Note on time reversible chaos in mixmaster dynamics

The Belinskii-Khalatnikov-Lifshitz (BKL) parametrization of the diagonal spatially homogeneous Bianchi IX vacuum (mixmaster) cosmology can be expressed as a discrete realization of a chaotic map. A well-known invertible two-dimensional version of their parametrization embodies the time reversibility of the model's Einstein equations and can be written (in both an epoch-era and era-era version) to emphasize the fact that the function that maps the first parameter is the inverse of that which maps the second. It is shown here that this can be understood in terms of the self-similar equilateral triangle minisuperspace equipotentials that describe the approximate dynamics in the anisotropy plane of minisuperspace. A new parameter is identified which has the same value in all the epochs of a given era. The BKL parameters of the two-dimensional map are different ratios of triangle scales to the new parameter. It is shown how one parameter controls the change of era (and thus the chaos) in the collapse direction while the other plays the same role for expansion.