On Logarithm Laws for Dynamical Systems

We generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a measure preserving system (X,μ, g) and a family of subsets {An | n ∈ N} of X under which #{1 ≤ n ≤ N | gnx ∈ An} is for μa.e. x ∈ X asymptotically (as N →∞) equal to PN n=1 μ(An). These conditions can be verified for various special cases of systems (X,μ, g, {An}), including actions of nonquasiunipotent elements g of semisimple Lie groups G on homogeneous spaces X = G/Γ, with {An} being certain subsets of X approaching infinity as n→∞.