## Entropy of polynomial and rational maps.(English)Zbl 0737.54006

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 $$X$$.

### MSC:

 54C70 Entropy in general topology 32A20 Meromorphic functions of several complex variables 37A99 Ergodic theory

rational map
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