Entropy and exponential growth of 𝜋₁ in dimension two

Franks, John M.; Handel, Michael

doi:10.1090/S0002-9939-1988-0929016-8

Entropy and exponential growth of $\pi _ 1$ in dimension two
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Proc. Amer. Math. Soc. 102 (1988), 753-760 Request permission

Abstract:

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.

References

Rufus Bowen, Entropy and the fundamental group, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 21–29. MR 518545

Traveaux de Thurston sur les surfaces

66-67

David Fried, Entropy and twisted cohomology, Topology 25 (1986), no. 4, 455–470. MR 862432, DOI 10.1016/0040-9383(86)90024-8
David Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), no. 4, 539–563. MR 829857, DOI 10.1017/S0143385700003151
Michael Handel and William P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, 173–191. MR 788938, DOI 10.1016/0001-8708(85)90028-3
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2

Lyapunov exponents, entropy and periodic points for diffeomorphisms

51

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6