  How can we interpret the infimum of Lipschitz constants in a conjugacy class
of interval maps? For positive entropy maps, the exponential of the topological
entropy gives a well-known lower bound. We show that for piecewise monotone
interval maps as well as for $C^{\infty}$ interval maps, these two quantities
are equal, but for countably piecewise monotone maps, the inequality can be
strict. Moreover, in the topologically mixing and Markov case, we characterize
the infimum of Lipschitz constants as the exponential of the Salama entropy of
a certain reverse Markov chain associated with the map. Dynamically, this
number represents the exponential growth rate of the number of iterated
preimages of nearly any point.
