Three remarks on geodesic dynamics and fundamental group.

*(English)*Zbl 1002.53028For 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.

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.

Reviewer: Ülo Lumiste (Tartu)

##### 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.) |