This is a talk in dynamical systems, and especially the dynamics of
maps of the unit interval f : [0,1] -> [0,1]. Classical interval
dynamics theory says that a transitive, piecewise-monotone map is
conjugate to a map with constant absolute value of slope, where the
logarithm of the slope is the topological entropy of the system. We
show that the story is vastly different for countably
piecewise-monotone maps (think of a function with infinitely many
local extreme points). For example, we produce such a map with no
conjugate map of constant slope on the interval [0,1], but with
conjugate maps of constant slope on the "interval" [0,\infty] for
every value of slope whose logarithm is greater than or equal to
the topological entropy.