Entropy for group endomorphisms and homogeneous spaces
Entropy for group endomorphisms and homogeneous spaces
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DOI：
10.1090/s0002-9947-1971-0274707-x
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发表时间：
1971
影响因子：
1.3
通讯作者：
R. Bowen
中科院分区：
文献类型：
--
作者：
R. Bowen
定义了度量空间上一致连续映射的拓扑熵há（T）.证明了这种熵的一般性陈述，并计算了李群和某些齐性空间的仿射映射的熵。我们比较了hd（T）与测度论熵h（T），特别是对于紧度量化群上的Haar测度和仿射映射T，h（T）= hd（T）.一个特殊的情况下，这产生了众所周知的公式h（T）时，T是一个toral自同构。导论.我们将研究拓扑熵，集中在它的关系，测量理论熵和代数的例子。本文对度量空间（X，d）上的一致连续映射F定义了拓扑熵hd（T）（见§2）。在[1]中，定义了紧拓扑空间上连续映射的拓扑熵h（T），若空间是紧度量，则h（T）-hd（T）.本文的一个重要部分是计算非紧空间上某些映射的hd（T）。设p是p（X）= l上的Borel测度，且p是F-不变的（即对每个Borel集A，p（T~x（A））=p（A））.然后可以如下定义测度理论熵hu（T）：Ar}是A”的（有限）可测划分，如果A…是覆盖X的X的不相交可测子集.现在设hu（T，a）= 2 - 4 mc（T-kAik）iogp（mn t-*a）则极限hu（T，a）= limm_00（1/m）hu（T，a）存在并且定义hu（T）= sup {hu（T，a）：a是X的有限可测划分}。(See[6]关于测量理论熵的详细信息。X中的两个点由a = {A±，.，假设它们位于不同的^i中。我们将使用以下事实来计算熵：事实（见[6]）。设{ak}k = 0是X的可测划分序列，满足以下性质：如果x，ye X是不同的，则存在一个n（x，y），使得当k S：n（x，y）时，ak分离x和y。则hu（f）= supfc h（T，ak）。众所周知，如果T：G -> G是紧致可度量化群的满射自同态，则F保持Haar测度p。对于这样的F，我们证明了编辑1969年10月17日和修订后的形式，1970年2月2日收到的。AMS 1969主题分类。小学2870，2875;中学5482。
Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism. Introduction. We shall study topological entropy, concentrating on its relation to measure theoretic entropy and algebraic examples. Our topological entropy hd(T) is defined (in §2) for a uniformly continuous map F on a metric space (X, d). In [1] a topological entropy h(T) was defined for a continuous map on a compact topological space; if the space is compact metric then h(T)—hd(T). An essential part of this paper is the computation of hd(T) for certain maps on noncompact spaces. Suppose p is a Borel measure on p(X) = l, and p is F-invariant (i.e. p(T~x(A)) =p(A) for every Borel set A). One can then define a measure theoretic entropy hu(T) as follows: Call a={Au ..., Ar} a (finite) measurable partition of A" if the A¡ are disjoint measurable subsets of X covering X. Now set Hja) = 2 -4mc\ T-kAik) iogp(mn t-*a\ Then the limit hß(T, a) = limm_00 (\/m)HJa) exists and one defines hu(T) = sup {hu(T, a) : a is a finite measurable partition of X}. (See [6] for details about measure theoretic entropy.) Two points in X are separated by a = {A±,..., Ar} provided they lie in different ^i's. We shall use the following fact to compute entropy : Fact (see [6]). Let {ak}k = 0 be a sequence of measurable partitions of X satisfying the following property: If x, ye X are distinct there is an n(x, y) such that ak separates x and y whenever k S: n(x, y). Then hu(f) = supfc h(T, ak). As is generally known, if T: G -> G is a surjective endomorphism of a compact metrizable group, then F preserves Haar measure p. For such a F we show that the Received by the editors October 17, 1969 and, in revised form, February 2, 1970. AMS 1969 subject classifications. Primary 2870, 2875; Secondary 5482.