[Paper Review] Quasi-compactness of transfer operators for contact Anosov flows

Masato Tsujii|arXiv (Cornell University)|Jun 4, 2008

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper establishes the quasi-compactness of transfer operators for $C^r$ contact Anosov flows with $r \geq 3$ by constructing a scale of Hilbert spaces of distributions that contain $C^r$ functions and on which the transfer operators extend boundedly. It provides explicit upper bounds on the essential spectral radii in terms of the differentiability $r$ and the hyperbolicity exponents, implying precise exponential decay rates of correlations without relying on Markov partitions.

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.

Motivation & Objective

• To establish the spectral regularity of transfer operators for $C^r$ contact Anosov flows with $r \geq 3$.
• To construct a scale of Hilbert spaces of distributions that contain $C^r$ functions and on which the transfer operators extend boundedly.
• To provide explicit upper bounds on the essential spectral radii of the extended operators in terms of $r$ and the hyperbolicity exponents.
• To demonstrate exponential decay of correlations with precise asymptotic estimates, independent of Markov partitions.

Proposed method

• Construct a scale of Hilbert spaces embedded in the space of distributions on the phase space, containing all $C^r$ functions.
• Define transfer operators $\mathcal{L}^t$ via pushforward: $\mathcal{L}^t(u)(z) = u \circ F^t(z)$ for $C^r$ functions.
• Use a dyadic decomposition of frequency space and microlocal analysis to define frequency-localized operators and associated norms.
• Introduce weighted norms $\|u\|_{\beta,\nu}^2 = \sum_n 2^{2\beta n} \sum_{\gamma: \widetilde{n}(\gamma) = n} \|d_\gamma^\nu u_\gamma\|_{L^2}^2$ to control decay and localization.
• Apply integration by parts and kernel estimates to bound the operator norms of frequency-localized components.
• Establish embeddings between the Hilbert spaces and Sobolev spaces $W^s(M)$ to relate regularity to spectral properties.

Experimental results

Research questions

• RQ1 Can transfer operators for $C^r$ contact Anosov flows be extended boundedly to a scale of Hilbert spaces of distributions?
• RQ2 What is the essential spectral radius of the extended transfer operators, and how does it depend on the differentiability $r$ and hyperbolicity exponents?
• RQ3 Can exponential decay of correlations be quantified via spectral properties without using Markov partitions?
• RQ4 How do frequency-localized operators and microlocal techniques enable spectral control in the absence of discrete hyperbolicity?

Key findings

• The transfer operators extend boundedly to a scale of Hilbert spaces of distributions that contain all $C^r$ functions for $r \geq 3$.
• The extensions are quasi-compact, implying that the spectrum consists of isolated eigenvalues of finite multiplicity and a spectral drop-off governed by the essential spectral radius.
• An explicit upper bound on the essential spectral radius is given in terms of $r$ and the hyperbolicity exponents $\lambda_0$ and $\Lambda_0$, with the bound decaying exponentially in $r$.
• The essential spectral radius is bounded above by $C \cdot 2^{-\delta r}$ for some $\delta > 0$ depending on $\lambda_0$ and $\Lambda_0$, ensuring exponential decay of correlations.
• The method is free from Markov partitions, enabling a direct spectral approach to correlation decay in smooth hyperbolic flows.
• The results imply a precise asymptotic estimate on the decay rate of correlations, extending Liverani's result to $C^r$ flows with $r \geq 3$.