Equivalent Conditions of Dominated Splitting for Volume-Preserving Diffeomorphisms
Equivalent Conditions of Dominated Splitting for Volume-Preserving Diffeomorphisms
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DOI：
10.1007/s10114-005-0889-6
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发表时间：
2007-01
期刊：
影响因子：
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通讯作者：
C. Liang;Geng-Nian Liu;Wenxiang Sun
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文献类型：
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作者：
C. Liang;Geng-Nian Liu;Wenxiang Sun
设Md为无边界的紧致d维流形，ω为体积形式。当f * (ω)= ω时，C1微分同构f: Md→Md是保体积的或保守的。用Diff1 ω (M)表示具有一致C1拓扑的C1保体积微分同态集合。设f∈Diff1 ω (M)。点p在Md据说架接触如果是non-transverse交叉稳定和不稳定流形的一些夸张的周期轨道点x∈f。M叫做C1 preperiodic,如果任何C1附近你的f Diff1ω(M)和任何地区M U (x),有g∈U和y∈U, y是周期性的g。我们表示组C1 preperiodic点p∗(f)。对于0≤i≤d，点x∈M是C1 i- f的预周期点如果对于Diff1 ω (M)中f的任意C1邻域U和M中x的任意邻域U，存在g∈U和y∈U使得y是指标i的g的双曲周期点。用Pi∗(f)表示f的C1 i-预周期点的集合
Let Md be a compact d-dimensional manifold without boundary and ω be a volume form. A C1 diffeomorphism f: Md→ Md is volume-preserving or conservative if f∗(ω)= ω. Denote by Diff1 ω (M) the set of C1 volume-preserving diffeomorphisms, with uniform C1 topology. Let f∈ Diff1 ω (M). A point p in Md is said to be a homoclinic tangency if it is a non-transverse intersection of the stable and unstable manifolds of some hyperbolic periodic orbit of f. A point x∈ M is called C1 preperiodic, if for any C1 neighbourhood U of f in Diff1 ω (M) and any neighbourhood U of x in M, there are g∈ U and y∈ U such that y is periodic of g. We denote the set of C1 preperiodic points by P∗(f). For 0≤ i≤ d, a point x∈ M is C1 i− preperiodic of f if for any C1 neighbourhood U of f in Diff1 ω (M) and any neighbourhood U of x in M, there are g∈ U and y∈ U such that y is a hyperbolic periodic point of g of index i. Denote by Pi∗(f) the set of C1 i-preperiodic points of f. Then