Table 3 :

 3 Sequential statements

Table 6 :

 6 The t (type) and v (value) semantic types. Boolean type | nat(n 1 , n 2 ) natural range n 1 to n 2 | array(t, n 1 , n 2 ) array of t with index range n 1 to n 2

	v ::= b	Boolean
	| n	natural number (limited to NATMAX)
	| a	array of values
	a ::

t ::= bool

  is_assgd_in_ss(id s ⇐ e, id s ) is_assgd_in_ss(id s (e 1 ) ⇐ e 2 , id s ) = true is_assgd_in_ss(for(id, e 1 , e 2 ){ss}, id s ) is_assgd_in_ss(falling{ss}, id s ) is_assgd_in_ss(rising{ss}, id s ) = is_assgd_in_ss(ss, id s ) is_assgd_in_ss(if(e){ss 1 } else {ss 2 }, id s ) is_assgd_in_ss(rst {ss 1 } else {ss 2 }, id s ) is_assgd_in_ss(ss 1 ;ss 2 , id s ) The is_assgd_in_omap function returns true if the signal identifier id s is an actual part (i.e. the right part of an association) in the output port map o. is_assgd_in_omap(o, id s ) = true if ∃name s.t. (name ⇒ id s ) ∈ o false otherwise

	=	is_assgd_in_ss(ss 1 , id s ) or is_assgd_in_ss(ss 2 , id s )
	otherwise = false
	Assignment in output port map
	Assignment in concurrent statement

is_assgd_in_cs(ps(id, vars, body), id s ) = is_assgd_in_ss(body, id s )

is_assgd_in_cs(comp(id c , id e , g, i, o), id s ) = is_assgd_in_omap(o, id s ) is_assgd_in_cs(cs 1 || cs 2 , id s ) = is_assgd_in_cs(cs 1 , id s ) or is_assgd_in_cs(cs 2 , id s ) is_assgd_in_cs(null, id s ) =

false Multiply-driven signal in output port map is_md_in_omap(o, id s ) = is_assgd_in_omap(o \ {(name ⇒ id s )}, id s ) if ∃name s.t. (name ⇒ id s ) ∈ o false otherwise Multiply-driven signal in concurrent statement is_md_in_cs(comp(id c , id e , g, i, o), id s ) = is_md_in_omap(o, id s ) is_md_in_cs(cs 1 || cs 2 , id s ) = is_md_in_cs(cs 1 , id s ) or is_md_in_cs(cs 2 , id s ) or (is_assgd_in_cs(cs 1 , id s ) and is_assgd_in_cs(cs 2 , id s )) otherwise = false


E H El

M Lema Ńczyk

APPROXIMATE TRANSITIVITY PROPERTY AND LEBESGUE SPECTRUM

Keywords: AMS Subject Classifications (2000): 37A15, 37A25, 37A30 ergodic theory, dynamical system, approximate transitivity property, Lebesgue spectrum

Exploiting a spectral criterion for a system not to be AT we give some new examples of zero entropy systems without the AT property. Our examples include those with finite spectral multiplicity -in particular we show that the system arising from the Rudin-Shapiro substitution is not AT. We also show that some nil-rotations on a quotient of the Heisenberg group as well as some (generalized) Gaussian systems are not AT. All known examples of non AT-automorphisms contain a Lebesgue component in the spectrum.

Introduction

In this article we deal with the approximate transitivity property (AT property for short) in ergodic theory. This property has been introduced by A. Connes and G.J. Woods in [START_REF] Connes | Approximately transitive flows and ITPFI factors, Ergodic Theory Dynam[END_REF] in connection with some classification problems of factors of type III 0 in the theory of von Neumann algebras. We recall now the definition and some basic facts.

Let G be an Abelian countable (discrete) group acting as measure preserving transformations: g → T g , on a standard probability Borel space (X, B, µ). This action is called AT (or AT [START_REF] Ageev | Dynamical systems with an even-multiplicity Lebesgue component in the spectrum[END_REF]) if for an arbitrary family of nonnegative functions f 1 , . . . , f l ∈ L 1 + (X, B, µ), l ≥ 2, and any ε > 0, there exist a positive integer s, g 1 , . . . , g s ∈ G, λ j,k ≥ 0, j = 1, . . . l, k = 1, . . . , s and f ∈ L 1 + (X, B, µ) such that

(1)

f j - s k=1 λ j,k f • T g k 1 < ε, 1 ≤ j ≤ l.
In fact, in [START_REF] Ageev | Dynamical systems with an even-multiplicity Lebesgue component in the spectrum[END_REF] we can restrict ourselves to take only l = 2; indeed, given ε > 0 we apply (1) to f 1 , f 2 and obtain f = f 12 , then we apply again (1) to f 12 and f 3 for some ε ′ sufficiently small and obtain f 123 and we conclude after l -1 steps.

Only few general facts about AT-systems in ergodic theory are known. The AT property forces the system to be ergodic and to have zero entropy [START_REF] David | Sur quelques problèmes de théorie ergodique non commutative[END_REF], [START_REF] Connes | Approximately transitive flows and ITPFI factors, Ergodic Theory Dynam[END_REF], [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF]. Moreover funny rank 1 systems enjoy the AT property [START_REF] Connes | Approximately transitive flows and ITPFI factors, Ergodic Theory Dynam[END_REF], [START_REF] Shoket | Les actions approximativement transitives dans la thérie ergodique[END_REF], [START_REF] Sokhet | Funny rank one and the approximate transitivity for induced actions[END_REF]. Clearly, the class of AT-systems is closed under taking factors and inverse limits (and roots for Z-actions).

The action of G on (X, B, µ) induces a (continuous) unitary representation, called Koopman representation, of G in the space L 2 (X, B, µ) given by U Tg f = f • T g , f ∈ L 2 (X, B, µ) and g ∈ G. Recall that such a representation is said to have simple spectrum if L 2 0 (X, B, µ) = G(f ) where G(f ) stands for the cyclic space generated by f , i.e. G(f ) = span{f • T g : g ∈ G}. In view of the definition of the AT property it is natural to ask whether it already implies simplicity of the spectrum -this question appeared (or is treated implicitly) in several papers, see David [START_REF] David | Sur quelques problèmes de théorie ergodique non commutative[END_REF], Hawkins [START_REF] Hawkins | Properties of ergodic flows associated to product odometers[END_REF], Hawkins-Robinson [START_REF] Hawkins | Approximately transitive (2) flows and transformations have simple spectrum, Dynamical systems[END_REF], Golodets [START_REF] Ya | Approximately Transitive Actions of Abelian Groups and Spectrum[END_REF] and Dooley-Quas [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF]. This is still an open problem also for G = Z. A stronger conjecture due to Dooley and Quas [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF] is that AT-systems are exactly funny rank-1 systems (this latter class is known to be a subclass of simple spectrum actions). This conjecture is based on the fact that a criterion for a system not be AT given in [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF] which we repeat in Section 2 is also sufficient for a system not to be of funny rank-1.

As in the definition of the AT property we deal with L 1 -functions, one can also ask about simplicity of the spectrum in L 1 (X, B, µ) for the induced action of G on L 1 , that is we ask whether there exists a function f ∈ L 1 (X, B, µ) for which the linear span of the functions f • T g , g ∈ G, is dense in L 1 . The conjecture of Thouvenot from the 1980th states that each ergodic automorphism has a simple L 1 -spectrum (see also related works on L p -multiplicities by Iwanik [START_REF] Iwanik | Positive entropy implies infinite L p -multiplicity for p > 1, Ergodic theory and related topics[END_REF][START_REF] Iwanik | The problem of L p -simple spectrum for ergodic group automorphisms[END_REF] and Iwanik-Sam de Lazaro [START_REF] Iwanik | Sur la multiplicité L p d'un automorphisme gaussien. (French) [On the L p multiplicity of Gaussian automorphisms[END_REF]). Thouvenot himself observed that for G = Z ATautomorphisms have simple L 1 -spectrum; indeed, all we need to show is that given g, h ∈ L 1 (X, B, µ) and ε > 0 we can find f ∈ L 1 (X, B, µ) such that d(g, f ) < ε and d(h, Z(f )) < ε because then the open set {f ∈ L 1 (X, B, µ) : d(h, Z(f )) < ε} is dense, and we can use a Baire type argument. Now by using the AT property, we can easily arrive at a situation that d(g, P (U T )(f ′ )) < ε ′ and d(h, Q(U T )(f ′ )) < ε ′ where P, Q are trigonometric polynomials. By replacing (in the space A(T)) P by a another trigonometric polynomial we can assume that P has no zeros on the circle and we simply put f = P (U T )f ′ noticing that the cyclic space generated by f is the same as the one generated by f ′ ; indeed 1/P (z) also belongs to the space A(T).

In fact, it was unknown until very recently that a system with zero entropy without AT property could exist (G = Z). In [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF] two examples of zero entropy non AT-systems are exhibited. For both of them the associated Koopman operator has a Lebesgue component in the spectrum, moreover the component has infinite multiplicity. In connection with that two natural questions arises. Can we find a non AT ergodic automorphism whose Koopman operator has a finite multiplicity? Does the AT property imply singularity of the spectrum? We will give the positive answer to the first question; we have been unable to answer the second one -recall that even in the class of rank-1 transformations it is unknown whether the spectrum has to be singular.

In the present paper we will prove a criterion (see Proposition 3.4 below) for a system not to be AT which is an elaborated version of an argument implicitly contained in [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF]. Our criterion looks spectral and should work in case of an automorphism with a "good" Lebesgue component in the spectrum. However we require for some function of type χ P0χ P1 ((P 0 , P 1 ) is a partition of X) to have the spectral measure absolutely continuous with a good control of its density which makes the use of the criterion "in practice" a delicate task. We will go through many known constructions of zero entropy dynamical systems having a Lebesgue component in the spectrum and show that they or systems "close" to them are not AT. It should be mentioned that for all known non AT systems the absence of approximate transitivity turned out to be a consequence of the criterion, including the non AT property of positive entropy automorphisms (see Corollary 4.6 below).

We show that the automorphism given by the Rudin-Shapiro substitution is not AT but it has a finite multiplicity (more precisely, the associated Koopman operator has the maximal spectral multiplicity equal to 2 [START_REF] Lemańczyk | Toeplitz Z 2 -extensions[END_REF], [START_REF] Mathew | A measure-preserving transformation whose spectrum has a Lebesgue component of multiplicity two[END_REF], [START_REF] Queffélec | Substitution Dynamical Systems-Spectral Analysis[END_REF]). Recently Giordano and Handelman [START_REF] Giordano | Matrix-valued Random Walks and Variations on Property AT[END_REF] introduced and studied the following notion of AT(n). Definition 1.1. Let (X, B, µ, T ) be a dynamical system and n be a positive integer. The transformation T is AT(n) if for any ε > 0, for any set of n+1 nonnegative functions {f i } n+1 i=1 ⊂ L 1 + (X, µ), there exist n nonnegative functions g m , m = 1, . . . , n, a positive integer N , positive coefficients (α (m) i,j ) m=1,...,n i=1...,n+1;j=1...n and a finite sequence of integers {t (m) j } m=1,...,n j=1,...,N such that

f i - n m=1 N j=1 α (m) i,j g m • T t (m) j 1 < ε for all i ∈ {1, . . . , n + 1}.
Clearly, each AT(n) system enjoys AT(n + 1) property. It is easy to see that the transformation with rank n (see e.g. [START_REF] Ferenczi | Systems of finite rank[END_REF] for the relevant definitions and properties) is AT(n) and it is well known that the system determined by a substitution on k symbols has the rank at most k. In the case of Rudin-Shapiro substitution the corresponding automorphism has rank 4 (see [31]). It follows that the automorphism given by the Rudin-Shapiro substitution is a natural example of a system which is AT(4) but not AT [START_REF] Ageev | Dynamical systems with an even-multiplicity Lebesgue component in the spectrum[END_REF], see [START_REF] Giordano | Matrix-valued Random Walks and Variations on Property AT[END_REF] for other examples of that type.

Moreover we show that each ergodic system has an ergodic distal (see [START_REF] Furstenberg | Recurrence in Ergodic Theory and Combinatorial Number Theory[END_REF]) extension which is not AT. Our method also shows that Helson and Parry's "random" construction from [START_REF] Helson | Cocycles and spectra[END_REF] of a 2-point extension of an arbitrary (aperiodic) system with a Lebesgue component is "almost" non-AT; actually (on a set of positive measure of parameters) its 4th power is not AT. We also show that some nil-translations on a quotient of the Heisenberg group are not AT. Furthermore we show that the non AT property for affine transformations of the torus enjoy some stability property. Finally we will deal with the non AT property in the class of Gaussian systems. We will give examples of zero entropy (mixing) Gaussian systems T such that T × T × T × T is not AT. Extensions of Gaussian systems via cocycles are treated in the last section.

A necessary combinatorial condition to have AT property

Let G be a countable Abelian (discrete) group acting measurably on a standard probability Borel space (X, B, µ) by measure-preserving transformations g → T g , g ∈ G. Let P = {P 0 , P 1 } be a (measurable) partition of X. Then each point x ∈ X has its P-name π(x) = (x g ) g∈G ∈ {0, 1} G , where

x g = 0 if T g (x) ∈ P 0 ; 1 if not.
Let Λ be a finite set in G. We define a funny word on the alphabet {0, 1} based on Λ as a finite sequence (W g ) g∈Λ , W g ∈ {0, 1}. For any two funny words W, W ′ based on the same set Λ their Hamming distance is given by

d Λ (W, W ′ ) = 1 |Λ| card {i ∈ Λ : W i = W ′ i } .
The proposition below gives a necessary condition for an action to be AT. The proof of it follows word by word the proof given by Dooley and Quas [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF] in the case of Z-actions. Proposition 2.1. ( [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF]) Let (X, B, µ, G) be an AT dynamical system. Then for any ε > 0, there exist a finite set Λ ⊂ G of arbitrary large cardinality and a funny word

W based on Λ such that |Λ|µ{x ∈ X : d(π(x)| Λ , W ) < ε} > 1 -ε.

Criterion to be non-AT

In this section we assume that G = Z and we put T 1 = T . We will give a certain criterion for a system not to be AT. It is a spectral extension of a criterion implicitly stated in [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF].

3.1. Strongly BH probability measures on the circle. Denote by ε 0 the unique zero in (0, 0.2) of the polynomial

P (t) = 2(1 -t)(1 -2t) 2 -1 -t.
Let µ be a probability measure on the circle group. Motivated by Theorem 3.1 below, we call µ a strongly Blum-Hanson measure (SBH measure) if the following holds lim sup

k→+∞ sup      1 √ k k j=1 (-1) ηj z nj 2 L 2 (µ) : n 1 < . . . < n k , η j ∈ {0, 1}, 1 ≤ j ≤ k      ≤ 1+ε 0 .
Clearly Lebesgue measure is an SBH measure. However more generally each absolutely continuous measure with sufficiently "flat" density g, i.e. the density satisfying sup z∈T g(z) < 1 + ε 0 is also an SBH measure.

Recall that a measure µ on the circle is called a Rajchman measure if lim n→∞ µ(n) = 0. Using some ideas of Lyons [START_REF] Lyons | The measure of nonnormal sets[END_REF] we will prove that each SBH measure is a Rajchman measure. It is well-known that Rajchman measure can be singular (see also Section 4.5 of the paper), however we have been unable to decide whether there exist singular SBH measures. We state also in the following the Blum-Hanson's theorem [START_REF] Blum | On the mean ergodic theorem for subsequences[END_REF] in ergodic theory as a characterization of Rajchman measures.

Theorem 3.1. µ is Rajchman measure if and only if for any infinite increasing sequence {n k } k∈N of integers, we have

1 k k j=1 z nj 2 L 2 (µ) -----→ k→∞ 0 (2)
Using some ideas of Lyons [START_REF] Lyons | The measure of nonnormal sets[END_REF] we will now show that SBH measures belong to the class of measures that annihilate all so called W * -sets (this class is known to be a proper subclass of Rajchman measures, [START_REF] Lyons | The measure of nonnormal sets[END_REF]). We need to recall some basic definitions. A sequence {t n } ⊂ T is said to be uniformly distributed if for all arcs

I ⊂ T lim N -→+∞ 1 N N k=1 χ I (n k x) = |I|.
Weyl's criterion (e.g. [START_REF] Kuipers | Uniform Distribution of Sequences[END_REF],pp. 1-3,7-8)) states that a sequence {t k } k∈N ⊂ T is uniformly distributed if and only if for every non-zero integer m,

lim N -→+∞ 1 N N k=1 e(mt k ) = 0,
where we use notation e(x) for e 2iπx . Now, following Kahane and Lyons we define the W * -sets. A Borel set B ⊂ T is called a W * -set [START_REF] Kahane | Sur les mauvaises répartitions modulo 1[END_REF] (or a non-normal set) if there exists an increasing sequence

{n k } k∈N such that for every x ∈ B, {n k x} ∞ k=1 is not uniformly distributed. The maximal W * -set corresponding to {n k } k∈N is the set W * ({n k }) := {x ∈ T : {n k x} is not uniformly distributed} (it is Borel).
By Weyl's Criterion, in order to show that some probability measure µ vanishes on all W * -sets, we need to show that for each m = 0 and each increasing sequence

{n k } 1 K K k=1 e(mn k t) = 0 for µ-a.e. t ∈ T. (3)
Actually, since {n k } is arbitrary, it is enough to establish (3) for m = 1. We need the following form of the strong law of large numbers for weakly correlated bounded random variables due to Lyons [START_REF] Lyons | The measure of nonnormal sets[END_REF] and for the convenience of the reader we include the proof. Lemma 3.2. Let {X n } be a sequence of random variables on a probability space (Ω, B, P). Suppose that |X n | ≤ 1 a.s. and there exists a positive constant c ≥ 1 such that for any {n k } ↑ +∞ and N large enough we have

1 K K k=1 X n k 2 < c K . ( 4
)
Then the strong law of large numbers holds:

lim N -→+∞ 1 K K k=1 X n k = 0 a. s. Proof. Assume that n k ↑ +∞. It follows from (4) that K≥1 1 K 2 K 2 k=1 X n k 2 dµ < c K≥1 1 K 2 = cπ 2 6
.

Therefore P-a.s.

K≥1 1 K 2 K 2 k=1 X n k 2 < ∞
and hence (P-a.s.) lim

K-→+∞ 1 K 2 K 2 k=1 X n k = 0. Now if m 2 ≤ K < (m + 1) 2 , then 1 K K k=m 2 +1 X n k ≤ 1 K (K -m 2 ) ≤ 1 K (2m + 1) ≤ 1 m 2 (2m + 1) → 0 when m → ∞. But 1 K 2 K k=1 X n k ≤ 1 m 2 m 2 k=1 X n k + 1 K K k=m 2 +1 X n k and hence lim K-→+∞ 1 K K k=1 X n k = 0 a.s.
which completes the proof.

Proposition 3.3. If µ is an SBH measure then µ(E) = 0 for each W * -set E ⊂ T.
Proof. Let µ be an SBH measure. Then, for any increasing sequence {n k } we have

1 √ k k j=1 z nj 2 L 2 (µ) < 1 + ε 0
for k large enough. It follows from Lemma 3.2 that the strong law of large numbers holds for the sequence {X n k } with X n k = e(n k •), k ≥ 1, and the result follows.

We now pass to our criterion for a system not to be an AT-system (we refer the reader to [START_REF] Cornfeld | Ergodic Theory[END_REF] to basic facts about spectral theory of dynamical systems). Proposition 3.4. Assume that a dynamical system (X, B, µ, T ) is ergodic and that there exists a (measurable) partition P = {P 0 , P 1 } with the following properties: i) There exists S in the centralizer C(T ) of T such that SP 0 = P 1 ; in particular µ(P 0 ) = µ(P 1 ) = 1 2 . ii) The spectral measure σ χ P 0 -χ P 1 of χ P0χ P1 is an SBH measure.

Then the system is not AT.

Proof. Let us take W a funny word based on a subset Λ :

n 1 < n 2 < • • • < n k . For x ∈ X put Θ W (x) = 1 k k j=1 A W j (x),
where A W j is defined as

A W j (x) = 1 if W nj = x nj -1 if not (recall that (x n ) = π(x) is the P-name of x). Then the distribution Θ * of Θ is symmetric. Indeed, we have π(x) = -π(Sx) and therefore Θ(π(x)) = -Θ(π(Sx))
and since S is measure-preserving the symmetry of Θ * follows.

Notice that (5)

A W j (x) = (-1) Wn j (χ P0 -χ P1 )(T nj x) and that (6) Θ W (x) = 1 -2d Λ (W, π(x)| Λ ).
In view of ( 6), the symmetry of Θ * and the Tchebychev inequality we obtain that

µ{x ∈ X : d Λ (W, π(x)| Λ ) < ε} = µ{x ∈ X : Θ W (x) > 1 -2ε} = 1 2 µ{x ∈ X : |Θ W (x)| > 1 -2ε} (7) ≤ 1 2(1 -2ε) 2 ||Θ W || 2 2 .
But, in view of ( 5) and the Spectral Theorem

Θ W 2 2 = X 1 k k j=1 A W j 2 dµ = 1 k 2 k i,j=1 X (-1) Wn i (χ P0 -χ P1 ) (T ni x) • (-1) Wn j (χ P0 -χ P1 ) (T nj x) dµ(x) = 1 k 2 k i,j=1 (-1) Wn i +Wn j σχ P 0 -χ P 1 (n i -n j ) = 1 k 1 √ k k i=1 (-1) Wn i z ni 2 dσ χ P 0 -χ P 1 (z).
It follows that for k large enough we have

(8) Θ 2 2 < 1 k (1 + ε 0 ).
Combining [START_REF] David | Sur quelques problèmes de théorie ergodique non commutative[END_REF] and ( 8) we obtain that

k µ{x ∈ X : d Λ (W, π(x)| Λ ) < ε} ≤ 1 + ε 2(1 -2ε) 2
and since W was arbitrary this contradicts Proposition 2.1.

Corollary 3.5. Under the assumptions of Proposition 3.4 assume that the Fourier transform of σ χP 0 -χP 1 is in l 1 . Then there exists m 0 ≥ 1 such that T m is not AT for all m ≥ m 0 .

Proof. In this case σ = σ χP 0 -χP 1 is absolutely continuous with the (continuous) density d given by d(z) = ∞ n=-∞ σ(n)z n and by the same token if instead of T we consider T m the spectral measure σ m of χ P0χ P1 is also absolutely continuous with the density d m given by

d m (z) = ∞ n=-∞ σm (n)z n = ∞ n=-∞ σ(mn)z n ,
so for m large enough σ m will be an SBH measure.

Remark 3.6. Notice that the L 2 -norm of χ P0 -χ P1 is 1. Moreover the spectral measure of χ P0 -χ P1 is Lebesgue if and only if the sequence of partitions {T n P} is pairwise independent. Indeed, for n ≥ 1, σχP 0 -χP 1 (n) = 0 implies that µ(T -n P 0 ∩ P 0 ) + µ(T -n P 1 ∩ P 1 ) = µ(T -n P 0 ∩ P 1 ) + µ(T -n P 1 ∩ P 0 ). Thus µ(T -n P 0 ∩ P 0 ) + µ(T -n P 1 ∩ P 1 ) = 1 2
and since µ(T -n P 0 ∩ P 1 ) + µ(T -n P 0 ∩ P 0 ) = 1 2 , we have

µ(T -n P 1 ∩ P 1 ) = µ(T -n P 0 ∩ P 1 ) = 1 4
and therefore we obtain pairwise independence (see [START_REF] Courbage | Decay of correlations and mixing properties in a dynamical system with zero K-S entropy[END_REF], [START_REF] Flaminio | Mixing k-fold independent processes of zero entropy[END_REF]). However, in general it is unclear that even for transformations with Lebesgue spectrum or the more with a Lebesgue component in the spectrum) we can always find a partition P satisfying the assumptions of Proposition 3.4 where in addition the spectral measure of σ χP 0 -χP 1 is exactly Lebesgue. A certain flexibility of Proposition 3.4 consists in the fact that for some natural partitions we need only to show that the corresponding spectral measure is absolutely continuous with the density sufficiently flat.

In the rest of the paper we will show how this can be applied in practice.

Remark 3.7. We can also define the notion of SBH measure for groups G more general than Z. Based on Propositon 2.1 we can then prove a relevant version of Proposition 3.4 as a criterion for a G-system to be non AT. It would be interesting to know which results of Section 4 have their natural generalizations.

Applications

Except for the Gaussian case considered in Section 4.5 all examples below of non AT-automorphisms will be given as group extensions. Recall briefly some basic facts. Let T : (X, B, µ) -→ (X, B, µ) be an ergodic automorphism. Let G be a compact metric Abelian group with Haar measure m. Denote by G the character group of G. By a cocycle we mean a measurable function ϕ : X → G; in fact such a ϕ generates a cocycle ϕ(n,

•) = ϕ (n) (•) by ϕ (n) (x) =    ϕ(x) + . . . + ϕ(T n-1 x) if n > 0, 0 if n = 0, -(ϕ(T n x) + . . . + ϕ(T -1 x)) if n < 0.
Then we define a µ ⊗ m-preserving automorphism

T ϕ : X × G → X × G, T ϕ (x, g) = (T x, ϕ(x) + g), x ∈ X, g ∈ G called a G-extension of T . Notice that T n ϕ (x, g) = (T n x, ϕ (n) (x) + g). The space L 2 (X × G, µ ⊗ m) can be decomposed as (9) L 2 (X × G, µ ⊗ m) = χ∈ b G L χ ,
where each subspace

L χ = {f χ : f ∈ L 2 (X, µ)} is U Tϕ -invariant, and the restriction of U Tϕ to L χ is unitarily equivalent to V ϕ,T,χ : L 2 (X, µ) -→ L 2 (X, µ) defined by V ϕ,T,χ (f )(x) = χ(ϕ(x))f (T x), x ∈ X. It follows that to describe spectral properties of T ϕ it is sufficient to study spectral properties of V ϕ,T,χ , χ ∈ G.
4.1. Two point extensions and the AT property, Rudin-Shapiro substitution. Assume that G = Z/2Z = {0, 1}. In this case we have a natural partition P given by P 0 = X × {0}, P 1 = X × {1}. The assumption (i) of Proposition 3.4 is satisfied as SP 0 = P 1 where S(x, i) = (x, i + 1) is in the centralizer of T ϕ . Notice that the decomposition (9) of L 2 (X × {0, 1}) is of the form L 0 L 1 where

L 0 = {f ∈ L 2 (X × {0, 1}) : f • S = f } and L 1 = {f ∈ L 2 (X × {0, 1} : f • S = -f }. Thus U Tϕ | L1 is spectrally isomorphic to the operator V defined by (V (f ))(x) = (-1) ϕ(x) f (T x).
The function χ P0χ P1 belongs to L 1 and under the above isomorphism it corresponds to the constant function 1 ∈ L 2 (X, B, µ). It follows that

σ χP 0 -χP 1 (n) = V n 1, 1 = µ   {x ∈ X : n-1 j=0 ϕ(T j x) = 0}   -µ   {x ∈ X : n-1 j=0 ϕ(T j x) = 1}   .
In case when T is the dyadic odometer such extensions were intensively studied in the 1980th. In particular, Mathew and Nadkarni [START_REF] Mathew | A measure-preserving transformation whose spectrum has a Lebesgue component of multiplicity two[END_REF] gave constructions of ϕ such that (10)

V n 1, 1 = 0 for all n = 0, that is, the corresponding spectral measure is equal to Lebesgue measure (in fact the Lebesgue component has multiplicity 2). Other examples with Lebesgue component with arbitrary even multiplicity are given by so called Toeplitz extensions in [START_REF] Lemańczyk | Toeplitz Z 2 -extensions[END_REF] -each time [START_REF] Flaminio | Mixing k-fold independent processes of zero entropy[END_REF] holds. In particular it is shown in [START_REF] Lemańczyk | Toeplitz Z 2 -extensions[END_REF] that the system given by Rudin-Shapiro substitution (see [START_REF] Queffélec | Substitution Dynamical Systems-Spectral Analysis[END_REF] where also it is shown that it has a Lebesgue component of multiplicity 2) is a particular member of Mathew-Nadkarni's family.

Corollary 4.1. Mathew-Nadkarni's maps as well as all examples from [START_REF] Lemańczyk | Toeplitz Z 2 -extensions[END_REF] having even Lebesgue multiplicity are not AT-systems. In particular the automorphism given by the Rudin-Shapiro substitution is not AT.

Remark 4.2. One may also use Ageev's construction [START_REF] Ageev | Dynamical systems with an even-multiplicity Lebesgue component in the spectrum[END_REF] to produce a continuum of weakly mixing automorphisms with spectral multiplicity equal to 2 and without AT property.

We now recall Helson and Parry's construction from [START_REF] Helson | Cocycles and spectra[END_REF] of "random" 2-point extensions. Given an aperiodic automorphism T of a standard probability Borel space they give a random construction of cocycles ϕ ω : X → {0, 1} (the parameter ω runs over a probability space (Ω, P )) such that for a.e. ω, T ϕω has absolutely continuous spectrum and on a set of positive P -measure T ϕω has Lebesgue spectrum. In particular, they prove that ( 11)

Ω X e πiϕ (n) ω (x) dµ(x) 2 dP (ω) < γ n for all n ≥ 4,
where {γ n } is an arbitrary set of positive numbers. Since, by [START_REF] Fraczek | On spectral stability of nil-translations[END_REF], on a set Ω n ⊂ Ω of measure at least 1 -1 2 n+1 we have

X e πiϕ (n) ω (x) dµ(x) 2 < 2 n+1 γ n , n ≥ 4,
by selecting {γ n } as small as we need we can obtain that the Fourier transform {σ χP 0 -χP 1 (n)} is absolutely summable with |n|≥4 |σ χP 0 -χP 1 (n)| as small as we need on the set ∩ n≥4 Ω n of positive measure of parameters. By Corollary 3.5 (or rather its proof) we obtain the following.

Corollary 4.3. For each ergodic (aperiodic) automorphism T there exists an ergodic 2-point extension T ϕ such that T 4 ϕ is not AT. Therefore we get a partial answer to the question from [START_REF] Dooley | Approximate transitivity for zero-entropy systems[END_REF] (see also next section). 4.2. Distal (ergodic) extension without AT property. The aim of this section is to prove the following. Proposition 4.4. For each ergodic transformation T acting on a standard Borel probability space (X, B, µ) there exists a 2-step group extension T which is ergodic and such that T is not AT.

Proof. We use the idea of affine extension of Glasner [START_REF] Glasner | Ergodic Theory via Joinings[END_REF] combined with Proposition 3.4. Let ϕ : X → T, T = [0, 1), be a cocycle such that T ϕ is ergodic and the groups of eigenvalues for T and T ϕ are the same; in particular if T is weakly mixing, so is T ϕ . Let ψ : X × T → T be the cocycle defined by (x, y) -→ y, and finally put T = (T ϕ ) ψ . It is easy to check (by considering the relevant functional equations and applying Fourier series arguments) that T is ergodic and it does not change the group of eigenvalues of T .

Consider

P 0 = X × T × [0, 1 2 ). Then P 1 def = P c 0 = R 1 2 P 0 , where R 1 2 (x, y, z) = (x, y, z + 1 2 ) and R 1 2
∈ C(T ). We have to compute the spectral measure of z) , where A = [0, 1 2 ). We have

f (x, y, z) = f (z) = χ P0 (x, y, z) -χ P1 (x, y, z) = e iπχA(
U n T f, f = X×T×T f (T (n) ϕ (x, y), ψ (n) (x, y) + z)f (x, y, z)dµ(x)dydz
= e iπχA(ψ(x,y)+ψ(T x,ϕ(x)+y)+•••+ψ(T n-1 x,ϕ (n-1) (x)+y)+z) e -iπχA(z) dµ(x)dydz

= X×T e -iπχA(z) T e iπχA(y+ϕ(x)+y+•••+ϕ (n-1) (x)+y+z) dy dµ(x)dz = X×T e -iπχA(z)
T e iπχA(ny) dy dµ(x)dz.

We will now show that T e iπχA(ny) dy = 0 for n = 0. We deduce that the spectral measure of χ P0χ P1 is exactly Lebesgue measure. Remark 4.5. Instead of ψ(x, y) = y we can take ψ(x, y) = my, m = 0. In the concluding argument we divide [0, 1) into intervals of length 1 2|m|n .

Corollary 4.6. ([4], [START_REF] David | Sur quelques problèmes de théorie ergodique non commutative[END_REF]). If the dynamical system (X, B, µ, T ) is AT then its entropy is zero.

Proof. First, note that no Bernoulli dynamical system is AT. Indeed, apply the proposition 4.4 to get the weakly mixing 2-step compact group extension (T φ ) ψ of T which is not AT. But, by [START_REF] Rudolph | Classifying the isometric extensions of a Bernoulli shift[END_REF] (T φ ) ψ is again Bernoulli with same entropy as T .

It follows from the Ornstein isomorphism theorem that T is not AT. Assume that the entropy of T is strictly positive. By the Kolmogorov-Sinai theorem there exists a Bernoulli factor with the same entropy. But the factor of the system with the AT property is AT. We get that T is not AT.

4.3.

Absolutely continuous cocycles over irrational rotations without AT property. Denote T = [0, 1) and let T x = x+α be an irrational rotation. Consider

F (x, y) = χ T×[0, 1 2 ) (x, y) -χ T×[ 1 2 ,1) (x, y) = 2χ T×[0, 1 2 ) (x, y) -1 = f (y),
where

f (y) = 2χ [0, 1 2 ) (y) -1. For m ∈ Z \ {0} we have f (m) = 1 0 f (y)e -2πimy dy = 2 1 0 χ [0, 1 2 ) (y)e -2πimy dy - 1 0 e -2πimy = 2 1/2 0 e -2πimy dy = 1 -πim e -2πimy | 1/2 0 = -1 πim e -πim -1 = 0 if m = 2k 2 πim if m = 2k + 1. Moreover F (x, y) = ∞ m=-∞ f (m)e 2πimy = ∞ k=-∞ 2 (2k + 1)πi e 2πi(2k+1)y .
Notice that functions

e 2πimy =: ξ m (x, y) ∈ L 2 (T) ⊗ e 2πimy
where the latter subspace is U T φ -invariant for each cocycle φ :

T → T.
In what follows we assume that φ(x) = e 2πi(x+g(x)) where g : T → R is a "smooth" function: we will precise conditions imposed on g later; it is however at least that g is absolutely continuous and its derivative is a.e. equal to a function of bounded variation. It follows that the spectral measure of F is equal to the sum of spectral measures of its orthogonal projections on subspaces L 2 (T) ⊗ e 2πimy . Let us compute the spectral measure of ξ m :

ξ m • T n φ • ξ m dxdy = 1 0 e 2πim(nx+ n(n-1) 2 α+g (n) (x)) dx.
For n = 0 we have (using integration by parts for Fourier-Stjeltjes integrals as in [START_REF] Iwanik | Absolutely continuous cocycles over irrational rotations[END_REF]):

1 0 e 2πim(nx+g (n) (x)) dx = 1 2πim 1 0 1 n + g ′ (n) (x) de 2πim(nx+g (n) (x)) = 1 2πim
• (-1)

1 0 e 2πim(nx+g (n) (x)) d 1 n + g ′ (n) (x) .
We will now assume additionally that g ′ > -1 + δ 0 (0 < δ 0 < 1) so that we can pass to a well-known estimation (see again [START_REF] Iwanik | Absolutely continuous cocycles over irrational rotations[END_REF]):

ξ m • T n φ • ξ m dxdy ≤ 1 2π|m| Var 1 n + g ′ (n) (x) ≤ 1 2π|m| Varg ′ (n) (1 -δ 0 ) 2 n 2 ≤ 1 2π|m| Var g ′ (1 -δ 0 ) 2 |n| .
Hence (for n = 0)

| σ F (n)| = F • T n φ • F dxdy = m =0 f (m) ξ m • T n φ • ξ m dxdy ≤ m =0 | f (m)| 1 2π|m| Var(g ′ ) (1 -δ 0 ) 2 |n| = Var(g ′ ) 2π(1 -δ 0 ) 2 |n| m =0 | f (m)| |m| = O 1 |n| .
Notice however that the constant (in the expression O 1 |n| ) which appears can be made as small as we need by assuming that Var(g ′ ) is small. It follows that σ F is an absolutely continuous measure whose density d is given by d(z) = ∞ n=-∞ a n z n , where a 0 = 1 and for n = 0,

a n = σ F (n); moreover |a n | ≤ C 1
|n| with C > 0 as small as we need.

Proposition 4.7. Assume that T x = x + α, Sx ′ = x ′ + β where α, β, 1 rationally independent. Let φ(x) = e 2πi(x+g(x)) , ψ(x ′ ) = e 2πi(x ′ +h(x)) where g, h : T → R are absolutely continuous, with the derivatives (a.e.) of bounded variation and bounded away from -1 + δ 0 for some 0 < δ 0 < 1. If the variations of g ′ and h ′ are sufficiently small then T φ × S ψ is not AT. [START_REF] Ageev | Dynamical systems with an even-multiplicity Lebesgue component in the spectrum[END_REF] . By repeating all above calculations we end up with

Proof. It is enough to show that the factor (T × S) e 2πi(x+x ′ +g(x)+h(x ′ )) is not AT. We consider F (x, x ′ , y) = χ T×T×[0, 1 2 ) -χ T×T×[ 1 2 ,
| σ F (n)| ≤ O 1 |n| 2 ,
where the bounding constant is as small as we need. It follows that the density d(z) = 1 + n =0 σ F (n)z n of σ F is as close to 1 as we need and therefore σ F is an SBH measure. 4.4. Nil-rotations without AT property. We consider nil-rotations S in dimension 3 only. Hence S is defined on (R 3 , * )/ * Z 3 where we recall that the multiplication (x, y, z) * (x ′ , y ′ , z ′ ) = (x + x ′ , y + y ′ , z + z ′ + xy ′ ) on R 3 is the same as the multiplication in the Heisenberg group of upper triangle matrices. Moreover S((x, y, z) * Z 3 ) = (α, β, 0) * (x, y, z) * Z 3 .

It is well-known (see e.g. [START_REF] Fraczek | On spectral stability of nil-translations[END_REF]) that each such nil-rotation is isomorphic to a skew product transformation T φ on T 2 × S 1 where T (x, y) = (x + α, y + β), φ = e 2πiϕ and ϕ(x, y) = α{y} -({x} + α)[{y} + β] + γ (the nil-rotation is hence ergodic if α, β, 1 are rationally independent; it follows that if a nil-rotation is ergodic so are all its non-zero powers). It is classical (Parry) that nil-rotations have countable Lebesgue spectrum in the orthocomplement of the subspace of eigenfunctions. Proposition 4.8. For every ergodic nil-rotation S on (R 3 , * )/ * Z 3 there exists q ≥ 1 such that S q is not AT. Moreover if in the above representation of S as a skew product T φ , 1 2 < β < 1 is sufficiently close to 1 then S is not AT. Proof. The method for spectral calculations which we apply below comes from [START_REF] Fraczek | On spectral stability of nil-translations[END_REF].

As before we consider the function

F (x, y, z) = χ T×T×[0, 1 2 ) (x, y, z) -χ T×T×[ 1 2 ,1) (x, y, z) = f (z), where f (z) = 2χ [0, 1
2 ) (z)-1, we look at its Fourier decomposition for L 2 (X)⊗e 2πimz where m ∈ Z \ {0} and we have

f (m) = 0 if m = 2k 2 πim if m = 2k + 1.

It follows that

F • T n φ , F = m =0 f (m) ξ m • T n φ • ξ m dxdydz = m =0 f (m) e 2πimϕ (n) (x,y) dxdy.
As F is real valued we only need to consider n > 0. We have

e 2πimϕ (n) (x,y) dxdy = c n (y) e 2πim(x• P n-1 j=0 [{y+jβ}+β]) dx dy.
Notice that if q ≥ 1 is so that

(12) qβ > 1 then q-1 j=0 [{y + jβ} + β] ≥ 1 for each y ∈ [0, 1).
It follows that for each n = 0

F • T nq φ , F = 0
and the spectral measure F for T q φ is purely Lebesgue (cf. Corollary 3.5), which completes the first part of the proposition.

If 1 2 < β < 1 in view of [START_REF] Furstenberg | Recurrence in Ergodic Theory and Combinatorial Number Theory[END_REF] for n = 2 (and for all n ≥ 2) we have F •T n φ , F = 0. Hence we have only to control the case n = 1. We have

F • T φ , F = m =0 f (m) e 2πimϕ(x,y) dxdy = m =0 f (m) e 2πim(α{y}-({x}+α)[{y}+β]) dxdy = m =0 f (m) 1-β 0 b m (y) e 2πim(x•[{y}+β]) dx dy + 1 1-β b m (y) e 2πim(x•[{y}+β]) dx dy) = m =0 f (m) 1-β 0 b m (y) e 2πim(x•[{y}+β]) dx dy = m =0 f (m) 1-β 0 b m (y) dy = m =0 f (m)e 2πimα 1-β 0 e 2πimα{y} dy = m =0 f (m)e 2πimα 1 2πimα e 2πimα{1-β} -1 .
Now, the series m =0 f (m)e 2πimα 1 2πimα is absolutely summable and the functions β → e 2πimα{1-β} -1 are continuous. It follows that if 1/2 < β < 1 is sufficiently close to 1 then the density of σ F (which is a trigonometric polynomial of degree 1) is as close to 1 as we want, and therefore σ F is SBH, so the corresponding nil-rotation is not AT. 4.5. Gaussian systems and the AT property. In this section we will study the non AT property in the class of zero entropy Gaussian dynamical systems. We will show the existence of mixing zero entropy Gaussian systems for which the 4th Cartesian product is not AT. Let us recall the definition of Gaussian systems. Definition 4.9. Given a symmetric Borel probability measure σ on the circle [0, 1) we call (real) Gaussian system of spectral measure σ the dynamical system (Ω, A, T σ , µ) where:

• Ω is R Z .

• A is the borelian σ-algebra.

• S is the shift : (T σ (ω)) n = ω n+1 .

• µ is defined on the cylinder by µ(ω

j1 ∈ A 1 , • • • , ω jn ∈ A n ) is the probability of visiting the set A 1 × • • • × A n for a Gaussian vector (X j1 , • • • , X jn )

of zero mean and covariances

Cov(X js , X jt ) = T z js-jt dσ(z).

Such a system is then generated by real stationary (centered) Gaussian process, namely X n = X 0 • T n σ , n ∈ Z, where X 0 (ω) = ω 0 . The basic account on the spectral analysis of Gaussian dynamical systems may be found in [START_REF] Cornfeld | Ergodic Theory[END_REF]. We recall that the maximal spectral multiplicity of every Gaussian dynamical systems is +∞ or 1.

Let us point out that the transformation S : (ω n ) -→ (-ω n ) preserves the Gaussian measure and commute with any Gaussian system. In addition, for the partition P = {P 0 , P 1 }, given by P 0 = {X 0 > 0}, we have SP 0 = P 1 . With this remark in mind we will compute the Fourier coefficients of the spectral measure σ χ P 0 -χ P 1 .

Lemma 4.10. Let X = (X n ) n∈Z be a stationary centered Gaussian process with spectral measure σ satisfying σ(n) ∈ (-1, 1) for n = 0. Then

µ{X 0 > 0, X n > 0) = 1 4 + 1 2π arcsin( σ(n)).
Proof. Put Z n = X nσ(n)X 0 . It follows that Z n and X 0 are independent, the distribution of Z n is Gaussian with variance 1σ(n) 2 and for any Borel function φ : R × R → R we have

E(φ(X 0 , Z n )|X 0 ) = h(Z n ),
where h(z) = E(φ(X 0 , z)).

Now, by taking φ(x, z) = 1 if x > 0 and z > -σ(n)x, and 0 otherwise we obtain that µ{X 0 > 0,

X n > 0} = E E(χ {X0>0} χ {Zn>-σ(n)X0} |X 0 ) = E χ {X0>0} (E(χ {Zn>-σ(n)X0} |X 0 ) = E χ {X0>0} µ Z n Var Z n > -σ(n)X 0 1 -σ(n) 2 |X 0 = E   χ {X0>0}    1 -G   -σ(n)X 0 1 -σ(n) 2        where G(x) = 1 √ 2π x -∞ e -u 2 2 du. But, since 1 -G(-x) = G(x), we get µ{X 0 > 0, X n > 0} = E   χ {X0>0}    G   (σ(n))X 0 1 -σ(n) 2        = ∞ 0    G   σ(n)u 1 -σ(n) 2      G ′ (u)du.
An easy calculation by taking the derivatives of the following functions yields

+∞ 0 G(au)G ′ (u)du = 1 2π arctan a + 1 4
, for any a ∈ R, and by taking a =

σ(n) √ 1-σ(n) 2 we conclude.
It follows from the lemma above that we have

σ χ P 0 -χ P 1 (n) = 2 π arcsin( σ(n)).
We now pass to a similar problem but in the Cartesian product (X × X, B, µ ⊗ µ, T × T ) in which we consider the process (Y n ) where

Y n = Y 0 • (T × T ) n and Y 0 (ω 1 , ω 2 ) = X 0 (ω 1 )X 0 (ω 2 )
. We first must extend Lemma 4.10. Lemma 4.11. Let X = (X n ) n∈Z be a stationary centered Gaussian process with spectral measure σ such that σ(n) ∈ (-1, 1) for n = 0 and let

Y n (ω 1 , ω 2 ) = X n (ω 1 )X n (ω 2 ). Then µ({Y 0 > 0, Y n > 0}) = 1 4 + 1 π 2 arcsin 2 ( σ(n)).
Proof. It is easy to see that

µ ⊗ µ{Y 0 > 0, Y n > 0} = (µ{X 0 > 0, X n > 0}) 2 + (µ{X 0 < 0, X n > 0}) 2 + (µ{X 0 > 0, X n < 0}) 2 + (µ{X 0 < 0, X n < 0}) 2 .
Using Lemma 4.10 and the fact that S ∈ C(T ) we obtain that

µ({X 0 < 0, X n < 0}) = µ({S{X 0 < 0, X n < 0}}) = µ({X 0 > 0, X n > 0}) = 1 4 + 1 2π arcsin( σ(n)),
and then

µ{X 0 < 0, X n > 0} = µ{X 0 < 0} -µ{X 0 < 0, X n < 0} = 1 4 - 1 2π arcsin( σ(n)), µ{X 0 < 0, X n > 0} = 1 4 - 1 2π arcsin( σ(n)).
and the proof of the lemma is complete.

Since X n (ω) = ω n (ω = (ω n ) n∈Z ∈ R Z ) and S × Id is in the centralizer of T × T , µ ⊗ µ({Y 0 < 0, Y n < 0}) = µ ⊗ µ(S × Id({Y 0 < 0, Y n < 0})) = µ ⊗ µ({Y 0 > 0, Y n > 0}) = 1 4 + 1 π 2 arcsin( σ(n)) and since µ ⊗ µ({Y 0 < 0}) = 1 2 , µ ⊗ µ({Y 0 < 0, Y n > 0}) = µ ⊗ µ({Y 0 > 0, Y n < 0}) = 1 4 - 1 π 2 arcsin( σ(n)) 2 .
Consider T = (T × T ) × (T × T ) with measure µ = (µ ⊗ µ) ⊗ (µ ⊗ µ) and the process (Z n ) n∈Z where Z n = Z 0 ⊗ T n and Z 0 (ω (1) , ω (2) , ω (3) , ω (4) ) = Y 0 (ω (1) , ω (2) )Y 0 (ω (3) , ω (4) )).

By the argument from the beginning of the proof of Lemma 4.11 it follows that

µ({Z 0 > 0, Z n > 0}) = 1 4 + 4 π 4 arcsin 4 (σ(n))
. Therefore if we put P ′ 0 = {Z 0 < 0} and P ′ 1 = {Z 0 ≥ 0} then for any n ∈ Z we have

σ χ P ′ 0 -χ P ′ 1 (n) = 16 π 4 (arcsin( σ(n))) 4 . Finally notice that P ′ 1 = S × Id × Id × Id(P ′ 0 )
, where S × Id × Id × Id ∈ C(T ). Before we formulate the main result of this section we recall the following result of Körner [START_REF] Körner | On the theorem of Ivasev-Musatov[END_REF] (which is a strengthening of a result of Ivašëv-Musatov [START_REF] Brown | Some new singular Fourier-Stieltjes series[END_REF]). Theorem 4.12. Assume that φ :

[1, ∞] → [1, ∞) is a continuous positive function such that (i) ∞ 1 φ(x) 2 dx = +∞, (ii) (∃K > 1) Kφ(x) ≥ φ(y) ≥ φ(x)/K for 2x ≥ y ≥ x ≥ 1.
Then there exists a singular probability measure σ on T such that for each n = 0,

| σ(n)| ≤ φ (|n|) .
We easily verify that for each c > 0 the function φ(x) = c/ √ x satisfies the assumptions of Theorem 4.12.

Notice that if σ satisfies the assertion of Theorem 4.12 then so does the measure σ(A) = σ(A) (since |σ(-n)| = |σ(n)|) and also [START_REF] Giordano | Matrix-valued Random Walks and Variations on Property AT[END_REF] 1 2 (σ + σ ˆ(n) ≤ φ(|n|).

In other words the assertion of Theorem 4.12 holds in the class of symmetric measures.

Proposition 4.13. There exists a mixing Gaussian zero entropy dynamical system (X, B, T, µ) such that T × T × T × T is not AT.

Proof. Using Theorem 4.12 (and ( 13)) we can find σ a probability symmetric singular measure on the circle T such that for n ≥ 1

| σ(n)| ≤ c √ n . ( 14
)
The constant c > 0 has to be small enough so that for |x| ≤ c is an SBH measure and the proof of the proposition is complete.

Question. Is it true that for every automorphism T its Cartesian square T × T does not have the AT property? Remark 4.14. In the recent paper [START_REF] Körner | On a theorem of Saeki concerning convolution squares of singular measures[END_REF], Körner shows that given α ∈ ( 1 2 , 1) there is a singular measure µ on the circle such that µ * µ is absolutely continuous with density of Lipshitz class α- 1 2 . However the measure µ is not symmetric. If Körner's construction can be "symmetrized" then we would obtain a zero entropy Gaussian automorphism whose square T × T is not AT. 4.6. Gaussian cocycles and non AT property. Following [START_REF] Lemańczyk | Multiplicative Gaussian cocycles[END_REF] in this section we consider extensions of Gaussian systems via Gaussian cocycles. So we assume that (X n ) n∈Z is a stationary centered real Gaussian process inducing the dynamical system T = T σ acting on (Ω, µ σ ), where Ω = R Z and the probability measure σ, always assumed to be continuous, is the spectral measure of the process (and we can assume that X n (ω) = ω n ). We then consider the skew product T e 2πiX 0 acting on (Ω × S 1 , µ σ ⊗ λ).

It has been proved in [START_REF] Lemańczyk | Multiplicative Gaussian cocycles[END_REF] that if σ(n) ≥ 0 for each n ∈ Z then T exp(2πiX0) has countable Lebesgue spectrum in the orthocomplement of L 2 (Ω, µ σ ) ⊗ 1. Based on the proof of this result we will now show the following. An elementary calculation using the fact that the Fourier transform of σ is positive (see [START_REF] Lemańczyk | Multiplicative Gaussian cocycles[END_REF]) shows that In view of [START_REF] Ya | Approximately Transitive Actions of Abelian Groups and Spectrum[END_REF] it follows that the Fourier transform of σ F still decreases exponentially and the assertion follows from Corollary 3.5. Remark 4.16. It follows that for each Gaussian system T σ where σ = η * η we have a Gaussian cocycle such that the corresponding skew product has a power which is not AT.

  Indeed T e iπχA(ny) dy = 2n-1 j=0 Ij e iπχA(ny) dy where I j = j 2n , j + 1 2n , and we consider the map y τn -→ny and the images of I j under this map. We have τ n (I 0 ) = A, τ n (I 1 ) = A c , τ n (I 2 ) = A, • • • . Hence Ij e iπχA(ny) dy = (-1) j+1 . Since j ∈ {0, • • • , 2n -1}, it follows that T e iπχA(ny) dy = 0.

log 2 4 .

 24 we have | arcsin x| ≤ 2|x| and we assume that c ≤ π 1/2 1 + ε 0 86 1/

Proposition 4 . 15 .

 415 Assume that σ(n) ≥ 0 for each n ∈ Z. Then there exists m 0 such that for each m ≥ m 0 , T m exp(2πiX0) is not AT.Proof. Take the partition (A, A c ) of S 1 into the upper and the lower semicircle and consider (Ω×A, Ω×A c ). As in Section 4.3 we notice that the Fourier decomposition of F (ω, z) = (χ Ω×Aχ Ω×A c )(ω, z) is of the formF (ω, z) = f (z) =

X 2 2

 2 e -2π 2 m 2 X (n) 0 ≤ e -Cm 2 |n| .

Acknowledgment 1. The first author would like to express thanks to J-P. Thouvenot and B. Host for formulating to him the question about existence of finite multiplicity non-AT transformation. It is also a pleasure for him to express thanks to the university of Toruń and to the organizers of the Grefi-Mefi colloque "From dynamics to statistical mechanics, Luminy, CIRM, 2008" where a part of this work has been done.

The second author would like to thank Emmanuel Lesigne and Anthony Quas for fruitful discussions on the subject.

1 Research partially supported by Polish MNiSzW grant N N201 384834, Marie Curie "Transfer of Knowledge" EU program -project MTKD-CT-2005-030042 (TODEQ) and MSRI (Berkeley) program "Ergodic