  This paper explores the range of bounded speedups in the topological
category. Bounded speedups represent both a strengthening of topological
speedups as defined in [A 16] and a generalization of powers of a
transformation. Here we show that bounded speedups preserve the structure of
two classical minimal Cantor systems. Specifically, a minimal bounded speedup
of an odometer is a conjugate odometer, and a minimal bounded speedup of a
primitive substitution is again a primitive substitution, though it is never
conjugate to the original substitution system. Further, we give bounds on the
topological entropy of bounded speedups, and in special cases we compute the
topological entropy of bounded speedups.
