[Paper Review] Spectra of differentiable hyperbolic maps

Viviane Baladi, Masato Tsujii|arXiv (Cornell University)|Sep 16, 2005

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper establishes improved bounds on the essential spectral radius of transfer operators for smooth hyperbolic dynamical systems using anisotropic Banach spaces of distributions. By applying Paley-Littlewood decomposition and approximation number techniques, it proves that the dynamical Fredholm determinant's holomorphic extension is controlled by the spectral properties, with eigenvalues in the determinant's zeros—offering a refined spectral theory for hyperbolic maps.

ABSTRACT

This note is about the spectral properties of transfer operators associated to smooth hyperbolic dynamics. In the first two sections (written in 2006), we state our new results relating such spectra with dynamical determinants, first announced at the conference ``Traces in Geometry, Number Theory and Quantum Fields" at the Max Planck Institute, Bonn, October 2005. In the last two sections, we give a reader-friendly presentation of some key ideas in our work in the simplest possible settings, including a new proof of a result of Ruelle on expanding endomorphisms. (These last two sections are a revised version of the lecture notes given during the workshop ``Resonances and Periodic Orbits: Spectrum and Zeta functions in Quantum and Classical Chaos" at Institut Henri Poincaré, Paris, July 2005.) (Revised version, submitted for publication)

Motivation & Objective

• To refine the spectral analysis of Ruelle transfer operators in smooth hyperbolic dynamics.
• To improve previous bounds on the essential spectral radius using new anisotropic Banach spaces of distributions.
• To establish a precise correspondence between eigenvalues of the transfer operator and zeros of the dynamical Fredholm determinant.
• To provide a new proof of Kitaev’s lower bound on the holomorphic extension radius of the dynamical determinant.

Proposed method

• Introduces a new family of anisotropic Banach spaces $\mathcal{C}^{p,q}(T,V)$ for distributions on a compact neighborhood $V$ of a hyperbolic basic set $\Lambda$, with parameters $p>0$, $q<0$ satisfying $p-q < r-1$.
• Applies Paley-Littlewood decomposition in Fourier space to analyze regularity and decay properties of transfer operators.
• Uses a decomposition of the Fourier space into stable and unstable cones, inspired by earlier work, to control operator norms.
• Employs approximation numbers from functional analysis to estimate spectral radii and establish variational bounds.
• Applies dyadic decomposition and integration by parts in phase space to derive $L^2$-type estimates for oscillatory integrals arising from the transfer operator.
• Establishes a bijection between eigenvalues of $\mathcal{L}_{T,g}$ on $\mathcal{C}^{p,q}(T,V)$ and zeros of the dynamical Fredholm determinant in a disk of radius $R(T,g)\max\{\lambda_s^p, \nu_u^q\}$.

Experimental results

Research questions

• RQ1 What are the sharp bounds on the essential spectral radius of the Ruelle transfer operator for $C^r$ hyperbolic diffeomorphisms with $r>1$?
• RQ2 How do the spectral properties of the transfer operator relate to the holomorphic extension of the dynamical Fredholm determinant?
• RQ3 Can the eigenvalues of the transfer operator be precisely located via the zeros of the dynamical zeta function?
• RQ4 What is the role of anisotropic function spaces in controlling the spectral gap and quasicompactness of transfer operators?
• RQ5 How do approximation numbers and Paley-Littlewood techniques improve spectral estimates in hyperbolic dynamics?

Key findings

• The essential spectral radius of $\mathcal{L}_{T,g}$ on $\mathcal{C}^{p,q}(T,V)$ is bounded above by $R(T,g)\max\{\lambda_s^p, \nu_u^q\}$, where $R(T,g)$ is a norm-related quantity and $\lambda_s, \nu_u$ are the stable and unstable expansion rates.
• The dynamical Fredholm determinant admits a holomorphic extension in a disk of radius $R(T,g)\max\{\lambda_s^p, \nu_u^q\}$, and this bound is sharp in the sense of Kitaev’s lower bound.
• The zeros of the dynamical Fredholm determinant in this disk are in one-to-one correspondence with the eigenvalues of $\mathcal{L}_{T,g}$ on $\mathcal{C}^{p,q}(T,V)$, establishing a spectral determinant identity.
• For expanding maps, a new proof of the spectral radius bound is given using dyadic decomposition, confirming known results with a novel analytical approach.
• The construction of $\mathcal{C}^{p,q}(T,V)$ ensures that $\mathcal{L}_{T,g}$ is quasicompact with a spectral gap when $g>0$, and the space contains $C^s(V)$ for $s>p$ and is contained in the dual of $C^s(V)$ for $s>|q|$, ensuring appropriate regularity and duality.
• The method yields variational expressions for the spectral radius bounds, linking spectral behavior to dynamical quantities like Lyapunov exponents and metric entropy.