Hamiltonian Cycle on the Set of Genotypes and Non-Ergodic Quadratic Stochastic Operators

Abstract

On the set of genotypes $\Phi =\{1,...,m\}$ we introduce a binary relation generated by Volterra quadratic stochastic operator $V$ on $(m-1)$ dimensional simplex $S^{m-1}$ and prove that the operator $V$ be non-ergodic if either there exists a Hamiltonian cycle or one of the vertices $M_i=({\delta }_{1i},{\delta }_{2i},...,{\delta }_{mi})$ of the simplex $S^{m-1}$ is a source and restriction of $V$ to the invariant face $F_i=\{x\in S^{m-1}:x_i=0\}$ is non-ergodic. In this paper we prove this result for $m=2,3,4.$

10 pages