Measure-preserving Dynamical System - Homomorphisms

Homomorphisms

The concept of a homomorphism and an isomorphism may be defined.

Consider two dynamical systems and . Then a mapping

is a homomorphism of dynamical systems if it satisfies the following three properties:

1. The map φ is measurable,
2. For each, one has ,
3. For μ-almost all x ∈ X, one has φ(Tx) = S(φ x).

The system is then called a factor of .

The map φ is an isomorphism of dynamical systems if, in addition, there exists another mapping

that is also a homomorphism, which satisfies

1. For μ-almost all x ∈ X, one has
2. For ν-almost all y ∈ Y, one has .