We have written here only the (potentially) relevant walls. There are other walls present in the
potential, but because these are behind the relevant walls, which are infinitely steep in the
BKL-limit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a
complete treatment where the BKL-limit is the zeroth order term in a gradient expansion yet to be
understood [47].
The scalar product dual to the scalar product in the space of the scale factors is
for two linear forms These recipes are all that we shall need for investigating the regularity properties of the billiards
associated with the class of actions Equation (2.1).
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