# $\omega$-limit set

## Primary tabs

Defines:
$\alpha$-limit, alpha-limit, $\omega$-limit, omega-limit
Synonym:
omega-limit set
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

37B99 None of the above, but in MSC2010 section 37Bxx

### alpha-limit set ???

The alpha-limit set seems rather glibly defined to me. In most cases, ''f'' won't be a bijection, for $f^{-1}$ won't be defined. So then $f^{-1}$ must mean the pre-image of ''f''? In this case, is the alpha-limit set the same thing as the Julia set ?? This needs clarification.

### merge articles.

Seems that PM has several articles on this topic. please consider merging them together:

http://planetmath.org/encyclopedia/OmegaLimitSet.html which is another article with the same title(!)

and http://planetmath.org/?op=getobj&from=objects&id=6722 (titled "limit cycle")

### Re: alpha-limit set ???

Note that $f$ is assumed to be a homeomorphism.

### Re: alpha-limit set ???

Thanks, OK, right. Since in dynamical systems, a lot of iterated maps are not homeomorphisms, this flew right past me. Certainly, the definition for the omega-limit can be extended to more general functions. The definition for the alpha-limit in such a case is trickier, and I'm wondering if there is a generally accepted definition in such a case.

### Re: alpha-limit set ???

The definition of omega-limit set is the same for non-invertible maps. The usual definition of alpha-limit set (altough not very useful) is by taking sequences of preimages, i.e. sequences {x_n} such that f(x_n)=x_{n-1} and x_0 = x, and consider the alpha-limit set of x to be the set of all acummulation points of all such sequences.
I decided not to add this to the entry for two reasons: A reader who is looking for the definition of omega-limit set will most likely not be discouraged by the fact that it is defined for homeomorphisms and assume that the definition is the same for non-homeomorphisms; on the other hand the definition of alpha-limit set is of such limited use that i didn't think it was worth mentioning. It is unlikely that anyone using alpha-limit sets of non-invertible maps will not define it first, so i don't see a good reason to obfuscate this entry by adding that definition.