  The folding entropy is a quantity originally proposed by Ruelle in 1996
during the study of entropy production in the non-equilibrium statistical
mechanics. As derived through a limiting process to the non-equilibrium steady
state, the continuity of entropy production plays a key role in its physical
interpretations. In this paper, we study the continuity of folding entropy for
a general (non-invertible) differentiable dynamical system with degeneracy. By
introducing a notion called degenerate rate, we prove that on any subset of
measures with uniform degenerate rate, the folding entropy, and hence the
entropy production, is upper semi-continuous. This extends the upper
semi-continuity result from endomorphisms to all $C^r(r>1)$ maps.
  We further apply in the one-dimensional setting. In achieving this, an
equality between the folding entropy and (Kolmogorov-Sinai) metric entropy, as
well as a general dimension formula are established. These admit their own
interests. The upper semi-continuity of metric entropy and dimension are then
valid when measures with uniform degenerate rate are considered. Moreover, the
sharpness of uniform degenerate rate is also investigated by examples in the
scope of positive metric (or folding) entropy.

Dynamical Systems