New chaotic planar attractors from smooth zero entropy interval maps

Abstract

We show that for every positive integer k there exists an interval map $f:I\to I$ such that (1) f is Li-Yorke chaotic, (2) the inverse limit space $I_{f}=\lim_{\leftarrow}\{f,I\}$ does not contain an indecomposable subcontinuum, (3) f is $C^{k}$ -smooth, and (4) f is not $C^{k+1}$ -smooth. We also show that there exists a $C^{\infty}$ -smooth f that satisfies (1) and (2). This answers a recent question of Oprocha and the first author from (Proc. Am. Math. Soc. 143(8):3659-3670, 2015), where the result was proved for $k=0$ . Our study builds on the work of Misiurewicz and Smital of a family of zero entropy weakly unimodal maps. With the help of a result of Bennett, as well as Blokh’s spectral decomposition theorem, we are also able to show that each $I_{f}$ contains, for every integer i, a subcontinuum $C_{i}$ with the following two properties: (i) $C_{i}$ is $2^{i}$ -periodic under the shift homeomorphism, and (ii) $C_{i}$ is a compactification of a topological ray. Finally, we prove that the chaotic attractors we construct are topologically distinct from the one presented by P Oprocha and the first author.