We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known results. The remaining and most challenging case is when the fibers are spheres of dimension 3. On the 3-sphere itself, using a result of Hamilton on the prescribed Ricci curvature equation, we establish existence and convergence of the Ricci iteration and confirm in this setting a conjecture on the relationship between ancient Ricci iterations and ancient solutions to the Ricci ow. In higher dimensions we obtain sufficient conditions for the solvability of the prescribed Ricci curvature equation as well as partial results on the behavior of the Ricci iteration.

我们研究球体和复杂投影空间上齐次度量的Ricci迭代。可以根据修改Hopf纤维的纤维上的规范度量来描述此类度量。当Hopf纤维的纤维是尺寸为2或7的圆形或球形时，我们观察到Ricci迭代以及所有古代Ricci迭代都可以使用已知结果完整描述。剩下的最具挑战性的情况是，当纤维是尺寸为3的球体时。在3球体本身上，使用汉密尔顿在规定的Ricci曲率方程式上的结果，建立Ricci迭代的存在性和收敛性，并在此设置下确认关于古代Ricci迭代与Ricci ow的古代解之间的关系的猜想。