  We present an extension of the ergodic, mixing and Bernoulli levels of the
ergodic hierarchy in dynamical systems, the information geometric ergodic
hierarchy (IGEH), making use of statistical models on curved manifolds in the
context of information geometry. We discuss the $2\times 2$ Gaussian Orthogonal
Ensembles (GOE) within a $2D$ correlated model. For $r$ vanishingly small, we
find that GOE belong to the information geometric (IG) mixing level having a
maximum negative value of scalar curvature. Moreover, we propose a measure of
distinguishability for the family of distributions of the $2D$ correlated model
that results to be an upper bound of the IG correlation.
