Semiconjugacy and factor map

Summary

A semiconjugacy from one dynamical system to another is a continuous surjection that intertwines the two maps. The target system is then called a factor of the source, and the source an extension of the target. The map itself is a factor map. A factor is a simplified picture of the system that may forget some information. Still, every orbit of the factor is the image of an orbit of the extension. A semiconjugacy that is a homeomorphism is a topological conjugacy; the same idea with measure-preserving maps gives factors in ergodic theory.