Li-Yorke sensitivity does not imply Li-Yorke chaos

TL;DR

The paper constructs a specific example of a dynamical system that is Li-Yorke sensitive but does not exhibit Li-Yorke chaos, disproving a previous conjecture linking the two concepts.

Contribution

It provides a counterexample in an infinite-dimensional setting showing Li-Yorke sensitivity does not necessarily lead to Li-Yorke chaos.

Findings

Constructed an infinite-dimensional compact metric space with a skew-product map.

Demonstrated the system is Li-Yorke sensitive but has at most countable scrambled sets.

Disproved the conjecture that Li-Yorke sensitivity implies 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 from the article [Akin E., Kolyada S., Li-Yorke sensitivity, Nonlinearity 16, (2003), 1421-1433].