On the almost sure spiraling of geodesics in CAT(0) spaces

We prove a logarithm law-type result for the spiraling of geodesics around certain types of compact subsets (e.g. quotients of periodic Morse flats) in quotients of rank one CAT(0) spaces.

💡 Research Summary

The paper studies the statistical behavior of geodesics in rank‑one CAT(0) spaces under the action of a discrete group Γ, focusing on how often and for how long a typical geodesic spends near a fixed compact convex subset C (for example, a quotient of a periodic Morse flat). The authors prove a logarithm law: for μ‑almost every boundary point ξ (with respect to the Patterson–Sullivan measure μ of Γ) and the associated unit‑speed geodesic ray γ_ξ, the maximal length p_{C,ε}(γ_ξ,t) of an interval containing time t that stays inside the ε‑neighbourhood of some Γ‑translate of C satisfies

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