Hyperkähler Kummer Rigidity and the Vieta Involutions on Tropical Markov Cubics

In this paper, we study holomorphic and algebraic dynamics on complex manifolds. It consists of two parts: one that discusses Kummer rigidity on hyperkähler manifolds, and another that discusses tropicalized actions of algebraic dynamics. In the part on Kummer rigidity, based on the fundamental structures that any hyperkähler manifold carries, we generalize some classical works done for complex surfaces. In particular, motivated by the study of Green currents on K3 surfaces, we have establised `Green-like currents' extracted from a singular measure under the assumption that the Green measure equals to the volume. By doing so, we have shown that the only such projective manifolds and dynamics should be constructed from linear actions on (complex) tori. In the part on tropical dynamics, we discuss the family of Markov cubics, which is a family of degree 3 affine surfaces that lives in the character variety of spheres with four punctures. By defining this family over a non-archimedean field, we define tropicalizations of a natural family of algebraic involutions: the Vieta involutions. These involutions, after tropicalization, exhibit a structure that resembles the hyperbolic plane with three independent reflections. By that we split the system in two parts: one that corresponds to hyperbolic reflections, and another that mimics the Euclidean algorithm on pairs of integers. We also include some introduction to the machinary used to perform the analysis.