Marked points of families of hyperbolic automorphisms of smooth complex projective varieties

Let $\pi : X\to \Lambda$ be a flat family of smooth complex projective varieties parameterized by a smooth quasi-projective variety $\Lambda$, and let $f: X\to X$ be a family of automorphisms with positive topological entropy. Suppose $\sigma : \Lambda \to X$ is a marked point, i.e., it is a rational section of $\pi$. We propose two methods to measure the stability, normality, or periodicity of the family given by $t \mapsto f_t^n(\sigma(t))$. First, from an algebraic perspective, we construct geometric canonical height functions that have desirable properties. Second, from an analytic viewpoint, we construct a positive closed $(1,1)$-current with continuous local potential. When $\Lambda$ is a curve, we demonstrate that these two constructions actually coincide, providing a unified approach to understanding the dynamical behavior of the family. As an application of the algebraic method, we prove a special case of the Kawaguchi-Silverman conjecture over complex function fields.