no-cycles condition

Let X be a metric space and let f:XX be a homeomorphism. Suppose ={Λ1,,Λk} is a family of compact invariant sets for f. Define a relation on by ΛiΛj if


that is, if the unstable set of Λi intersects the stable set of Λj outside the union of the Λl’s.

A cycle for is a sequence {ni:i=1,,j} such that


for 1i<j and


With some abuse of notation, we can write this as


If has no cycles, then we say that it satisfies the no-cycles condition.

Title no-cycles condition
Canonical name NocyclesCondition
Date of creation 2013-03-22 14:30:53
Last modified on 2013-03-22 14:30:53
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Definition
Classification msc 37-00
Classification msc 37C75
Synonym no-cycles
Synonym no-cycle
Synonym no cycles condition
Synonym cycle