Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing
Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing
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DOI：
10.3934/dcds.2006.16.871
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发表时间：
2006-09
影响因子：
1.1
通讯作者：
S. Pilyugin;K. Sakai;O. Tarakanov
中科院分区：
文献类型：
--
作者：
S. Pilyugin;K. Sakai;O. Tarakanov
设Int$^1WS(M)$为具有弱阴影性质的光滑闭流形$M$的微分同态集合的$C^1$内。第二作者证明了如果$\dim M = 2$，并且$ Int$^1WS(M)$中一个微分同态$f \的所有源和汇都是平凡的，则$f$是结构稳定的。本文证明了在$ Int$^1WS(M)$, $\dim M = 2$中存在微分同态$f \，使得$(i)$ $f$属于$C^0$-横性条件不满足的$C^1$-内模，$(ii)$ $f$具有鞍形连接。这些结果基于以下定理：如果任意维流形$M$的$\ ω $稳定微分同构$f$的相图不包含长度$M > 3$的链，则$f$具有弱阴影性质。
Let Int$^1WS(M)$ be the $C^1$-interior of the set of diffeomorphisms of a smooth closed manifold $M$ having the weak shadowing property. The second author has shown that if $\dim M = 2$ and all of the sources and sinks of a diffeomorphism $f \in$ Int$^1WS(M)$ are trivial, then $f$ is structurally stable. In this paper, we show that there exist diffeomorphisms $f \in$ Int$^1WS(M)$, $\dim M = 2$, such that $(i)$ $f$ belongs to the $C^1$-interior of diffeomorphisms for which the $C^0$-transversality condition is not satisfied, $(ii)$ $f$ has a saddle connection. These results are based on the following theorem: if the phase diagram of an $\Omega$-stable diffeomorphism $f$ of a manifold $M$ of arbitrary dimension does not contain chains of length $m > 3$, then $f$ has the weak shadowing property.