# no-cycles condition

Let $X$ 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 $f$. Define a relation $\rightarrow$ on $\mathcal{F}$ by $\Lambda_{i}\rightarrow\Lambda_{j}$ if

 $W^{u}(\Lambda_{i})\cap W^{s}(\Lambda_{j})-\bigcup_{l=1}^{k}\Lambda_{l}\neq\emptyset,$

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

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

for $1\leq i and

 $\Lambda_{n_{j}}\rightarrow\Lambda_{n_{1}}.$

With some abuse of notation, we can write this as

 $\Lambda_{n_{1}}\rightarrow\Lambda_{n_{2}}\rightarrow\cdots\rightarrow\Lambda_{% n_{j}}\rightarrow\Lambda_{n_{1}}.$

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

Title no-cycles condition NocyclesCondition 2013-03-22 14:30:53 2013-03-22 14:30:53 Koro (127) Koro (127) 5 Koro (127) Definition msc 37-00 msc 37C75 no-cycles no-cycle no cycles condition cycle