Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy (Q2788595)

Let \(M\) be a complex projective manifold. A dominant self map \(f: M \dasharrow M\) of \(M\) is called imprimitive if and only if there exists a complex projective manifold \(B\) with \(\dim B < \dim M\), and dominant rational maps \(g : B \dasharrow B\) and \(\phi : M \dasharrow B\) such that \(\phi \circ f = g \circ \phi\). A dominant self map that is not imprimitive is called primitive. It is known that if \(M\) is a projective complex threefold which admits a primitive birational automorphism of infinite order then \(M\) is either rational or a weak Calabi-Yau threefold or a rationally connected threefold [\textit{D.-Q. Zhang}, J. Differ. Geom. 82, No. 3, 691--722 (2009; Zbl 1187.14061)].NEWLINENEWLINEThe entropy of \(f\) and the \(k\)-dynamical degree \(\lambda_k(f)\) of \(f\) are invariants of the map \(f\) which carry information about the action of \(f\) on \(M\) [\textit{S. Friedland}, Ann. Math. (2) 133, No. 2, 359--368 (1991; Zbl 0737.54006)], [\textit{V. Guedj}, Ergodic Theory Dyn. Syst. 25, No. 6, 1847--1855 (2005; Zbl 1087.37015)].NEWLINENEWLINEIn this paper the authors give examples of rational and weak Calabi-Yau threefolds which admit primitive biregular automorphisms of positive entropy and whose first and second dynamical degrees are not the same. Moreover, the authors prove that if \(f: M \dasharrow M\) is a dominant rational map of a complex projective manifold such that \(\lambda_1(f) > \lambda_2(f)\) then \(f\) is primitive.