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Entropy and exponential growth of $$\pi _ 1$$ in dimension two. (English) Zbl 0644.58016
Summary: The authors show that if $$f: M\to M$$ is a $$C^{1+\alpha}$$ diffeomorphism of a compact surface and if the topological entropy of f is positive then there is a finite invariant set P such that the map induced by f on $$\pi_ 1(M-P)$$ has exponential growth.

MSC:
 37D99 Dynamical systems with hyperbolic behavior 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 28D20 Entropy and other invariants
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