interval map, arc-like, Li-Yorke chaotic, indecomposable continuum.

We exhibit a Li-Yorke chaotic interval map F such that the corresponding inverse limit does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if F has positive entropy then the inverse limit space contains an indecomposable subcontinuum. From a result of Barge and Martin it follows that our space is a chaotic attractor of a planar homeomorphism. In addition, F can be modified to give a cofrontier that is a chaotic attractor of a planar homeomorphism but contains no indecomposable subcontinuum. Finally, F can be modified, without removing or introducing new periods, to give a chaotic zero entropy interval map, such that the corresponding inverse limit contains the pseudoarc.