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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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