Mixing constructions with infinite invariant measure and spectral multiplicities

We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $M\subset \Bbb N\cup \{\infty \}$ as the set of essential values of the multiplicity function for the Koopman operator o...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2011-06, Vol.31 (3), p.853-873
Main Authors: DANILENKO, ALEXANDRE I., RYZHIKOV, VALERY V.
Format: Article
Language:English
Subjects:
Automorphisms
Construction
Dynamical systems
Ergodic processes
Invariants
Preserving
Spectra
Spectrum analysis
Transformations
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Summary:We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $M\subset \Bbb N\cup \{\infty \}$ as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, and (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385710000052