2019/01/15 by Gang Liao, Liao, Gang, Shirou Wang +1
Mathematics · Physics and Astronomy · #37A60 (Primary) 37A35 #37C40 #82C05 (Secondary) #Advanced Mathematical Theories and Applications #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1901.04612
openalex publication_date 2019/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Cr(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.