A weak ergodic theorem for infinite products of Lipschitzian mappings

Abstract

Let $K$ be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self-mapping $A$ of $K$, we denote by $\text{Lip}\left(A\right)$ its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self-mappings of $K$. We consider the set of all sequences ${\left\{A_t\right\}}_{t=1}^{\infty }$ of such self-mappings with the property ${\text{limsup}}_{t\to \infty }\text{Lip}\left(A_t\right)\le 1$. Endowing it with an appropriate topology, we establish a weak ergodic theorem for the infinite products corresponding to generic sequences in this space.