<i>G,μ</i>)-quadratic stochastic operators
Jochen BlathFakultät II, Institut für Mathematik, MA 7-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, GermanyUygun JamilovInstitute of Mathematics, National University of Uzbekistan, 29, Do'rmon Yo'li Street, 100125 Tashkent, UzbekistanMichael ScheutzowFakultät II, Institut für Mathematik, MA 7-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
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We consider a new subclass of quadratic stochastic (evolutionary) operators on the simplex indexed by a finite Abelian group G with heredity law . With the help of the notion of -invariant subgroups, where denotes the support of in G, we prove that the trajectories of such operators always converge either to a periodic orbit or to a fixed point (where the latter holds for Lebesgue-almost all initial values). In particular, these operators are ergodic and almost regular. We also consider the speed of convergence to the limit, respectively, to the periodic orbit and give criteria to distinguish between regularity and periodicity.
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