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FYI. This article was copied from PlanetMath but appears to contain a rather dubious definition of the alpha-limit set. In general, f is not a bijection, so ${\displaystyle f^{-1}}$ as an inverse doesn't exist. However, it is common in dynamical systems for ${\displaystyle f^{-1}}$ to denote the preimage. However, I think its rather glib to define the alpha-limit set as the set of preimages... that would certainly make the alpha-limit set a much more complex and complicated beast than the omega-limit set. (I think it would make the alpha-limit set be a Julia set). Needs clarification. linas 14:22, 9 June 2006 (UTC) reply

Never mind. If f is restricted to be a homeomorphism, then its a bijection, and that takes care of that. I think a more general definition of the alpha-limit set is possible, but do not wish to invent one here. linas 14:42, 10 June 2006 (UTC) reply

" if X is compact then lim_ω γ and lim_α γ are nonempty, compact and simply connected"

lim_ω could be a cycle but how is a cycle simply-connected? Novwik ( talk) 08:57, 6 December 2007 (UTC) reply