[Paper Review] A spectral gap for transer operators of piecewise expanding maps

Damien Thomine|arXiv (Cornell University)|Jun 14, 2010

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper establishes a spectral gap for transfer operators of piecewise expanding maps using Sobolev spaces and bounded variation functions, proving the existence of finitely many mixing physical measures with exponential mixing rates. It generalizes Lasota-Yorke and Cowieson-type results by using a robust framework based on Sobolev norms and interpolation theory, yielding sharp estimates on the essential spectral radius via iteration and Hennion's theorem.

ABSTRACT

We provide a simplified proof of the existence, under some assumptions, of a spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding maps on Riemannian manifolds when acting on some Sobolev spaces. Its consequences include, among others, the existence of invariant physical measures, and an exponential decay of correlations for suitable observables. These features are then adapted to different function spaces (functions with bounded variation or bounded oscillation), so as to give a new insight of - and generalize - earlier results.

Motivation & Objective

• To establish a spectral gap for transfer operators of piecewise expanding maps under minimal regularity assumptions.
• To generalize Lasota-Yorke and Cowieson's results on physical measures using standard function spaces like Sobolev and bounded variation spaces.
• To simplify and unify proofs of spectral gap results by relying on interpolation theory and Fourier analysis instead of ad hoc function spaces.
• To extend the framework to the limit case of C^1+Lip maps with bounded variation functions, strengthening Cowieson's theorem.

Proposed method

• Uses Sobolev spaces defined via Fourier transform and interpolation theory to ensure regularity while allowing discontinuities.
• Applies elementary inequalities for composition with C^1+α diffeomorphisms and localization techniques to control operator norms.
• Employs Hennion's theorem on essential spectral radius and iterates the transfer operator to refine spectral estimates.
• Derives a Lasota-Yorke type inequality in Sobolev spaces to bound the essential spectral radius of the Perron-Frobenius operator.
• Adapts the cone method and compactness arguments to prove existence and mixing properties of physical measures.
• Uses iteration of the map to improve spectral radius estimates, leading to the limit expression in equation (6.6).

Experimental results

Research questions

• RQ1 Can a spectral gap be established for transfer operators of piecewise expanding maps using standard Sobolev spaces rather than ad hoc function spaces?
• RQ2 What is the sharp estimate for the essential spectral radius of the Perron-Frobenius operator in terms of expansion rate and discontinuity complexity?
• RQ3 Does the method yield exponential mixing for physical measures when the map is piecewise C^1+α with bounded variation densities?
• RQ4 Can the framework be extended to the C^1+Lip case with functions of bounded variation, improving on Cowieson's theorem?
• RQ5 How does the spectral gap depend on the combinatorial complexity of discontinuities and expansion rate?

Key findings

• The essential spectral radius of the transfer operator is bounded by an expression involving the expansion rate λ_n and the discontinuity complexity D_n^b, with the bound (λ_n^{-α} + 4γ_d D_n^b / ((λ_n - 1)γ_{d-1}))^{1/n} for each iterate n.
• The limit of this bound as n → ∞ yields the spectral gap estimate ρ_ess ≤ sup{ lim (λ_n^{-1/n} (D_n^b)^{1/n}), lim (λ_n^{-α/n}) }, which ensures exponential mixing.
• For piecewise C^1+α uniformly expanding maps, the Perron-Frobenius operator has a spectral gap on Sobolev spaces, implying finitely many mixing physical measures with densities in these spaces.
• In the C^1+Lip case, the method yields a stronger version of Cowieson's theorem, proving existence of finitely many physical measures with bounded variation densities.
• The framework is robust and can be adapted to other function spaces such as V_α (Saussol's space), suggesting broader applicability beyond Sobolev and BV spaces.
• The use of iteration and Hennion's theorem allows for tighter spectral radius estimates than the initial bound in Saussol's theorem, improving convergence rates.