Exact enumeration of the Critical States in the Oslo Model

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We determine analytically the number Nᵣ(L) of recurrent states in the 1d Oslo model as a function of system size L. The solution Nᵣ(L) = 1+√52√5(3+√52)ˡ + √5-12√5(3-√52)ˡ is in exact agreement with the number enumerated in computer simulations for L = 1 - 10. For L ≫ 1, the number of allowed metastable states in the attractor increases exponentially as Nᵣ(L) ≈ c+ λ+ˡ, where λ+ = 3+√52 is the golden mean. The system is non-ergodic in the sense that the states in the attractor are not equally probable.