A non-singular, invertable, ergodic transformation is the quadriple $(X,\mathcal B, \mu, T)$ where $(X,\mathcal B, \mu)$ is a measure space and $T$ is an invertable, measurable automorphism where $\mu$ and $\mu\circ T$ equivalent measures.


Two such systems $(X,\mathcal B, \mu, T)$ and $(Y,\mathcal C, \nu, S)$ are isomorphic when there exists isomorphism $\phi: X \mapsto Y$ where
$$ S\phi x = \phi T x$$


It seems it should be obvious that $(X,\mathcal B, \mu, T)$ and $(X,\mathcal B, \mu, T^{-1})$ are isomorphic. If I assume $X$ is a product space and $T$ is the $+1$ odometer action then I can prove it myself. But I would rather quote someone and be done with it.


Do you know of a source to quote for this?



Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse?
You cannot obtain an example from odometers because they have discrete point spectrum. 


It is not true in the topological setting- look at example 7.4.19 from <http://books.google.ca/books/about/An_Introduction_to_Symbolic_Dynamics_and.html?hl=zh-CN&id=qSkNs3jr-DIC&redir_esc=y>


or in the measure setting-look at
 <http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pja/1195571227&page=record> 


