Automorphisms of minimal entropy on supersingular K3 surfaces

In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that logλd, where λd is the minimal Salem number of degree d, is realized in characteristic 5 if and only if d⩽22 is even and d≠18. In the complex projective setting we settle the case of entropy logλ12, left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, that is, preserves a chamber of the positive cone.