  We define the notion of canonical boundedness among rank-one transformations
and use it to characterize the class of all bounded rank-one transformations
with trivial centralizer. We also explicitly characterize totally ergodic
rank-one transformations with bounded cutting parameter. Together with a recent
result of Ryzhikov our results provide a simple procedure for determining
whether a bounded rank-one transformation has minimal self-joinings of all
orders purely in terms of the cutting and spacer parameters for the
transformation.
