Strict inequalities for the entropy of transitive piecewise monotone maps

Let T : [0, 1] ? [0, 1] be a piecewise differentiable piecewise monotone map, and let r > 1. It is well known that if |T'| = r (respectively |T'| = r) then htop(T) = log r (respectively htop(T) = log r). We show that if additionally |T'| <r (respectively |T'| > r) on some subinterval and T is topologically transitive then the inequalities for the entropy are strict. We also give examples that the assumption of piecewise monotonicity is essential, even if T is continuous. In one class of examples the dynamical dimension of the whole interval can be made arbitrarily small.