Bergman iteration and $C^{\infty}$-convergence towards K\"ahler-Ricci flow

On a polarized manifold $(X,L)$, the Bergman iteration $\phi_k^{(m)}$ is defined as a sequence of Bergman metrics on $L$ with two integer parameters $k, m$. We study the relation between the K\"ahler-Ricci flow $\phi_t$ at any time $t \geq 0$ and the limiting behavior of metrics $\phi_k^{(m)}$ when $m=m(k)$ and the ratio $m/k$ approaches to $t$ as $k \to \infty$. Mainly, three settings are investigated: the case when $L$ is a general polarization on a Calabi-Yau manifold $X$ and the case when $L=\pm K_X$ is the (anti-) canonical bundle. Recently, Berman showed that the convergence $\phi_k^{(m)} \to \phi_t$ holds in the $C^0$-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman's result and show that this convergence actually holds in the smooth topology.