Study of dipion bottomonium transitions and search for the hb(1P) state

We study inclusive dipion decays using a sample of 108 x 10(6)Y(3S) events recorded with the BABAR detector. We search for the decay mode Y(3S) -> pi(+)pi(-) h(b)(1P) and find no evidence for the bottomonium spin-singlet state h(b)(1P) in the invariant mass distribution recoiling against the pi(+)pi(-) system. Assuming the h(b)(1P) mass to be 9.900 GeV/c(2), we measure the upper limit on the branching fraction B[Y(3S) -> pi(+)pi(-) h(b)(1P)] < 1.2 x 10(-4), at 90% confidence level. We also investigate the chi(bJ)(2P) -> pi(+)pi(-) chi(bJ)(1P), Y(3S) -> pi(+)pi(-) Y(2S), and Y(2S) -> pi(+)pi(-) Y(1) dipion transitions and present an improved measurement of the branching fraction of the Y(3S) -> pi(+)pi(-) Y(2S) decay and of the Y(3S) - Y(2S) mass difference.