On divergence of any order Cesàro mean of Lotka--Volterra operators

Abstract

Based on some numerical calculations, S.M. Ulam has conjectured that the ergodic theorem holds true for any quadratic stochastic operator acting on the finite dimensional simplex. However, M.I. Zakharevich showed that Ulam's conjecture is false in general. Later, N.N. Ganikhodjaev and D.V. Zanin have generalized Zakharevich's example in the class of quadratic stochastic Volterra operators acting on 2D simplex. In this paper, we provide a class of Lotka--Volterra operators for which any order Cesàro mean diverges. This class of Lotka--Volterra operators encompasses all previously presented operators in this context.