Effective results on nonlinear ergodic averages in CAT spaces

Abstract

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\unicode[STIX]{x1D705})$ spaces (with $\unicode[STIX]{x1D705}>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.