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We summarize our recent work on spin motion in storage rings *. In fact we return to our study ** of spin motion in storage rings. We again focus on spin tunes, polarization fields etc. but in contrast to ** we base the description on one turn maps and refine and expand our toolset from that in * by using a rather modern method from Dynamical-Systems theory, developed in the 1980's by R. Zimmer and others based on bundles **, ***. With this we obtain new insights into invariant spin fields,doi:10.18429/jacow-ipac2014-thpro061 fatcat:ilr7dsnvrrh7lpu2bnw22ywi4q