On indecomposability in chaotic attractors

Abstract

We exhibit a Li-Yorke chaotic interval map F such that the inverse limit XF = lim ←−{F, [0, 1]} does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if φ has positive entropy then Xφ contains an indecomposable subcontinuum. Each subcontinuum of XF is homeomorphic to one of the following: an arc, or XF , or a topological ray limiting to XF . From a result of Barge and Martin it follows that XF 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.