The Li–Yorke theorem and infinite discontinuities

An unbounded mapping of the half-line which has no 3-cycles is continuously extended to the one-point compactification of the set of nonnegative real numbers. The extension is shown to have a 3-cycle so that the Li–Yorke theorem may be applied to the extension to efficiently obtain a scrambled set that is then relativized to the original domain. More generally, this approach may invoke other known chaos theorems and can be applied to various types of mappings with infinite discontinuities.  2004 Elsevier Inc. All rights reserved