[Paper Review] Banach spaces for piecewise cone hyperbolic maps

Viviane Baladi, Sébastien Gouëzel|arXiv (Cornell University)|Jul 8, 2009

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper establishes a spectral gap for transfer operators associated with piecewise cone hyperbolic maps under a bunching condition, using anisotropic Sobolev spaces to unify and extend prior results. It proves that when complexity is subexponential, the essential spectral radius is strictly less than one, implying exponential decay of correlations and statistical stability for physical measures in many cases, including volume-preserving systems and systems with stable dimension one.

ABSTRACT

We consider piecewise cone hyperbolic systems satisfying a bunching condition and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani and our previous work. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal to 1 and the unstable dimension is not zero).

Motivation & Objective

• To unify and extend the functional analytic approach to statistical properties of piecewise cone hyperbolic maps with singularities.
• To provide a general framework for transfer operators acting on anisotropic Sobolev spaces that handles discontinuities and singularities.
• To prove a spectral gap for the transfer operator under a bunching condition, ensuring exponential decay of correlations for physical measures.
• To generalize results from Demers-Liverani (2008) and Baladi-Gou"ezel (2009) into a single, coherent framework.
• To lay the foundation for proving exponential decay of correlations in discrete-time Sinai billiards via spectral gap in transfer operators.

Proposed method

• The authors construct anisotropic Sobolev spaces $\mathbf{H}_{p}^{t,s}$ in the Triebel-Lizorkin class, adapted to the hyperbolic structure of the system.
• They define a transfer operator $\mathcal{L}$ acting on distributions via $\mathcal{L}\omega = \frac{\omega \circ T^{-1}}{|\det DT \circ T^{-1}|}$, extending it to the Banach space $\mathbf{H}$.
• The essential spectral radius of $\mathcal{L}$ on $\mathbf{H}$ is bounded using the bunching condition and transversality assumptions on stable and unstable manifolds.
• The proof relies on the boundedness and compactness of characteristic functions of shrinking neighborhoods of discontinuity sets in $\mathbf{H}$, ensuring that physical measures give zero mass to discontinuities.
• A duality argument is used: the dual of the transfer operator $\mathcal{L}$ is shown to preserve Lebesgue measure, linking the fixed point of $\mathcal{L}$ to the physical measure.
• The spectral gap is established by proving that the iterates of $\mathcal{L}$ are uniformly bounded and that the essential spectral radius is strictly less than one.

Experimental results

Research questions

• RQ1 Can a unified functional-analytic framework be developed for piecewise cone hyperbolic maps with singularities, subsuming prior results?
• RQ2 Under what conditions does the transfer operator on anisotropic Sobolev spaces exhibit a spectral gap?
• RQ3 Does the essential spectral radius of the transfer operator remain strictly less than one under the bunching condition?
• RQ4 Can the spectral gap result imply exponential decay of correlations for physical measures in systems with subexponential complexity?
• RQ5 Does the framework ensure that physical measures are supported away from discontinuity sets?

Key findings

• The essential spectral radius of the transfer operator $\mathcal{L}$ on the anisotropic Sobolev space $\mathbf{H}$ is strictly less than one under the bunching condition.
• The transfer operator $\mathcal{L}$ has a fixed point in $\mathbf{H}$, corresponding to the physical measure, and 1 is the only eigenvalue on the unit circle, which is simple.
• For systems with subexponential complexity, the spectral gap implies exponential decay of correlations for physical measures and Hölder observables.
• The framework ensures that physical measures give zero mass to the discontinuity set of $T$, as shown by the convergence of characteristic functions of shrinking neighborhoods to zero in $\mathbf{H}$.
• The result generalizes and unifies earlier work by Demers-Liverani (2008) and Baladi-Gou"ezel (2009), applying to volume-preserving systems and systems with stable dimension one.
• The method avoids Markov partitions and artificial expanding maps, preserving the smoothness of the original dynamics.