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Three remarks on geodesic dynamics and fundamental group. (English) Zbl 1002.53028
For a complete simply connected Riemannian manifold $$X$$ of negative curvature by $$\text{Cl}(X)$$ is denoted its compactification (closure), by $$\partial(X)$$ the complement $$\text{Cl}(X)\setminus X$$, and by $$\text{St}_2 (X)$$ the space of all tangential orthonormal 2-frames in $$X$$. Let $$V,W$$ be closed manifolds of negative curvature with isomorphic fundamental groups. The Cheeger homeomorphism theorem states that the spaces $$\text{St}_2(V)$$ and $$\text{St}_2(W)$$ are homeomorphic. Now the set of triples $$(x_1,x_2,x_3)$$, $$x_1, x_2, x_3 \in\partial (X)$$ with $$x_i\neq x_j$$ for $$i\neq j$$ is denoted by $$\partial^3 (X)$$. It is proved that if $$X$$ has strictly negative curvature, then $$\text{St}_2(X)$$ is canonically homeomorphic to $$\partial^3(X)$$. This leads to a new proof of the Cheeger homeomorphic theorem.
The results by M. Shub [Am. J. Math. 91, 175-199 (1969; Zbl 0201.56305)] and by J. Franks [Proc. Sympos. Pure Math. 14, 61-93 (1970; Zbl 0207.54304)] on endomorphisms are discussed and slightly generalized in Appendices. A covering lemma is given which immediately implies A. Manning’s estimate of the topological entropy of an $$f:S\to S$$ (for a compact manifold $$S)$$ in terms of the spectral radius of $$f_*:H_1(S; \mathbb{R})\to H_1(S;\mathbb{R})$$ [Lect Notes in Math. 468, 185-190 (1975; Zbl 0307.54042)].
Smale’s horseshoe is a space $$X$$ with three subspaces $$A,B,Z$$ and a map $$f:X\to X$$ with the properties: (a) $$f$$ sends $$A\cup B$$ into $$A$$ and $$Z$$ into $$B$$, (b) $$Z$$ separates $$A$$ from $$B$$, i.e. there exists a function $$a:X\to\mathbb{R}$$ which is positive on $$A$$, negative on $$B$$ and with $$a^{-1}(0)\subset Z$$. A Smale’s estimate is proved: if $$X,A$$ and $$B$$ are closed balls, then $$\text{card (Fix}(f^m)) <{2^m-1\over 2}$$. Examples are given which demonstrate connections between fundamental groups and closed geodesics. According to the Note of the Editors this paper was written and circulated as a SUNY preprint in 1976, and has been reproduced here without change, only some appendices with new references, written in May 2000, are added.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)