Continuity properties of folding entropy

The folding entropy is a quantity originally proposed by Ruelle in 1996 during the study of entropy production in the non-equilibrium statistical mechanics [53]. 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, the continuity of folding entropy is studied for a general (non-invertible) differentiable dynamical system with degeneracy. By introducing a notion called degenerate rate, it is proved 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 in [53] from endomorphisms to all C^r (r > 1) maps.

We further apply our result 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. The upper semi-continuity of metric entropy and dimension are then valid when measures with uniform degenerate rate are considered. Moreover, the sharpness of the uniform degenerate rate condition is shown by examples of C^r interval maps with positive metric (and folding) entropy.