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Well, if it is connected to iTunes when it shuts off then it could possibly be iTunes. You could also try restoring your device to factory settings and see if it does it again. Be sure to update your iTunes!
You turn it back on
Try connecting it to your computer and restoring it with itunes
In a mueseum
Replace it +++++++++++++++ Try to look for a bloated capacitor in the main panel. Just replace it with the same one with same label on it.
anti theft going out
after a couple weeks/days the ipod finally shuts down and restores which means basically everything is restarted on that device
Replace the ignition module and see if that don't fix your problem.
you need to replace your idle jets, take it to a mechanic
either your hard drive, logic board or cpus are going bad
Duke's. It shuts down and he starts going to McGinty's (sp?)
Your ignition switch is worn out. you will need to replace it.