For non-invertible dynamical systems, we investigate how ‘non-invertible’ a system is and how the ‘non-invertibility’ contributes to the entropy from different viewpoints. For a continuous map on a compact metric space, we propose a notion of pointwise metric preimage entropy for invariant measures. For systems with uniform separation of preimages, we establish a variational principle between this version of pointwise metric preimage entropy and pointwise topological entropies introduced by Hurley [On topological entropy of maps. Ergod. Th. & Dynam. Sys.15 (1995), 557–568], which answers a question considered by Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113]. Under the same condition, the notion coincides with folding entropy introduced by Ruelle [Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys.85(1–2) (1996), 1–23]. For a $C^{1}$-partially hyperbolic (non-invertible and non-degenerate) endomorphism on a closed manifold, we introduce notions of stable topological and metric entropies, and establish a variational principle relating them. For $C^{2}$ systems, the stable metric entropy is expressed in terms of folding entropy (namely, pointwise metric preimage entropy) and negative Lyapunov exponents. Preimage entropy could be regarded as a special type of stable entropy when each stable manifold consists of a single point. Moreover, we also consider the upper semi-continuity for both of pointwise metric preimage entropy and stable entropy and give a version of the Shannon–McMillan–Breiman theorem for them.

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关于原像熵、折叠熵和稳定熵

对于不可逆动力系统，我们从不同的角度研究系统的“不可逆性”以及“不可逆性”如何影响熵。对于紧凑度量空间上的连续映射，我们提出了用于不变度量的逐点度量原像熵的概念。对于原像均匀分离的系统，我们在此版本的逐点度量原像熵和 Hurley 引入的逐点拓扑熵之间建立了变分原理 [关于地图的拓扑熵。埃尔戈德。钍。&动态。系统。15(1995), 557–568]，它回答了 Cheng 和 Newhouse 考虑的一个问题 [原像熵。埃尔戈德。钍。&动态。系统。25(2005), 1091–1113]。在同样的条件下，这个概念与 Ruelle 提出的折叠熵相吻合[Positivity of entropy production in nonequilibrium 统计力学。J.统计。物理。85(1-2) (1996), 1-23]。为一个$C^{1}$- 闭流形上的部分双曲（不可逆和非退化）自同态，我们引入了稳定拓扑和度量熵的概念，并建立了与它们相关的变分原理。为了$C^{2}$系统中，稳定的度量熵用折叠熵（即逐点度量原像熵）和负李雅普诺夫指数来表示。当每个稳定流形由一个点组成时，原像熵可以被视为一种特殊类型的稳定熵。此外，我们还考虑了逐点度量原像熵和稳定熵的上半连续性，并为它们给出了香农-麦克米兰-布雷曼定理的一个版本。