Li–Yorke sensitivity does not imply Li–Yorke chaos

Abstract

We construct an infinite-dimensional compact metric space $X$ , which is a closed subset of $\mathbb{S}\times \mathbb{H}$ , where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li–Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li–Yorke sensitivity implies Li–Yorke chaos [Akin and Kolyada. Li–Yorke sensitivity. Nonlinearity 16, (2003), 1421–1433].

Sensitivities in Models with Backward Dynamics

“…In [1], the author also characterized that certain properties of the models with backward dynamics can be obtained by using the backward maps of the systems and introduced the concept of chaos for a multivalued dynamical systems where a concept of sensitivity of the multivalued map was raised as well. In [3] it was proved that Li-Yorke sensitivity does not imply Li-Yorke chaos. Inspired by the work of the authors mentioned above, in this paper our main purpose is to consummate the theoretical framework for analysis models with backward dynamics.…”