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Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry.

Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry on a compact metric space (equivalently it has discrete spectrum)?

I understand Krieger representation theorem states that ergodic MPT are measure theoretically isomorphic to minimal systems on a compact metric spaces. I would like to know if structure like isometry can be conserved.

The general question I am interested in is: do ergodic isometries have discrete spectrum?

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  • $\begingroup$ @FelipeG, can you add a few words on the equivalence between minimal isometries and discrete spectra? These sound interesting but I am not familiar with either concept in the context of general metric spaces. $\endgroup$
    – Tom LaGatta
    Feb 7, 2013 at 4:15
  • $\begingroup$ Sure, a theorem by Halmos and Von Neumann state that T is transitive (one dense orbit) and isometric, iff it is topologically isomoprhic to a minimal rotation on a compact abelian group iff T is minimal and has discrete topological spectrum. (topological because in this case the operator associated to the dynamical system is acting on the space of continuous functions. ) A good reference for this is An introduction to ergodic theory by Peter Walters. $\endgroup$
    – FelipeG
    Feb 7, 2013 at 8:41

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