A measure-preserving formalism is applied to topological spin/band models and yields observations about pumping. It occurs at topological phase transition (TPT), i.e., a $Z_2$-flip, suggesting that $Z_2$ can imply bulk effects. The models asymptotic behavior is analytically solved via ergodicity. The pumping probability is geometric, fractional, and has a ceiling of $frac{1}{2}$. Intriguingly, theorems are proved about dephasing associated with this pumping, which is linked to the systems dimension and the distinction between rational and irrational numbers. Experimental detection is discussed.
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