To understand the low-energy structure of the neutron deficient iodine isotopes, lifetimes for the low-lying 9/2+ and 11/2+ positive-parity states in 113I have been measured as τ = 28(4) ps and τ = 3.7(7) ps, respectively. The lifetime for the 11/2− state, which feeds the 9/2+ and 11/2+ states, was remeasured with improved accuracy as τ = 216(7) ps. The reduced transition probability, B(E2) = 32(5) W.u., for the 9/2+ → 5/2+ transition agrees with that calculated within the shell model using a Hamiltonian based on the charge-dependent Bonn nucleon-nucleon interaction. In contrast, the much larger transition probability, B(E2) = 209(39) W.u., measured for the 11/2+ → 7/2+ transition has been interpreted, with the aid of configuration-constrained total Routhian surface calculations, as resulting from a slightly γ -soft rotor with an associated quadrupole deformation of β2 ≈ 0.18. Remarkably similar reduced E1 transition probabilities of 5.5(5) × 10−4 and 4.9(5) × 10−4 W.u. were deduced for the 11/2− → 9/2+ and 11/2− → 11/2+ transitions, respectively, which feed apparently dissimilar but competing structures.