Quasi-compactness of transfer operators for contact Anosov flows

TL;DR

This paper constructs a hierarchy of Hilbert spaces for contact Anosov flows, enabling the transfer operators to extend boundedly with quasi-compactness, and provides explicit spectral radius bounds based on flow properties.

Contribution

It introduces a new scale of Hilbert spaces for contact Anosov flows, ensuring transfer operators are bounded and quasi-compact, with explicit spectral bounds.

Findings

Transfer operators extend boundedly on constructed Hilbert spaces.

Extensions of transfer operators are quasi-compact.

Explicit bounds on spectral radii depend on flow smoothness and hyperbolicity.

Abstract

For any $C^r$ contact Anosov flow with $r\ge 3$, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all $C^r$ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow.