Spectral analysis of time changes for horocycle flows

TL;DR

This paper proves that all time changes of horocycle flows have purely absolutely continuous spectrum, answering a longstanding question about their spectral properties using positive commutator methods.

Contribution

It establishes the spectral nature of time changes for horocycle flows under certain conditions, advancing understanding in dynamical systems and spectral theory.

Findings

All time changes have purely absolutely continuous spectrum.

The result applies under the condition of A. G. Kushnirenko.

The proof uses positive commutator methods.

Abstract

We prove (under the condition of A. G. Kushnirenko) that all time changes for the horocycle flow have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes for horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators.