Metric entropy and degeneracy of smooth interval maps

We consider $C^{r}(r>1)$ maps on an interval (or a circle). By introducing the notions of folding and degenerate rate, we investigate the mechanisms of upper semi-continuity of metric entropy. To be specific, we prove that on any subset of measures with uniform folding or degenerate rate, the metric entropy is upper semi-continuous. Moreover, the sharpness of the conditions on the uniformity of folding and degenerate rate are also investigated.