A proof of the existence and simplicity of a maximal eigenvalue for Ruelle-Perron-Frobenius operators
In this note the author gives a new proof of the existence and simplicity of a unique maximal eigenvalue for a Ruelle-Perron-Frobenius operator acting on a certain Hölder continuous function space without using fixed-point theorems and any properties for convex sets. To this end the author defines local expanding and mixing maps from a compact metric space into itself and Ruelle-Perron-Frobenius operators. As an application of this result, one can prove the existence and uniqueness of the smooth probability invariant measure for a \(C^{1+\alpha}\) locally expanding map from a compact \(C^2\) Riemannian manifold ito itself.
scientific article; zbMATH DE number 1347970