The present paper introduces a class of stochastic Lotka–Volterra type mappings on the two-dimensional simplex, called \(\textbf{p}\) -Stein–Ulam maps. The distinguished interior point \(\textbf{p}\) plays the role of a coexistence state, while the cyclic boundary motion is modeled on the classical Stein–Ulam dynamics and is analogous to the rock–paper–scissors mechanism in zero-sum evolutionary games. We show that the structural conditions defining this class are preserved under convex combinations. We also prove that a map in this class is non-ergodic whenever its orbit visits the three vertex neighborhoods infinitely often in the prescribed cyclic order. The proof identifies the dynamical mechanism responsible for non-ergodicity: a Lyapunov-type quantity drives the orbit toward the boundary, while increasingly long residence times near the vertices prevent the Cesàro averages from converging. Finally, we explain how this framework connects with Takens’ last problem.
