  We study phases of five-dimensional three-charge black holes with a circle in
their transverse space. In particular, when the black hole is localized on the
circle we compute the corrections to the metric and corresponding
thermodynamics in the limit of small mass. When taking the near-extremal limit,
this gives the corrections to the constant entropy of the extremal three-charge
black hole as a function of the energy above extremality. For the partial
extremal limit with two charges sent to infinity and one finite we show that
the first correction to the entropy is in agreement with the microscopic
entropy by taking into account that the number of branes shift as a consequence
of the interactions across the transverse circle. Beyond these analytical
results, we also numerically obtain the entire phase of non- and near-extremal
three- and two-charge black holes localized on a circle. More generally, we
find in this paper a rich phase structure, including a new phase of
three-charge black holes that are non-uniformly distributed on the circle. All
these three-charge black hole phases are found via a map that relates them to
the phases of five-dimensional neutral Kaluza-Klein black holes.
