鄒 成

(四川化工職業(yè)技術(shù)學(xué)院，四川 瀘州 646005)

關(guān)于熵的可交換性

鄒 成

(四川化工職業(yè)技術(shù)學(xué)院，四川 瀘州 646005)

不管在測度空間還是拓?fù)淇臻g上，兩個連續(xù)映射復(fù)合后，其熵與復(fù)合的先后次序有關(guān)，但滿足一定條件后，有些復(fù)合的順序是可以交換的，即交換秩序后的熵保持不變.詳細(xì)回顧了一些關(guān)于熵的定義，討論了兩個映射復(fù)合后其測度熵、測度序列熵、拓?fù)潇亍⑼負(fù)湫蛄徐?、二維映射的拓?fù)潇?、旋轉(zhuǎn)熵及拓?fù)鋲旱目山粨Q性.

測度熵；拓?fù)潇?；序列熵；旋轉(zhuǎn)熵；拓?fù)鋲海豢山粨Q性

1 預(yù)備知識

1967年，Kusnhirenko A在文獻(xiàn)[3]中給出了測度序列熵的定義.

1974年，Goodman TNT在文獻(xiàn)[4]中給出了拓?fù)湫蛄徐氐亩x.

引理1[5]若A是X的一個有限的開覆蓋，f:X→X是連續(xù)映射,則N(A)=N(f-1(A))(此處N(A)為從A選取的有限覆蓋的最小基數(shù)).

證明 見文獻(xiàn)[5].

證明 對于μ∈M(X)和E∈β(X)，有

命題2 若記

由命題1和命題2，很容易得到：

命題4 把命題2中的M(X,f°g)換成遍歷可測集E(X,f°g)，有同樣的結(jié)果.

2 幾類熵的可交換性

2.1 測度熵的可交換性

推論1 若μ∈Μ(X,f)∩Μ(X,g)，則有hμ(f°g)=hμ(g°f).

若記Mmax(X,f)為連續(xù)映射f:X→X最大熵的可測集，則有：

由命題2和定理1得到定理2.

2.2 拓?fù)潇氐慕粨Q性

定理3[1]若f,g:X→X是連續(xù)映射，則有h(f°g)=h(g°f).

余下的證明與定理3類似.

2.3 條件拓?fù)潇氐慕粨Q性

定理4 若f,g:X→X是同構(gòu)，則有h*(f°g)=h*(g°f).

2.4 二維映射熵的可交換性

記F:X×X→X×X(X是緊的)，對?(x,y)∈X×X，有F(x,y)=(f(x),g(y)),當(dāng)X=[0,1]時，它有一個Sharkovskii型的序[7].

命題6 若F:X×X→X×X，則有h(F)=h(f°g)=h(g°f).

證明 考慮F2(x,y)=(f°g(x),g°f(y)).

由f:Xi→Xi,i=1,2，則由h(f1×f2)=h(f1)×h(f2)[8]和命題5知，h(F2)=2h(f°g)+2h(g°f)，眾所周知,(X,f)是緊系統(tǒng)時,拓?fù)潇貪M足性質(zhì)ent(fm)=m·ent(f)[9],知h(F)=h(f°g)=h(g°f),再由h(F2)=2h(F)即得結(jié)論.

2.5 旋轉(zhuǎn)熵的可交換性

1999年， GELLER W和MISIUREWICZ M在文獻(xiàn)[10]中定義了旋轉(zhuǎn)熵.

υ∈Rd，此時的旋轉(zhuǎn)熵定義為hυ(f,φ)=sup{hμ(f):μ在υ方向遍歷}.

證明 令E=suppμ，于是E=suppμ閉，且μ(E)=1，f-1(E)=β(X)，則gμ(f-1(E))=μ(g-1(f-1E)))=μ(g-1(f-1(E)))=μ(f°g)-1(E))=μ(E)=1.

證明 由定理1和引理3即得.

2.6 拓?fù)鋲旱目山粨Q性

命題7 若f,g:X→X，φ:X→R連續(xù)，則P(f°g,φ)=P(g°f,φ°f).

證明

推論3 如果M(X,g°f)=Μ(X,f)，則P(f°g,φ)=P(g°f,φ).

相反方類似證明.

易知Mφ(X,f°g)=Φ當(dāng)且僅當(dāng)Mφ°f(X,g°f)=Φ.

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責(zé)任編輯：李翠薇

On the Commutativity of Entropy

ZOU Cheng

(Sichuan College of Chemical Technology, Sichuan Luzhou 646005, China)

Regardless of a measurement space or a topological space, after two consecutive mappings composite, their entropy is correlated to the composite order, however, when some conditions are met, some of composite orders can be exchanged but the entropy keeps constant after the orders are exchanged. This paper reviews in detail the definitions related to the entropy, and discusses the commutativity of the metric entropy after the composite of two mappings, metric sequence entropy, topological entropy, topological sequence entropy, topological entropy of two-dimensional mapping, rotational entropy and topological pressure.

metric entropy; topological entropy; sequence entropy; rotational entropy; topological pressure; commutativity

10.16055/j.issn.1672-058X.2017.0001.010

2016-01-08；

2016-05-25.

鄒成(1974-)，男，四川宜賓人，副教授，碩士，從事拓?fù)鋵W(xué)及數(shù)學(xué)教育的研究.

O189

A

1672-058X(2017)01-0048-04