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Volume growth and entropy. (English) Zbl 0641.54036
This paper and its addendum [see the following review] are concerned with inequalities involving topological entropy, growth rates of the volumes of iterates of smooth submanifolds and the entropy conjecture. Let \(f: N\to N\) be a continuous mapping on a compact m-dimensional \(C^{\infty}\)-smooth manifold. The logarithm of the spectral radius of \(f_*: H_{\ell}(N,{\mathbb{R}})\to H_{\ell}(N,{\mathbb{R}})\) will be denoted by \(S_{\ell}(f)\) for \(\ell =0,1,...,m\) and \(S(f)=\max_{\ell}S_{\ell}(f)\). If h(f) is the topological entropy of f [R. Bowen, Trans. Am. Math. Soc. 153, 401-414 (1971; Zbl 0212.292)] then it is proved in the first paper that if f is a \(C^{\infty}\)- smooth map, then S(f)\(\leq h(f)\). To give a precise statement about results on the growth rates of volumes, we need additional invariants of \(C^ k\)-smooth maps \(f: N\to N\) for \(k\geq 1\). Let \(\Sigma\) (k,\(\ell)\) be the set of \(C^ k\)-smooth maps \(\sigma:[0,1]^{\ell}\to N\) and v(\(\sigma)\) be the \(\ell\)-dimensional volume of the image of \(\sigma\) in N, counted with multiplicities. For n a natural number, define \(v(f,\sigma,n)=v(f^ n\circ \sigma)\). Then for \(k\geq 1\) and \(\ell \leq m\) \(v_{\ell,k}(f)=\sup_{\sigma \in \Sigma (\ell k)}\overline{\lim}_{n\to \infty}1/n \log v(f,\sigma,n)\), \(v_ k(f)=\max_{\ell}v_{\ell,k}(f)\), and \(v(f)=v_{\infty}(f)\). S. Newhouse recently obtained the following result: For \(f\in C^{1- \epsilon}\), \(\epsilon >0\), then h(f)\(\leq v(f)\). Here the following opposite inequality is proved: For \(f\in C^ k\), \(k=1,...,\infty\); \(\ell \leq m\) \[ (*)\quad v_{\ell,k}(f)\leq h(f)+\frac{2\ell}{k}R(f)... \] where \(R(f)=\lim_{n\to \infty}\log \max_{x\in N}\| df^ n(x)\|\).
Reviewer: D.Hurley

MSC:
54H20 Topological dynamics (MSC2010)
54C70 Entropy in general topology
37A99 Ergodic theory
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