Spectral types of random M-extensions

In this paper we study some spectral properties of a class of transformations called M-extensions, for a random construction. Such transformations are skew products over rank one transformations and their class contains transformations arising from generalized Morse sequence as defined by M. Keane. The same methods as for rank one transformations allow the determination of the spectral type of M-extensions, in terms of generalized Riesz products. For a random construction of M-extensions, we then prove their almost sure spectral singularity and mutual singularity on the orthocomplement in L-2 of the basis. We also show tbt: almost sure spectral simplicity of these random M-extensions. Nevertheless we shall investigate conditions for almost sure spectral continuity on the orthocomplement of the basis of these transformations. (C) Elsevier, Paris.