  Jonsson and Reschke showed that birational selfmaps on projective surface
defined over a number field satisfy the energy condition of Bedford and Diller
so their ergodic properties are very well understood. Under suitable hypotheses
on the indeterminacy loci, we extend that result to birational maps
$\mathbb{P}^k\dashrightarrow\mathbb{P}^k$, $k\geq2$, defined over a number
field, showing that they satisfy a similar energy condition introduced by De
Th\'elin and the second author. As a consequence, we can construct for such
maps their Green measure and deduce several important ergodic consequences.
  Under a mild additional hypothesis, we show that generic sequences of Galois
invariant subset of periodic points equidistribute toward the Green measure.
