Ergodic theory of accessible partially hyperbolic endomorphisms

In 1995 at the International Conference on Dynamical Systems a conjecture was posed. That among invertible partially hyperbolic systems ergodicity was the general case. But not all dynamical systems are invertible, a system is non-invertible when different past states lead to the same future state. Non-invertible partially hyperbolic systems are the object of our study. To obtain our main result we first had to carefully study the current theory of invertible partially hyperbolic systems. By adapting these techniques to the non-invertible setting, we show what are sufficient conditions for ergodicity.