Diffeomorphisms with the Average-Shadowing Property on Two-Dimensional Closed Manifolds
Diffeomorphisms with the Average-Shadowing Property on Two-Dimensional Closed Manifolds
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DOI：
10.1216/rmjm/1021477263
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发表时间：
2000-09
影响因子：
0.8
通讯作者：
K. Sakai
中科院分区：
文献类型：
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作者：
K. Sakai
研究了二维闭流形上所有仿射的平均伪轨和平均跟踪性质，并将满足平均跟踪性质的所有仿射的集合的C1内部刻画为所有Anosov仿射的集合.伪轨的概念经常出现在现代动力系统理论的几个分支中，特别是伪轨跟踪性质在稳定性理论的研究中起着重要的作用。Blank在[1]中引入了平均伪轨道的概念，作为伪轨道概念的某种推广（另见[2，p.19]），并在那里证明了，对于某种双曲系统f，f的每一个平均伪轨道平均被f的某个真轨道所遮蔽（平均遮蔽性质）。设M是C∞闭流形，即M是紧连通的，且M = M，d是由一个黎曼度量M·M在TM上导出的距离.用Diff（M）表示M上所有赋C拓扑的超同态的集合。对于δ > 0，M中的点序列{xi}i=−∞称为f ∈ Diff（M）的δ-平均伪轨道，如果存在一个数N = N（δ）> 0使得对所有n ≥ N，k ∈ Z，
The average pseudo-orbits and the averageshadowing property of diffeomorphisms on two-dimensional closed manifolds are considered, and the C1 interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. The notion of pseudo-orbits very often appears in several branches of the modern theory of dynamical systems, and, especially, the pseudoorbit shadowing property usually plays an important role in the investigation of the stability theory. In [1] Blank introduced the notion of average pseudo-orbits as a certain generalization of the notion of pseudo-orbits (see also [2, p. 19]) and it was proved there that, for a certain kind of hyperbolic system f , every average pseudo-orbit of f is shadowed in average by some true orbit of f (the average-shadowing property). Let M be a C∞ closed manifold, that is, M is compact connected and ∂M = ∅, and let d be the distance induced from a Riemannian metric ‖ · ‖ on TM . Denote by Diff (M) the set of all diffeomorphisms on M endowed with C topology. For δ > 0, a sequence {xi}i=−∞ of points in M is called a δ-average pseudo-orbit of f ∈ Diff (M) if there is a number N = N(δ) > 0 such that for all n ≥ N , k ∈ Z,