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Entropy and volume. (English) Zbl 0638.58016
The main result of this paper shows that the topological entropy of a \(C^{1+\alpha}\) self map of a compact manifold M is bounded above by the maximum volume growth of \(C^{1+\alpha}\) disks in M. Applications of this result are given; e.g. simple proofs of theorems of Gromov estimating the entropy of algebraic maps. In particular, Gromov’s theorem that the entropy of a holomorphic self map of complex projective space equals its topological degree is proved. A recent result of Y. Yomdin [Isr. J. Math. (to appear)] states that for \(C^{\infty}\) maps the topological entropy is bounded below by the volume growths of \(C^{\infty}\) disks, so in the \(C^{\infty}\) case one has the fact that topological entropy equals the maximum volume growth.
Reviewer: S.Newhouse

37A99 Ergodic theory
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