Take the 2-minute tour ×  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?
share|improve this question @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. –  Tom LaGatta
Feb 7 '13 at 4:15 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 r