In the present we introduce a concept of doubly stochastic quadratic operator. We prove necessary and sufficient conditions for doubly stochasticity of operator. Besides, we prove that the set of all doubly stochastic operators forms convex polytope. Finally, we study analogue of Birkhoff's theorem for the class of doubly stochastic operators. Mathematics Subject Classification: 15A51, 15A63, 46T99, 46A55.

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The purpose of the paper is to extend the notion of dissipativity of maps on infinite-dimensional simplex. We study the fixed points of dissipative quadratic stochastic operators on infinite-dimensional simplex. Besides, we study the limit behavior of the trajectories of such operators. We also show the difference of dissipative operators defined on finite and infinite-dimensional spaces. We obtain the results by using majorization for infinite vectors and 1 convergence. MSC: 15A51; 47H60; 46T05; 92B99

Generalizing the concept of a quadratic doubly stochastic operator, we introduce the concept of an arbitrary doubly stochastic operator. We give a necessary condition for double stochasticity. Moreover, we prove an ergodic theorem for doubly stochastic operators.

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