Exact enumeration of the Critical States in the Oslo Model

We determine analytically the number $N_{\mathcal{R}}(L)$ of recurrent states in the 1d Oslo model as a function of system size L. The solution $N_{\mathcal{R}}(L) = \frac{1+\sqrt{5}}{2\sqrt{5}}(\frac{3+\sqrt{5}}{2})^L + \frac{\sqrt{5}-1}{2\sqrt{5}}(\frac{3-\sqrt{5}}{2})^L$ is in exact agreement with the number enumerated in computer simulations for $L = 1 - 10$. For $L \gg 1$, the number of allowed metastable states in the attractor increases exponentially as $N_{\mathcal{R}}(L) \approx c_+ {\lambda}_+^L$, where $\lambda_+ = \frac{3+\sqrt{5}}{2}$ is the golden mean. The system is non-ergodic in the sense that the states in the attractor are not equally probable.