Abstract
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.
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Yomdin, Y. Volume growth and entropy. Israel J. Math. 57, 285–300 (1987). https://doi.org/10.1007/BF02766215
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DOI: https://doi.org/10.1007/BF02766215