Locating Ruelle–Pollicott resonances*

Abstract

We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on the discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding or simply monotone maps, hyperbolic diffeomorphisms. We believe this approach could be greatly generalised.

Discontinuities cause essential spectrum

“…The present results don't say anything about the point spectrum and a high degree of smoothness of the system doesn't prevent the existence of plenty of it [20]. Sometimes it is possible to use other arguments in order to identify the point spectrum [13,7] but control on the essential spectrum, as obtained here, is the key prerequisite to further investigation.…”

“…6 In particular, ϕ admits a continuous extension up to ω, for all ω ∈ Ω. 7 We prove the slightly stronger statement EssSpec(Lϕ) ⊇ {|λ| ≤ Λ} (see Section 4).…”

“…The aim is to choose a Banach space for which the essential spectrum is as small as possible. Once this is done, one may proceed to determine the point spectrum or "resonances" of the system (see, e.g., [7,13] and references within). Moreover the point spectrum is essentially independent of the choice of Banach space, it is inherent in the system studied [5].…”

“…Similarly, for smooth hyperbolic systems, the essential spectrum can be made as small as desired for a specific choice of the Banach space [15]. In the case when the piecewise monotone interval transformation is Markov, it is possible to make the essential spectrum as small as desired (see e.g., [7]). For this reason, the requirement in Theorem 1 that the transformation has a non-trivial discontinuity, is essential.…”

Discontinuities Cause Essential Spectrum

“…The aim is to choose a Banach space for which the essential spectrum is as small as possible. Once this is done, one may proceed to determine the point spectrum or "resonances" of the system (see, e.g., [12,18] and references within). Moreover the point spectrum is essentially independent of the choice of Banach space, it is inherent in the system studied [7].…”