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I will briefly explain my question.
I want to use the wiggle expression on the circle shape layer 1 and I want the circle shape layer 2 to synchronize the position change with the circle shape layer 1. That way both circular layers would float synchronously, but in different positions.
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Wiggle layer 1, then make Layer 2 the child of layer 1. Use the dropdown menu or the pickwhip in the Parent/Link column of the timeline. (Shift + F4 shows and hides the column)
Or, put both circles on the same shape layer.
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In addition to Rick's excellent suggestions, if you want circle shape layer 2 to keep its own keyframed/expression motion but also inherit the motion of circle shape layer 1, you could use a position expression like this:
p = thisComp.layer("circle shape layer 1").transform.position;
value + p - p.valueAtTime(0)