On a necessary condition for the ergodicity of quadratic operators defined on a two-dimensional simplex. (English. Russian original) Zbl 1160.37307

Ganikhodzhaev, N. N.; Zanin, D. V.

Russ. Math. Surv. 59, No. 3, 571-572 (2004); translation from Usp. Mat. Nauk 59, No. 3, 161-162 (2004).

From the text: Let \(S^{m-1}=\{x=(x_1,x_2,\dots,x_m)\in\mathbb R^m\colon x_ i\geq 0,\;\sum^m_{i=1}x_i=1\}\) be an \((m-1)\)-dimensional simplex. A mapping \(V\) of \(S^{m-1}\) into itself is called a quadratic stochastic operator (q.s.o.) if \((Vx)_k=\sum^m_{i,j=1}p_{ij,k}x_ix_j\) for any \(x\in S^{m-1}\) and for all \(k=1,\dots,m\), where \(p_{ij,k}\geq 0\), \(\sum^m_{k=1}p_{ij,k}=1\), and \(p_{ij,k}=p_{ji,k}\) for all \(i\), \(j\), \(k\). On the basis of numerical computations, S. Ulam [Unsolved mathematical problems (Russian), ‘Nauka’, Moscow (1964; Zbl 0161.00101)] conjectured that the ergodic theorem holds for any q.s.o. \(V\), i.e., the limit \(\lim_{n\to\infty}\frac 1n\sum^{n-1}_{k=0}V^kx\) exists for any \(x\in S^{m-1}\). M. I. Zakharevich [Usp. Mat. Nauk 33, No. 6(204), 207–208 (1978; Zbl 0407.58030)] proved that this conjecture is false in general. A q.s.o. \(V\) defined on \(S^{m-1}\) is called a Volterra operator if \(p_{ij,k}=0\) whenever \(k\) differs from \(i\) and \(j\), i.e., \(k\notin\{i,j\}\).
In the present paper we establish a necessary condition for the ergodic theorem to hold for a class of Volterra q.s.o. defined on \(S^2\).

MSC:

37A30	Ergodic theorems, spectral theory, Markov operators
28D05	Measure-preserving transformations
37C99	Smooth dynamical systems: general theory

Citations:

Zbl 0161.00101; Zbl 0407.58030