We give conditions for the Bernoullicity of the v-dimensional Markov processes. 1. Symbols and Definitions Z v is the v-dimensional lattice of the points with integral coordinates and K = I = γ[ 1,1 — {0,1}, is the space of sequences of 0's and Γs ξeZlabelled with the points ξ s Z . The space K is compact if endowed with the topology obtained as product of the discrete topologies on the factors /. Similarly if Θ C Z we define the compact space KΘ = I Θ = f\ I. ξeΘ We shall identify the elements X e KΘ as subsets of Θ: so that X = (χί9χ2...χp)eKΘ means the sequence XeKΘ with values 1 in x 1 ? x 2 , ...,xp and 0 in Θ\X. If XeK and ξeZ we put τξX = X + ξ = {xί + ξ,x2 + ξ,...) if X = (x 1 ? x 2 . . .) The transformations τξ:K-^K form a v-dimensional group which we denote with the symbol τ; τ transforms Borel sets into Borel sets. If μ is a Borel probability measure on K which is τ-invariant and A C Z v is a finite set (i.e. \Λ\ 0 QA-&*. (1.4) ii) fΛ(X\Y) = fA(X\Y ) if YndίA=Tnδ1A (1.5) the last equation being understood QΛ x QΛ — a.e. Define, next, \X\ = number of points in X [X] = number of nearest neighbours in X (1.6) ί(X\ Y) = number of couples of nearest neighbours (ξ, η) s u c h t h a t ξeX,ηeY then the following very remarkable theorem holds [1]: Theorem 1. A τ-invariant probability measure on K is a non singular Markov process if and only if there are two real parameters z 0, β such that V X C A V Y C Z\A (QΛ a.e.) z\X\e4.βi(X\Y)e4.β[X] f(\Y) (17) Λ( )= y z\X'\e4rβί(X'\Y)e4β[X'] ' because of this theorem we shall refer to a Markov process as to a (z, β)Markov process. There is a natural two set partition 0> of the space K on which the above Markov processes act: P0 = {X\XeK, OφX}, (1.8) Pi = {X\XeK9 OeX}. If yl C Z v is a finite region the 2 atoms of the partition 9>A = \/ τ ξ ^ are of the form Ayl(X) = {Y| Y G X , 7 n τ l = X}, and their measure will be denoted (1.9) Bernoulli Schemes 261 2. Description of the Results It has been recently shown that the non-singular Markov chains (i.e. 1-dimensional non-singular Markov processes) are uniquely determined by their conditional probabilities [2] and are Bernoulli schemes for all values of (z, β) [3]. In two or more dimensions the same questions are more difficult. It happens that the conditional probabilities do not necessarily determine the process which generates them [4]. It might even happen that a measure μ with conditional probabilities (1.7) is not necessarily τ-invariant [5]. It is, therefore, particularly interesting to ask whether a τ-invariant Markov process (z, β) is a Bernoulli scheme. In this paper we consider two extreme situations and show that the corresponding Markov processes are actually of Bernoulli type. The two situations correspond to the cases: i) β fixed and z l. ( } These two cases are extreme in the sense that in case i) the conditional probabilities uniquely determine a measure μ which is, furthermore, known to be τ-invariant, ergodic and, better, a K-system [6] in case ii) the conditional probabilities do not determine μ [4] and it is known that the corresponding τ-invariant ergodic measures are just two [7] (and furthermore they are both mixing). The proof will consist in showing that the partition & is "finitely determinate" (see next section) in a Markov process (z, β) verifying i), ii). It is known that this fact together with the fact that & is a τ-generator for (z, β) implies that (z, β) is a Bernoulli scheme [8]. The finite determinability of & relative to (z, β) is deduced from the strong cluster property Σ Σ \fMuΛ2{X^X2)-fΛί{Xχ)fΛl{X2)\^n{Λι,Λ2), (2.2) XiCΛi X2CΛ2 valid for \AX\, \A2\ We assume, from now on, that the Markov process (z,/?) on X, denoted by μ, verifies (2.2), (2.3) (hence is mixing). For simplicity we shall also fix v = 2. More generally if (K\ μ') is a Lebesgue measure space and τ' is a group of measure preserving transformations of K' and if &' is a partition of K' we shall call the couple (&', τ') a process on (K\ μ'). Thus a Markov process could be regarded as a process (^, τ) on (X, μ). Definition. A process (^, τ) will be called a weak Bernoulli process of exponential type (wbe-process) if there is a function F(a): R -+R + such that lim F(a) = 0 and, for any two disjoint regions AX,A2CZ 2 the α » 0 + two partitions — \l τ op ΰ) _ \ / i — V ξr > ^Λ2~ V ξeΛ, ξeΛ2 Bernoulli Schemes 263 are such that Σ Σ Iμfai °42) μ{qi) μ{q2)\ where d(Λ1,Λ2) = distance of y^ from /1 2 , |3i Λ| = number of points ξ in Z 2 neighbouring A and 9Ά)= £ μ{PAQt) Let us define a useful family of subsets of Z 2 : a) A — finite square = {ξ\ξeZ a1Sζiύb1,a2^ ξ2ύb2} with ahbt integers; i=ί,2," b) A°n = c) Λn = d) if A is the set in a) above we put A~ = {ξ\ξeZ either ξi