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Entropy and volume

Published online by Cambridge University Press:  10 December 2009

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Abstract

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An inequality is given relating the topological entropy of a smooth map to growth rates of the volumes of iterates of smooth submanifolds. Applications to the entropy of algebraic maps are given.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 283 - 299
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Fathi, A., Herman, M. & Yoccoz, J. C.. A Proof of Pesin's Stable Manifold Theorem. Lecture Notes in Mathematics 1007. Springer-Verlag, 177–216.Google Scholar
Gromov, M.. On the entropy of holomorphic maps. Preprint (1977).Google Scholar
Hirsch, M. & Pugh, C.. Stable manifolds and hyperbolic sets. Proc. AMS Symp. in Pure Math. 14 (1970), 133–163.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic points for diffeomorphisms. Publ. Math. IHES 51 (1980), 137–173.10.1007/BF02684777Google Scholar
Ljubich, M. Ju.. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys. 3 (1983), 351–387.Google Scholar
Mañé, R.. Lyapunov Exponents and Stable Manifolds for Compact Transformations. Lecture Notes in Mathematics 1007. Springer-Verlag, 522–577.Google Scholar
Misiurewicz, M. & Przytycki, F.. Topological entropy and degree of smooth mappings. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 25 (1977), 573–574.Google Scholar
Mumford, D.. Algebraic Geometry I, Complex Projective Varieties. Grundlehren der Mathematischen Wissenschaften 221. Springer-Verlag (1976).Google Scholar
Oseledec, V.. A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Pesin, Ja.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR Izv. 10 (1976), 1261–1305.10.1070/IM1976v010n06ABEH001835Google Scholar
Przytycki, F.. An upper estimation for the topological entropy of diffeomorphisms. Invent. Math. 59 (1980), 205–213.10.1007/BF01453234Google Scholar
Raghunathan, M.. A proof of Oseledec's multiplicative ergodic theorem. Israel J. Math. 32 (1979), 356–362.10.1007/BF02760464Google Scholar
Rohlin, V.. Exact endomorphisms of a Lebesgue space. AMS Trans. 39, Series 2 (1964), 1–37.Google Scholar
Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50 (1979), 275–306.Google Scholar
Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1980), 137–173.Google Scholar
Ruelle, D.. Characteristic exponents and invariant manifolds in Hilbert space. Ann. Math. 115 (1982), 243–291.10.2307/1971392Google Scholar
Ruelle, D. & Shub, M.. Stable Manifolds for Maps. Lecture Notes in Mathematics 819. Springer-Verlag (1980), 389–393.Google Scholar
Stolzenberg, G.. Volumes, Limits, and Extensions of Analytic Varieties. Lecture Notes in Mathematics 19. Springer-Verlag (1966).Google Scholar
Yomdin, Y.. Volume growth and entropy. Preprint. IHES (1986).Google Scholar
Also: C k-resolution of semialgebraic mappings—addendum to ‘Volume growth and entropy’. Preprint. Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel (1986).Google Scholar