[Paper Review] Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows

Livio Flaminio, Giovanni Forni|arXiv (Cornell University)|Nov 12, 2018

Mathematical Dynamics and Fractals3 references4 citations

TL;DR

This paper establishes that non-trivial smooth time-changed horocycle flows on compact quotients satisfy the AOP (Asymptotic Orthogonal Powers) property, which implies Sarnak’s Möbius orthogonality conjecture. By combining Ratner’s classification of joinings for time-changed horocycle flows with the authors’ cohomological results, the paper proves that if distinct powers of such a flow have a non-trivial joining, the time-change function must be cohomologous to a constant—otherwise, the AOP property holds, leading to Möbius orthogonality for all topological uniquely ergodic systems conjugated to the time-one map.

ABSTRACT

We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.

Motivation & Objective

• To establish the AOP (Asymptotic Orthogonal Powers) property for non-trivial smooth time-changes of horocycle flows on compact quotients.
• To prove that such flows satisfy Sarnak’s Möbius orthogonality conjecture when the time-change function is not cohomologous to a constant.
• To answer a question posed by Kanigowski, Lemańczyk, and Ulcigrai by deriving the result directly from Ratner’s classification of joinings and the authors’ cohomology characterization.
• To show that the Möbius orthogonality conjecture holds for all topological uniquely ergodic systems measurably conjugated to the time-one map of such time-changed flows.

Proposed method

• Leverages Ratner’s classification of joinings for time-changed horocycle flows to analyze the structure of joint distributions between distinct powers of the flow.
• Applies the authors’ cohomological characterization of coboundaries in horocycle flows to determine conditions under which a time-change function is cohomologous to a constant.
• Uses the fact that the time-change function τ is in W^s(Γ\G) with s > 2, ensuring sufficient regularity for Sobolev space techniques.
• Analyzes the action of the geodesic flow on invariant distributions and uses spectral decomposition to study the behavior of the flow’s transfer operators.
• Applies the theory of Sobolev spaces and G-equivariant isometric embeddings to relate distributions on covering spaces to those on the base space.
• Employs duality and projection techniques on invariant subspaces of distributions to show that any invariant distribution annihilating τ must vanish identically.

Experimental results

Research questions

• RQ1 Under what conditions do distinct powers of a smooth time-changed horocycle flow admit a non-trivial joining?
• RQ2 When is a smooth time-change function τ on a compact quotient cohomologous to a constant?
• RQ3 How does the AOP property relate to the Möbius orthogonality conjecture in the context of time-changed horocycle flows?
• RQ4 Can Ratner’s classification of joinings be combined with cohomological results to derive orthogonality properties for time-changed flows?
• RQ5 What is the role of Sobolev regularity s > 2 in ensuring the validity of the AOP property for such flows?

Key findings

• If distinct powers of a smooth time-changed horocycle flow have a non-trivial joining, then the time-change function τ is cohomologous to a constant.
• For smooth time-changed horocycle flows with τ not cohomologous to a constant, the AOP property holds, implying asymptotic orthogonality of powers.
• The Möbius orthogonality conjecture holds for all topological uniquely ergodic systems measurably conjugated to the time-one map of such flows.
• The result is derived directly from Ratner’s classification of joinings and the authors’ cohomological characterization of horocycle flows, answering an open question posed by Kanigowski, Lemańczyk, and Ulcigrai.
• The analysis shows that all invariant distributions annihilating τ must vanish, proving that τ cannot be cohomologous to a constant unless the joining is trivial.
• The proof relies on spectral analysis of the geodesic flow action on invariant distributions and the structure of Jordan blocks in the case of the eigenvalue μ = 1/4.