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.


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.)