Eigenvalues of transfer operators for dynamical systems with holes

Abstract

For real-analytic expanding open dynamical systems in arbitrary finite dimension, we establish rigorous explicit bounds on the eigenvalues of the corresponding transfer operators acting on spaces of holomorphic functions. In dimension 1 the eigenvalue decay rate is exponentially fast, while in dimension d it is O(θn1/d) as n → ∞ for some 0 < θ < 1.