Non-ergodicity and Li–Yorke chaos for the family W_alpha

Let $S^2$ be the two-dimensional simplex, and let $W_\alpha:S^2\to S^2$ be the quadratic stochastic operator defined in the source by equation $W_\alpha$. Assume $0<\alpha<1-\frac{\sqrt{3}}{2}$. The W_alpha chaos claim. The operator $W_\alpha$ is non-ergodic and exhibits Li–Yorke chaos. The surrounding results describe this parameter range as having an infinite compact omega-limit set in $\operatorname{int}S^2$ for every initial point other than the fixed point $C$, providing context for the claimed behavior.