In this talk, we study discrete dynamical systems under reversible mappings. In particular, we examine the orbit statistics when reduced to a finite space. We will briefly talk about previous results regarding distribution and bounds of orbit lengths.
We will then look at the number of periodic orbits and see that there is a strong relationship between this and the number of algebraic integrals that the mapping has. By using this number, we develop a test to extract the number of integrals in reversible mappings.