Volume growth and entropy
type: Item
instance of: scholarly article
title: Volume growth and entropy
zbMATH Open document ID: 0641.54036
DOI: 10.1007/BF02766215
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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 [\textit{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 \(R(f)=\lim_{n\to \infty}\log \max_{x\in N}\| df^ n(x)\|\).
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 4044735
zbMATH Keywords: volume growth
zbMATH Keywords: entropy conjecture
zbMATH Keywords: compact m-dimensional \(C^{\infty }\)- smooth manifold
zbMATH Keywords: \(C^{\infty }\)-smooth map
author: Yosef Yomdin
reviewed by: D. J. Hurley
MaRDI profile type: Publication
cites work: Entropy for Group Endomorphisms and Homogeneous Spaces
cites work: Q3963137
cites work: ON THE RELATIONS AMONG VARIOUS ENTROPY CHARACTERISTICS OF DYNAMICAL SYSTEMS
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cites work: Entropy and volume
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cites work: Global bounds for the Betti numbers of regular fibers of differentiable mappings
cites work: \(C^ k\)-resolution of semialgebraic mappings, addendum to volume growth and entropy
full work available at URL: https://doi.org/10.1007/bf02766215
OpenAlex ID: W2049920220
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description: scientific article; zbMATH DE number 4044735