An extension which is relatively twofold mixing but not threefold mixing

We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold. 1. Factors, extensions and relative mixing 1.1. Factors, extensions and Rokhlin cocycle. We are interested in dynamical systems (X,A , μ, T ), where T is an ergodic automorphism of the Lebesgue space (X,A , μ). We will often designate such a system by simply the symbol T . A factor of T is a sub-σ-algebra H of A such that H = TH . The canonical example of a system with factor is given by the skew product, constructed from a dynamical system (XH ,AH , μH , TH) (called the base of the skew product) and a measurable map x 7−→ Sx from XH to the group of automorphisms of some Lebesgue space (Y,B, ν) (such a map is called a Rokhlin cocycle). The transformation is defined on the product space (XH × Y,AH ⊗ B, μH ⊗ ν) by T̃ (x, y) = (TH x, Sx y). In this context, the sub-σ-algebra AH ⊗ {Y, ∅} is clearly a factor of T̃ . Since the work of Abramov and Rokhlin [1], this kind of construction is known to be the general model for a system with factor: If H is a factor of T , then there exists an isomorphism φ between T and a skew product T̃ constructed as above, with φ(H ) = AH ⊗ {Y, ∅}. In such a situation, we say that T is an extension of TH . 1.2. Mixing relative to a factor. To understand precisely the way a factor is embedded in the dynamical system, one is led to study the behaviour of the system relative to the factor ; to this end, relative properties are defined which are generalizations of absolute properties of dynamical systems. For example, one can define weak-mixing relative to a factor (see e.g. [2]), or the property of being a K-system relative to a factor [4]. We are interested in this work in the property of being mixing relative to a factor. Definition 1.1. Let H be a factor of the system (X,A , μ, T ). T is said H relatively mixing if