  In this article and in its sequel we propose the study of certain
discretizations of geometric evolution equations as an approach to the study of
the existence problem of some elliptic partial differential equations of a
geometric nature as well as a means to obtain interesting dynamics on certain
infinite-dimensional spaces. We illustrate the fruitfulness of this approach in
the context of the Ricci flow, as well as another flow, in Kahler geometry. We
introduce and study dynamical systems related to the Ricci operator on the
space of Kahler metrics that arise as discretizations of these flows. We pose
some problems regarding their dynamics. We point out a number of applications
to well-studied objects in Kahler and conformal geometry such as constant
scalar curvature metrics, Kahler-Ricci solitons, Nadel-type multiplier ideal
sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy
functionals and the geometry and structure of the space of Kahler metrics.
E.g., we obtain a new sharp inequality strengthening the classical
Moser-Trudinger-Onofri inequality on the two-sphere.
