Points périodiques d'applications birationnelles de \({\mathbb P}^2\). (Periodic points for birational mappings of \({\mathbb P}^2\).)
The paper under review is concerned with dynamics of birational maps of \(\mathbb{P}^2\) of degree at least 2. Over the last 10 years, polynomial automorphisms on \(\mathbb{C}^2\) and holomorphic endomorphisms of \(\mathbb{P}^k\) have received considerable attention (see a series of work by E. Bedford and J. Smillie (in part with M. Lyubich), by N. Sibony and by J. Diller). The author obtains an estimate the cardinal number, \(\text{Fix}^n\), of the set of periodic points with period \(n\), which has direct relations with the entropy of this system. The main theorem is as follows: Let \(f: \mathbb{P}^2\to\mathbb{P}^2\) be a `generic' birational map with degree \(d\geq 2\). If \(\text{Fix}^n\) is finite for any \(n\geq 1\), then there exists a constant \(D>0\) such that for any \(n\geq 1\), \[ d^n+ 2\geq \text{Fix}^n\geq d^n- D. \]
scientific article; zbMATH DE number 1220350