Within the framework of dispersion theory, we analyze the dipion transitions between the lightest $Upsilon$ states, $Upsilon(nS) rightarrow Upsilon(mS) pipi$ with $m < n leq 3$. In particular, we consider the possible effects of two intermediate bottomoniumlike exotic states $Z_b(10610)$ and $Z_b(10650)$. The $pipi$ rescattering effects are taken into account in a model-independent way using dispersion theory. We confirm that matching the dispersive representation to the leading chiral amplitude alone cannot reproduce the peculiar two-peak $pipi$ mass spectrum of the decay $Upsilon(3S) rightarrow Upsilon(1S) pipi$. The existence of the bottomoniumlike $Z_b$ states can naturally explain this anomaly. We also point out the necessity of a proper extraction of the coupling strengths for the $Z_b$ states to $Upsilon(nS)pi$, which is only possible if a Flatte-like parametrization is used in the data analysis for the $Z_b$ states.
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