Automorphisms of compact K\"ahler manifolds with slow dynamics

Serge Cantat (1) , Olga Paris-Romaskevich (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 I2M - Institut de Mathématiques de Marseille

We study the automorphisms of compact K\"ahler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions $2$ and $3$. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the $C^{\infty}$ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove they exist only on tori. We conjecture that this is true for any dimension.

Domaines: Mathématiques [math]; Systèmes dynamiques [math.DS]

Serge Cantat, Olga Paris-Romaskevich. Automorphisms of compact K\"ahler manifolds with slow dynamics. Transactions of the American Mathematical Society, 2021, 374 (2), pp.1351-1389. ⟨10.1090/tran/8229⟩. ⟨hal-03089240⟩