Asymptotic periodicity of densities and ergodic properties for nonsingular systems
Asymptotic periodicity of densities and ergodic properties for nonsingular systems
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非奇异系统密度的渐近周期性和遍历特性
DOI：
10.32917/hmj/1206128723
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发表时间：
1991
影响因子：
0.2
通讯作者：
Hiroshi Ishitani
中科院分区：
文献类型：
--
作者：
Tomoki Inoue;Hiroshi Ishitani
有限区间上的每个一维分段光滑扩张变换T具有如下遍历性（A），这是Li和Yorke [11]和瓦格纳[15]的结果（也见Morita [13]）。在一维情况下，m表示勒贝格测度。(A)存在m-绝对连续的T-不变概率测度序列{y>i，-">Hi}，其中Lt：= supp | i{ for i= I，-，/，它具有以下性质。(1)fii（Li）=lfori=l，.~，L. (2)（T，/Zj）是遍历的，其中i = 1，··-，/. (3)m（LfnL，.）= 0 i / i # j。(4)T-^ ^ L（m-a.e.对于i = I，，i，i，(5)如果rj是m-绝对连续的T-不变概率测度，则rj可以写成/xf的凸组合。(6)设C= Un°°=o{*; T”（x）*Uu^}>，则m（C）= 0。(7)对于i= l，· · ·，l，存在集合L ∈ 1，···，Lfr（f）的集合，具有以下性质：（a）Li = [j ^1 Lij. (b)m（Lij（Lik））= 0 (c)T-（LiJ+1）ID Ltj m-a.e.对于j = l，r（0 - 1，和T-1 - 1）=> LUr（1）m-a. e. (d)（T，|iy）是精确的，其中fitj = r（0，|我。.另一方面，Lasota，Li和Yorke [8]指出与T相关联的Frobenius-Perron算子P的行为是渐近周期的.即它具有以下性质（B）。(B)存在一个密度序列gi，…，gr和一个有界线性泛函序列X1，…，Xr，使得
Each one dimensional piecewise smooth expanding transformation T on a finite interval has the following ergodic property (A), which is the result of Li and Yorke [11] and Wagner [15], (see also Morita [13]). In one dimensional case m denotes the Lebesgue measure. (A) There exists a sequence of m-absolutely continuous T-invariant probability measures {y>i,-">Hi} with Lt:= supp\i{ for i= I,---,/, which has the following properties. (1) fii(Li)=lfori=l,.~,l. (2) (T, /Zj) is ergodic for i = 1,••-,/. (3) m(LfnL,.) = 0 i / i # j . (4) T-^ ^ L( m-a.e. for i = I , , / . (5) If rj is an m-absolutely continuous T-invariant probability measure, rj can be written as a convex combination of /xf's. (6) Put C= Un°°=o{*; T"(x)*UU^}> then m(C) = 0. (7) For i= l , • • • , / , there exists a collection of sets L£1,---,Lfr(f) with the following properties: (a)Li = [j^1Lij. (b) m(Lij(]Lik) = 0 ifj^k. (c) T-(LiJ+1) ID Ltj m-a.e. for j = l,--.,r(0 1, and T " 1 ^ ) => LUr{i) m — a.e. (d) (T, |iy) is exact, where fitj = r (0 '^ | L i . . On the other hand, Lasota, Li and Yorke [8] pointed out that the behavior of the Frobenius-Perron operator P associated with T is asymptotically periodic. Namely it has the following property (B). (B) There exists a sequence of densities gi,--,gr and a sequence of bounded linear functional X1,-",Xr such that