Decay of Correlations for Non-Uniformly Hyperbolic Diffeomorphisms via Transfer Operators on Anisotropic Banach Spaces

We establish exponential decay of correlations for a broad class of non-uniformly hyperbolic diffeomorphisms on compact manifolds, utilizing a functional analytic framework. By constructing a family of anisotropic Banach spaces of distributions, adapted to the underlying hyperbolic splitting, we prove that the associated Ruelle-Perron-Frobenius transfer operator admits a spectral gap. Unlike the classical uniform setting, our construction accommodates singularities and unbounded derivatives characteristic of non-uniform systems by introducing weight functions that penalize orbits near the singular set. We derive a Lasota-Yorke type inequality, ensuring the quasi-compactness of the transfer operator. Consequently, we obtain precise estimates on the essential spectral radius, yielding exponential mixing rates for Hölder continuous observables. This approach unifies and extends previous results derived via Young towers, offering a direct spectral interpretation of the thermodynamic formalism for systems with weak hyperbolicity.