Dynamical systems of quadratic operators on set of idempotent measures

I. T. Juraev^a, U. A. Rozikov<sup>bcd

a</sup> Namangan State University, Boburshox str. 161, 160107, Namangan, Uzbekistan
^b V.I. Romanovskiy Institute of Mathematics, Universitet str. 9, 100174, Tashkent, Uzbekistan
^c National University of Uzbekistan, Universitet str. 4, 100174, Tashkent, Uzbekistan
^d Karshi State University, Kuchabag str. 17 180119, Karshi, Uzbekistan

Abstract: We consider quadratic operators, which map the $n$–dimensional simplex of idempotent measures into itself. We introduce the concept of Volterra quadratic operator on the simplex of idempotent measures and provide some general properties of such operators.
We also consdier a special Volterra quadratic operator and provide a comprehensive analysis of the dynamical system generated by this operator. Moreover, the dynamical systems generated by general Volterra operators defined on 2 and 3–dimensional simplices of idempotent measures are studied. For each case, we find fixed points and limits of trajectories.

Keywords: quadratic operator, Volterra operator, simplex, idempotent measure, fixed point, trajectory, dynamical system, attracting, repelling.

MSC: 37C25, 47Axx

Received: 03.07.2024

Language: English

 English version: Ufa Mathematical Journal, 2025, 17:3, 121–136