Entropy of polynomial and rational maps (Q1174928)

Let \(F: X\to X\) be a rational map of an irreducible smooth projective variety \(X\subset\mathbb{C}\mathbb{P}^ n\) of complex dimension \(n\). The present paper proposes a definition of an entropy \(H(F)\), which measures the growth rates of volumes of algebraic subvarieties of \(X\). It is proved that for each closed \(\Omega\subset X\) such that \(F: \Omega\to\Omega\) and \(F\) is holomorphic on \(\Omega\) the standard entropy \(h(F,\Omega)\) of \(F\) on \(\Omega\) satisfies \(h(F,\Omega)\leq H(F)\). Using a result of Yomdin, the author proves \(h(F)=H(F)\) in case \(F\) is holomorphic on all of \(X\).