An inequality is proved, bounding the growth rates of the volumes of iterates of smooth submanifolds in terms of the topological entropy. ForC x-smooth mappings this inequality implies the entropy conjecture, and, together with the opposite inequality, obtained by S. Newhouse, proves the coincidence of the growth rate of volumes and the topological entropy, as well as the upper semicontinuity of the entropy.
KeywordsVolume Growth Topological Entropy Algebraic Function Opposite Inequality Smooth Submanifolds
Unable to display preview. Download preview PDF.
- 2.M. Coste,Ensembles semi-algébriques, Lecture Notes in Math.959, Springer-Verlag, Berlin, 1982, pp. 109–138.Google Scholar
- 4.D. Fried,Entropy and twisted cohomology, Topology, to appear.Google Scholar
- 6.L. D. Ivanov,Variations of Sets and Functions, Nauka, 1975 (in Russian).Google Scholar
- 8.S. Newhouse,Entropy and volume, preprint.Google Scholar
- 10.A. G. Vitushkin,On Multidimensional Variations, Gostehisdat, 1955 (in Russian).Google Scholar