Search

Let $$S^{m-1}$$ be the simplex in $${{\mathbb {R}}}^m$$ , and $$V:S^{m-1}\rightarrow S^{m-1}$$ be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^n V^k(x) \end{aligned}$$ exists for every $$x\in S^{m-1}$$ . It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

We showed that one form of generalized Catalan numbers is the solution to the problem of finding different connected component with finite vertices containing a fixed root for the semi-infinite Cayley tree of order 3. We give the formula for the full graph, Cayley tree of order 3 which is derived from the generalized Catalan numbers. Using ratios of Gamma functions, two upper bounds are given for problem defined on semi-infinite Cayley tree of order 3 as well as the full graph.

We consider q-component models on both the integer lattice ℤ2 and a second-order Cayley tree and study phase diagrams of these models.