← 研踪 PaperTrailarXiv:2611.13904 · v1
Thermodynamic Formalism for Non-Invariant Probabilities: Entropy, Divergence, Channels
首次提交 2026-11-10 19:00
版本更新 2026-11-10 19:00
雷达观测 2026-11-11 11:04
摘要
Let $\mathcal{M}$ and $\mathcal{M}_σ$ denote, respectively, the sets of Borel and $σ$-invariant probabilities on the uncountable set $Ω=\{1,2,...,d\}^\mathbb{N}$. Given a Hölder Jacobian $J$, the dual Ruelle operator $μ\in \mathcal{M} \mapsto \mathcal{L}_{J}^* (μ)$ will play the role of a dynamical channel; generalizes the action of a stochastic matrix on probability vectors. Since $\mathcal{L}_{J}^*$ generally breaks $σ$-invariance, we introduce the \emph{ni-entropy} $h_{ni}(μ)$ for non-invariant measures, employing the concept of the expected value function on cylinders of size one. $h_{ni}(μ)$ differs from Shannon entropy but extends the Kolmogorov-Sinai entropy from $\mathcal{M}_σ$ to $\mathcal{M}$; $h_{ni}$ is concave and u.s.c. This leads us to the concept of ni-pressure $\mathcal{P}(A)$ of a potential $A:Ω\to \mathbb{R}$. $μ\in \mathcal{M}$ is not a density with respect to a dominating measure. We define an {\it ni-Kullback-Leibler divergence} $D_{ni}(ρ|| ν)$ not assuming absolute continuity. We investigate concepts of Information Theory such as Minimal Output Entropy, Donsker-Varadhan variational principle, a variational formula for $h_{ni}$, bounds for divergence, information projections, and their connections to the ni-pressure $\mathcal{P}$.
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v1 · 2026-11-10 19:00 · 最新观测版本
Thermodynamic Formalism for Non-Invariant Probabilities: Entropy, Divergence, Channels
Let $\mathcal{M}$ and $\mathcal{M}_σ$ denote, respectively, the sets of Borel and $σ$-invariant probabilities on the uncountable set $Ω=\{1,2,...,d\}^\mathbb{N}$. Given a Hölder Jacobian $J$, the dual Ruelle operator $μ\in \mathcal{M} \mapsto \mathcal{L}_{J}^* (μ)$ will play the role of a dynamical channel; generalizes the action of a stochastic matrix on probability vectors. Since $\mathcal{L}_{J}^*$ generally breaks $σ$-invariance, we introduce the \emph{ni-entropy} $h_{ni}(μ)$ for non-invariant measures, employing the concept of the expected value function on cylinders of size one. $h_{ni}(μ)$ differs from Shannon entropy but extends the Kolmogorov-Sinai entropy from $\mathcal{M}_σ$ to $\mathcal{M}$; $h_{ni}$ is concave and u.s.c. This leads us to the concept of ni-pressure $\mathcal{P}(A)$ of a potential $A:Ω\to \mathbb{R}$. $μ\in \mathcal{M}$ is not a density with respect to a dominating measure. We define an {\it ni-Kullback-Leibler divergence} $D_{ni}(ρ|| ν)$ not assuming absolute continuity. We investigate concepts of Information Theory such as Minimal Output Entropy, Donsker-Varadhan variational principle, a variational formula for $h_{ni}$, bounds for divergence, information projections, and their connections to the ni-pressure $\mathcal{P}$.
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