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On the almost sure spiraling of geodesics in negatively curved manifolds. (English) Zbl 1229.53050

Let \(C\) be a closed geodesic in a compact Riemannian manifold \(M\) with negative sectional curvature. Because the geodesic flow on \(M\) is ergodic (with respect to the Bowen-Margulis measure \(\mu\)), every geodesic will become close to \(C\) and stay within an \(\epsilon\)-tubular neighborhood \(N_\epsilon C\) of \(C\) for a long time. The main results of this paper quantify this behavior, and establish analogous results in very general contexts. In particular, the authors obtain results when \(C\) is instead a totally geodesic submanifold, and when \(M\) is instead a CAT\((-1)\) geodesic metric space, e.g., a hyperbolic building, or a locally finite tree.
Restricting attention to compact, connected Riemannian manifolds \(M\), let \(h\) be the topological entropy of the geodesic flow on \(T^1 M\). Define the penetration \(p(v,t)\) to measure how long the geodesic in direction \(v \in T^1 M\) remains inside \(N_\epsilon C\) around time \(t\): \(p(v,t)\) is the duration of the time-interval containing \(t\) when the geodesic is inside the neighborhood, and \(p(v,t)=0\) if it is outside at \(t\). A corollary of their result on closed geodesics is that, for \(\mu\)-almost every \(v\), \({\lim \sup}_{t \rightarrow \infty} \frac{p(v,t)}{\log t} = \frac{1}{h}\). This result was known when \(M\) has constant curvature.
When the role of \(C\) is replaced by a cycle in a graph based on the automorphisms of a locally finite tree, they obtain an explicit expression lower-bounding the number of spiraling turns almost every path executes around \(C\). They are also able to apply their results to achieve bounds for Diophantine approximation of elements of a non-Archimedean local field \(\hat{K}\) by quadratic irrational elements: almost every element of \(\hat{K}\) is poorly approximated by such elements.
A novel technical tool they use is a new “distance-like” function \(d_C\) for every convex subset \(C\) of a CAT\((-1)\) space \(X\), preserved by every isometry of \(X\) that also preserves \(C\). When \(C\) is a point, \(d_C\) reduces to the Gromov visual distance; when \(C\) is a horoball, \(d_C\) reduces to Hamenstädt’s distance.

MSC:

53C22 Geodesics in global differential geometry
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
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