Complex Dynamics of birational maps of $\mathbb{P}^k$ defined over a number field

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.
Comment: 16 pages, commets welcome!