## On quadratic stochastic operators generated by Gibbs distributions.(English)Zbl 1164.37309

Summary: We give a constructive description of quadratic stochastic operators which act to the set of all probability measures on some measurable space. Our construction depends on a probability measure $$\mu$$ and cardinality of a set of cells (configurations) which here can be finite or continual. We study behavior of trajectories of such operators for a given probability measure $$\mu$$ which coincides with a Gibbs measure. For the continual case we compare the quadratic operators which correspond to well-known Gibbs measures of the Potts model on $$\mathbb{Z}^d$$. These investigations allows a natural introduction of thermodynamics in studying some models of heredity. In particular, we show that any trajectory of the quadratic stochastic operator generated by a Gibbs measure $$\mu$$ of the Potts model converges to this measure.

### MSC:

 37C20 Generic properties, structural stability of dynamical systems 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 82B26 Phase transitions (general) in equilibrium statistical mechanics
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