Chaotic functions with zero topological entropy

Abstract

Recently Li and Yorke introduced the notion of chaos for mappings from the class C 0 ( I , I ) {C^0}(I,I) , where I I is a compact real interval. In the present paper we give a characterization of the class M ⊂ C 0 ( I , I ) M \subset {C^0}(I,I) of mappings chaotic in this sense. As is well known, M M contains the mappings of positive topological entropy. We show that M M contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of M M consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) δ \delta -chaos, distinguishable on a certain level δ > 0 \delta > 0 .