On the spectrum of stochastic perturbations of the shift and Julia sets (Q2895964)

The authors study the spectrum of some stochastic perturbation of the shift operator introduced by \textit{P. R. Killeen} and \textit{T. J. Taylor} [Nonlinearity 13, No. 6, 1889--1903 (2000; Zbl 0978.37001)], who considered a stochastic adding machine as a stochastic perturbation of the shift.NEWLINENEWLINEThe main results of the present paper are stated in the following three theorems.NEWLINENEWLINETheorem 3.1. The spectrum of the operator \(S_p\) acting on \(c_0, c\), and \(\ell^{\alpha }\), \(\alpha \geq 1\), is equal to the filled Julia set \(K_f\) of some quadratic map.NEWLINENEWLINETheorem 3.2. In \(\ell^1\), the point spectrum of \(S_p\) is empty. The residual spectrum of \(S_p\) is not empty and contains a countable dense subset of the Julia set \(J_f = \partial K_f\). The continuous spectrum of \(S_p\) is the complement of the residual spectrum in the Julia set \(K_f\).NEWLINENEWLINETheorem 3.3. The spectra of \(S_p\) acting, respectively, in \(\ell^{\infty }, c_0, c\) and \(\ell^{\alpha }\), \({\alpha } \geq 1\), associated to stochastic Fibonacci adding machines, contain a certain set (and are conjectured to equal this set).