Li-Yorke sensitive and weak mixing dynamical systems

TL;DR

This paper discusses Li-Yorke sensitivity in dynamical systems, focusing on the relationship between weak mixing and minimal systems, and explores a conjecture about their connection.

Contribution

The paper provides arguments supporting the conjecture that minimal systems with a weak mixing factor are Li-Yorke sensitive, advancing understanding in dynamical systems theory.

Findings

Supported the conjecture that minimal systems with weak mixing factors are Li-Yorke sensitive

Reviewed existing examples and theoretical background on Li-Yorke sensitivity

Identified challenges in proving the conjecture

Abstract

Akin and Kolyada in 2003 [E. Akin, S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003) 1421 - 1433] introduced the notion of Li-Yorke sensitivity. They proved that every weak mixing system $(X, T)$, where $X$ is a compact metric space and $T$ a continuous map of $X$ is Li-Yorke sensitive. An example of Li-Yorke sensitive system without weak mixing factors was given in [M. \v{C}iklov\'a, Li-Yorke sensitive minimal maps, Nonlinearity 19 (2006) 517 - 529] (see also [M. \v{C}iklov\'a-Ml\'{\i}chov\'a, Li-Yorke sensitive minimal maps II, Nonlinearity 22 (2009) 1569 -1573]). In their paper, Akin and Kolyada conjectured that every minimal system with a weak mixing factor, is Li-Yorke sensitive. We provide arguments supporting this conjecture though the proof seems to be difficult.