This paper investigates the conditional entropy, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h _{\mu }(T\mid \langle G \rangle )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>G</mi> <mo stretchy="false">⟩</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which is induced by the T-invariant finite partition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\langle G \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>G</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>. Several fundamental propositions are obtained and a corresponding variational principle is proven as follows <Equation ID="Equ1"> <EquationSource Format="TEX">$\begin{aligned} \sup _{\mu \in \mathcal {M}(X,T)}h_{\mu }(T\mid \langle G\rangle )= h_{top}(T\mid \langle G \rangle ), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo movablelimits="true">sup</mo> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </munder> <msub> <mi>h</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>G</mi> <mo stretchy="false">⟩</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mrow> <mi mathvariant="italic">top</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>G</mi> <mo stretchy="false">⟩</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {M}(X,T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the collection of all invariant measures of X, which is an extension of the usual variational principle. Moreover, this study also consists of its presentation of the commutativity proposition for those conjugate invariants.

On conditional entropy of the T-invariant finite partition

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发表日期：

2026年4月13日

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Wen-Chiao Cheng

摘要

This paper investigates the conditional entropy, $h _{\mu }(T\mid \langle G \rangle )$ $h_{μ} (T ∣ ⟨ G ⟩)$ , which is induced by the T-invariant finite partition $\langle G \rangle $ $⟨ G ⟩$ . Several fundamental propositions are obtained and a corresponding variational principle is proven as follows $\begin{aligned} \sup _{\mu \in \mathcal {M}(X,T)}h_{\mu }(T\mid \langle G\rangle )= h_{top}(T\mid \langle G \rangle ), \end{aligned}$ $\begin{matrix} sup_{μ \in M (X, T)} h_{μ} (T ∣ ⟨ G ⟩) = h_{top} (T ∣ ⟨ G ⟩), \end{matrix}$ where $\mathcal {M}(X,T)$ $M (X, T)$ is the collection of all invariant measures of X, which is an extension of the usual variational principle. Moreover, this study also consists of its presentation of the commutativity proposition for those conjugate invariants.