If N denotes a Poisson process, a splitting of N is formed by two point processes N_1 and N_2 such that N=N_1+N_2.
If N_1 and N_2 are independent Poisson processes then the splitting is said to be Poisson and such a splitting is always available (We allow the possibility to enlarge the ambient probability space).
In general, a splitting is not Poisson but the situation changes if we require that the distributions of the point processes involved are left invariant by a common underlying map that acts at the level of each point of the processes.
We will prove that if this map has infinite ergodic index, then a splitting is necessarily Poisson if the environment is ergodic.
This is a work in progress, with Elise Janvresse and Thierry de la Rue.