On indecomposability in chaotic attractors

Boroński, Jan; Oprocha, Piotr

doi:10.1090/S0002-9939-2015-12526-9

On indecomposability in chaotic attractors
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by Jan P. Boroński and Piotr Oprocha PDF

Proc. Amer. Math. Soc. 143 (2015), 3659-3670 Request permission

Abstract:

We exhibit a Li-Yorke chaotic interval map $F$ such that the inverse limit $X_F=\varprojlim \{F,[0,1]\}$ does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if $\varphi$ has positive entropy then $X_\varphi$ contains an indecomposable subcontinuum. Each subcontinuum of $X_F$ is homeomorphic to one of the following: an arc, or $X_F$, or a topological ray limiting to $X_F$. Through our research, we found that it follows that $X_F$ 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.

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