Criteria for the existence of equivariant fibrations on algebraic surfaces and hyperk\"ahler manifolds and equality of automorphisms up to powers - a dynamical viewpoint

Let $X$ be a projective surface or a hyperk"ahler manifold and $G \le Aut(X)$. We give a necessary and sufficient condition for the existence of a non-trivial $G$-equivariant fibration on $X$. We also show that two automorphisms $g_i$ of positive entropy and polarized by the same nef divisor are the same up to powers, provided that either $X$ is not an abelian surface or the $g_i$ share at least one common periodic point. The surface case is known among experts, but we treat this case together with the hyperk"ahler case using the same language of hyperbolic lattice and following Ratcliffe or Oguiso. This arXiv version contains proofs omitted in the print version.