Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms

Let $K$ be a number field, $X$ a smooth projective variety over $K$ and $f:X\to X$ a polarized endomorphism of degree $d\ge 2$. We prove an exponential lower bound on $[K({\text{Per}}_n):K]$, where ${\text{Per}}_n$ is the set of $n$-periodic points, extending results of [Yap24] to higher dimensions. We also prove a quantitative rate of equidistribution for ${\text{Per}}_n$ to the equilibrium measure.