### 2.7 Chaos and billiard volume

With our rules for writing down the billiard region, one can determine in which case the volume of the
billiard is finite and in which case it is infinite. The finite-volume, chaotic case is also called “mixmaster
case”, a terminology introduced in four dimensions in [137].
The following results have been obtained:

- Pure gravity in dimensions is chaotic, but ceases to be so for [63, 62].
- The introduction of a dilaton removes chaos [15, 3]. The gravitational four-derivative action
in four dimensions, based on , is dynamically equivalent to Einstein gravity coupled to a
dilaton [160]. Hence, chaos is removed also for this case.
- -form gauge fields () without scalar fields lead to a finite-volume
billiard [44].
- When both -forms and dilatons are included, the situation is more subtle as there is a
competition between two opposing effects. One can show that if the dilaton couplings are in a
“subcritical” open region that contains the origin – i.e., “not too big” – the billiard volume is
infinite and the system is non chaotic. If the dilaton couplings are outside of that region, the
billiard volume is finite and the system is chaotic [49].