For bounded Cantor numeration systems, the spectrum of the continuous-state stochastic adding machine equals the fibered filled Julia set in the null recurrent case and its boundary in the transient case.
Davies, Linear Operators and Their Spectra , Cambridge University Press, Cam- bridge Studies in Advanced Mathematics 106, 2007
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Operators of stochastic adding machines and Julia sets
For bounded Cantor numeration systems, the spectrum of the continuous-state stochastic adding machine equals the fibered filled Julia set in the null recurrent case and its boundary in the transient case.