10.3 Interpretation of the asymptotic velocity as a rate of convergence to the boundary in hyperbolic space

Similarly to the discussion carried out in the polarized case, the asymptotic velocity v ∞ can be interpreted as the rate at which the solution goes to the boundary of hyperbolic space; see Equation (36View Equation). Given a solution (Q, P ), we shall let ρ(τ,𝜃) denote the hyperbolic distance from a reference point to [Q (τ, 𝜃),P(τ,𝜃)] (the particular choice of reference point is not of importance). Then
lim ρ-(τ,-𝜃0) = v (𝜃 ); τ→∞ τ ∞ 0

see [79Jump To The Next Citation Point, Theorem 1.2, p. 982].


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