On the spectrum of stochastic perturbations of the shift and Julia sets

We extend the Killeen-Taylor study in \cite{KT} by investigating in different Banach spaces ($\ell^\alpha(\N), c_0(\N),c_c(\N)$) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base 2 and in Fibonacci base. For the base 2, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectra involve the Julia set of an endomorphism of \C^2.Comment: 24 pages, 8 figures and 10 reference

Approximately transitive dynamical systems and simple spectrum

For some countable discrete torsion Abelian groups we give examples of their finite measure-preserving actions which have simple spectrum and no approximate transitivity property.Comment: 9 pages, Key words and phrases: ergodic theory, dynamical system, AT property, funny rank one, Haar spectru