# Continuity properties of folding entropy

## TL;DR

This paper investigates the continuity of folding entropy in non-invertible dynamical systems, introducing the degenerate rate concept to establish upper semi-continuity results and exploring their implications in one-dimensional cases.

## Contribution

It extends upper semi-continuity of folding entropy to all $C^r$ maps with a new degenerate rate concept and establishes key equalities and dimension formulas.

## Key findings

- Folding entropy equals metric entropy in certain settings.
- Upper semi-continuity holds for measures with uniform degenerate rate.
- Examples demonstrate the sharpness of the degenerate rate condition.

## Abstract

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.