PlanetMath: no-cycles condition

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no-cycles condition

(Definition)

Let be a metric space and let $f\colon X\to X$ be a homeomorphism. Suppose $\mathcal F= \{\Lambda_1,\dots,\Lambda_k\}$ is a family of compact invariant sets for . Define a relation $\rightarrow$ on $\mathcal{F}$ by $\Lambda_i \rightarrow \Lambda_j$ if

$\displaystyle W^u(\Lambda_i)\cap W^s(\Lambda_j) - \bigcup_{l=1}^k \Lambda_l \neq \emptyset,$

that is, if the unstable set of $\Lambda_i$ intersects the stable set of $\Lambda_j$ outside the union of the $\Lambda_l$ 's.

A cycle for $\mathcal F$ is a sequence $\{n_i:i=1,\dots,j\}$ such that

$\displaystyle \Lambda_{n_i}\rightarrow \Lambda_{n_{i+1}}$

for $1\leq i<j$ and

$\displaystyle \Lambda_{n_j}\rightarrow \Lambda_{n_1}.$

With some abuse of notation, we can write this as

$\displaystyle \Lambda_{n_1}\rightarrow\Lambda_{n_2}\rightarrow\cdots\rightarrow\Lambda_{n_j}\rightarrow\Lambda_{n_1}.$

If $\mathcal{F}$ has no cycles, then we say that it satisfies the no-cycles condition.

"no-cycles condition" is owned by Koro.

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Other names:

no-cycles, no-cycle, no cycles condition, cycle

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Cross-references: sequence, union, intersects, unstable set, relation, invariant, compact, homeomorphism, metric space
There are 17 references to this entry.

This is version 2 of no-cycles condition, born on 2004-08-01, modified 2004-10-18.
Object id is 6054, canonical name is NoCyclesCondition.
Accessed 4120 times total.

Classification:

AMS MSC:	37-00 (Dynamical systems and ergodic theory :: General reference works )
	37C75 (Dynamical systems and ergodic theory :: Smooth dynamical systems: general theory :: Stability theory)

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