Separating Solution of a Quadratic Recurrent Equation

and set Λ1 = y > 0. We shall occasionally write Λp(y) to emphasize the dependence of Λp on the initial value y. It is clear that Λp(cy) = c Λp(y). Therefore if Λp(y) → ∞ as p → ∞ and c > 1, then Λp(y ) → ∞ as p → ∞ where y = cy. On the other hand if Λp(y) → 0 and 0 < c < 1, then Λp(y ) → 0. Thus there exist y and y such that Λp(y) → ∞ for y ∈ (y,∞) with y as small as possible and Λp(y) → 0 for y ∈ (0, y ) with y as large as possible. It is a natural question whether y = y = y and whether Λp(y ) → const Mathematics Department, Princeton University and Landau Institute of Theoretical Physics, Russian Academy of Sciences Mathematics Department, Princeton University