We give a classification of birational transformations on smooth projective surfaces which have a Zariski-dense set of noncritical periodic points. In particular, we show that if the first dynamical degree is greater than one, the union of all noncritical periodic orbits is Zariski-dense.
Xie Junyi. "Periodic points of birational transformations on projective surfaces." Duke Math. J. 164 (5) 903 - 932, 1 April 2015. https://doi.org/10.1215/00127094-2877402