Topological entropy on saddle sets in \(\mathbb{P}^2\)
The authors establish concepts of a saddle set and a terminal for a holomorphic map \(f\) from \({\mathbb P}^2\) into itself. A saddle set is an invariant set \(\Lambda\) which is transitive and hyperbolic (with nontrivial expansion and nontrivial contraction), and in some sense it is both maximal and isolated to the latter property. A terminal is an invariant set \(\Lambda\) such that for any negative semi-orbit \(\hat{p} \subset \Lambda\), the iterates of \(f\) act normally on \(W^u_{loc}(\hat{p})\smallsetminus \Lambda\). They prove that for a holomorphic map of degree \(d\) the topological entropy of \(f\) restricted to a saddle set \(\Lambda\) is no greater than \(\log d\) and the equality holds, iff, \(\Lambda\) is terminal.
scientific article; zbMATH DE number 1484734