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In this article and in its sequel we propose the study of certaindiscretizations of geometric evolution equations as an approach to the study ofthe existence problem of some elliptic partial differential equations of ageometric nature as well as a means to obtain interesting dynamics on certaininfinite-dimensional spaces. We illustrate the fruitfulness of this approach inthe context of the Ricci flow, as well as another flow, in Kahler geometry. Weintroduce and study dynamical systems related to the Ricci operator on thespace of Kahler metrics that arise as discretizations of these flows. We posesome problems regarding their dynamics. We point out a number of applicationsto well-studied objects in Kahler and conformal geometry such as constantscalar curvature metrics, Kahler-Ricci solitons, Nadel-type multiplier idealsheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energyfunctionals and the geometry and structure of the space of Kahler metrics.E.g., we obtain a new sharp inequality strengthening the classicalMoser-Trudinger-Onofri inequality on the two-sphere.

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I