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 .
The scalar product dual to the scalar product in the space of the scale factors is
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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