On the spectrum of stochastic perturbations of the shift and Julia sets
Volume 218 / 2012
Fundamenta Mathematicae 218 (2012), 47-68 MSC: Primary 47A10, 47A35; Secondary 37F50, 37A30. DOI: 10.4064/fm218-1-3
We extend the Killeen–Taylor study [Nonlinearity 13 (2000)] by investigating in different Banach spaces ($\ell^\alpha(\mathbb N), c_0(\mathbb N),c(\mathbb N)$) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base $2$ and in the Fibonacci base. For the base $2$, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectrum is related to the Julia set of an endomorphism of $\mathbb C^2$.