Computability of the Julia set. Nonrecurrent critical orbits
Artem Dudko
Discrete & Continuous Dynamical Systems - A 2014, 34(7): 2751-2778 doi: 10.3934/dcds.2014.34.2751
We prove, that the Julia set of a rational function $f$ is computable in polynomial time, assuming that the postcritical set of $f$ does not contain any critical points or parabolic periodic orbits.
keywords: computational complexity Computability Julia set.
On spectra of Koopman, groupoid and quasi-regular representations
Artem Dudko Rostislav Grigorchuk
Journal of Modern Dynamics 2017, 11(1): 99-123 doi: 10.3934/jmd.2017005

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

keywords: Koopman representation groupoid construction quasi-regular representation spectrum weak containment weakly branch groups

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