Examples of natural extensions of nonsingular endomorphisms

Abstract

We extend a nonsingular noninvertible map T T on ( X , B , μ ) (X,\mathcal {B},\mu ) to a minimal nonsingular automorphism containing T T as a factor and preserving some measure theoretic properties of T T . We call the automorphism the natural extension of T T . We compute some examples of natural extensions: first when T T is a shift with μ \mu a product measure, and then generalize the result to some nonexact examples as well.