Absolutely continuous spectrum for parabolic flows/maps

Lucia D. Simonelli

doi:10.3934/dcds.2018013

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2018, Volume 38, Issue 1: 263-292. Doi: 10.3934/dcds.2018013

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Absolutely continuous spectrum for parabolic flows/maps

Lucia D. Simonelli

Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy

Received: November 2016

Revised: August 2017

Published: October 2017

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Abstract

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.

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Mathematics Subject Classification: Primary:37A30, 37C40;Secondary:37A25, 81Q10.

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