The first application of algebraic geometry to dynamical systems that comes to my mind is the following preprint of Gromov -- very old one : ON THE ENTROPY OF HOLOMORPHIC MAPS
https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1024.pdf
A recent, excellent survey of dynamics on algebraic surfaces is :
http://perso.univ-rennes1.fr/serge.cantat/Articles/dyn-aut.pdf
