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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?

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    $\begingroup$ It isn't true in general, even when the measure is invariant; however, I don't know a reference, except that the construction is bound to be difficult. There even exist AT (approximately transitive) examples (again with invariant measures)—I found one myself, but it is too complicated to give here. $\endgroup$ May 3 '14 at 13:33
  • $\begingroup$ Why is it obvious that $T$ and $T^{-1}$ should be isomorphic? $\endgroup$
    – Ian Morris
    May 6 '14 at 15:07
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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

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  • $\begingroup$ Over the next few years these links might become broken but people may still wish to read this answer, so I suggest that you might write the names and publication data of these articles in your answer as well as providing links to them. $\endgroup$
    – Ian Morris
    May 6 '14 at 15:14
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    $\begingroup$ Sure. The first reference is the book titled "AN introduction to Symbolic dynamics" by Brian Marcus and Doug Lind. The second reference is a paper titled "On an example of a measure preserving transformation which is not conjugate to its inverse" by Hirotada Anzai published in 1951. $\endgroup$ May 8 '14 at 7:41
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Page 108 of Peter Walters' classic textbook An Introduction to Ergodic Theory suggests the reference "An uncountable family of $K$-automorphisms" by Donald Ornstein and Paul Shields [Advances in Mathematics 10 63--88 (1973)] for the fact that there exists an invertible measure-preserving tranformation with the Kolmogorov property which is not isomorphic to its inverse.

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