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 → I such that (1) f is Li-Yorke chaotic, (2) the inverse limit space I f = lim ← {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 ∞ -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 Smítal 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.

Locations

• Advances in Difference Equations - View - PDF