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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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##### References:
 [1] R. Bowen,Entropy for group automorphisms and homogeneous spaces, Trans. Am. Math. Soc.153 (1971), 401–414. · Zbl 0212.29201 · doi:10.1090/S0002-9947-1971-0274707-X [2] M. Coste,Ensembles semi-algébriques, Lecture Notes in Math.959, Springer-Verlag, Berlin, 1982, pp. 109–138. [3] E. I. Dinaburg,On the relations among various entropy characteristics of dynamical systems, Math. USSR-Isv.5 (1971), 337–378. · Zbl 0248.58007 · doi:10.1070/IM1971v005n02ABEH001050 [4] D. Fried,Entropy and twisted cohomology, Topology, to appear. · Zbl 0611.58036 [5] D. Fried and M. Shub,Entropy, linearity and chain-recurrence, Publ. Math. IHES50 (1979), 203–214. · Zbl 0423.58010 [6] L. D. Ivanov,Variations of Sets and Functions, Nauka, 1975 (in Russian). [7] O. D. Kellogg,On bounded polynomials in several variables, Math. Z.27 (1928), 55–64. · JFM 53.0082.03 · doi:10.1007/BF01171085 [8] S. Newhouse,Entropy and volume, preprint. · Zbl 0638.58016 [9] M. Shub,Dynamical systems, filtrations and entropy, Bull. Am. Math. Soc.80 (1974), 27–41. · Zbl 0305.58014 · doi:10.1090/S0002-9904-1974-13344-6 [10] A. G. Vitushkin,On Multidimensional Variations, Gostehisdat, 1955 (in Russian). [11] Y. Yomdin,Global bounds for the Betti numbers of regular fibers of differentiable mappings, Topology24 (1985), 145–152. · Zbl 0566.57014 · doi:10.1016/0040-9383(85)90051-5 [12] Y. Yomdin,C k -resolution of semialgebraic mappings. Addendum to ”Volume growth and entropy”, Isr. J. Math.57 (1987), 301–317 (this issue). · Zbl 0641.54037 · doi:10.1007/BF02766216
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