Geodesic
$F$-Quadratic Stochastic Operators
Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 606-612.

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In this paper, we introduce the notion of an $F$‑quadratic stochastic operator. It is shown that each $F$-quadratic operator has a unique fixed point. Besides, it is proved that any trajectory of an $F$-quadratic stochastic operator exponentially rapidly converges to this fixed point.
Keywords: quadratic stochastic operator, non-Volterra stochastic operator, ergodic theorem, mathematical genetics, fixed point of an operator. 
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U. A. Rozikov; U. U. Zhamilov. $F$-Quadratic Stochastic Operators. Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 606-612. http://geodesic.mathdoc.fr/item/MZM_2008_83_4_a11/

[1] S. N. Bernshtein, “Reshenie odnoi matematicheskoi problemy, svyazannoi s teoriei nasledovannosti”, Uchenye zapiski nauchno-issled. kaf. Ukr. (otd. matem.), 1 (1924), 83–115

[2] R. N. Ganikhodzhaev, “Kvadratichnye stokhasticheskie operatory, funktsiya Lyapunova i turniry”, Matem. sb., 83:8 (1992), 119–140 | MR | Zbl

[3] R. N. Ganikhodzhaev, “K opredeleniyu kvadratichnykh bistokhasticheskikh operatorov”, UMN, 48:4 (1993), 231–232 | MR | Zbl

[4] R. N. Ganikhodzhaev, “Karta nepodvizhnykh tochek i funktsii Lyapunova dlya odnogo klassa diskretnykh dinamicheskikh sistem”, Matem. zametki, 56:5 (1994), 40–49 | MR | Zbl

[5] H. Kesten, “Quadratic transformations: a model for population growth. I”, Adv. in Appl. Probab., 2:1 (1970), 1–82 | DOI | MR | Zbl

[6] Yu. I. Lyubich, Mathematical Structures in Population Genetics, Biomathematics, 22, Springer-Verlag, Berlin, 1992 | MR | Zbl

[7] R. N. Ganikhodzhaev, D. B. Eshmamatova, “Kvadratichnye avtomorfizmy simpleksa i asimptoticheskoe povedenie ikh traektorii”, Vladikavkaz. matem. zhurn., 8:2 (2006), 12–28 ; http://vmj.ru/articles/2006_2_2.pdf | Zbl

[8] R. N. Ganikhodzhaev, “Semeistvo kvadratichnykh stokhasticheskikh operatorov deistvuyuschikh na $S^2$”, Dokl. AN UzSSR, 1989, no. 1, 3–5 | MR | Zbl

[9] U. A. Rozikov, N. B. Shamsiddinov, On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure, arXiv: math/0608201v1

[10] P. R. Stein, S. M. Ulam, Nonlinear Transformations Studies on Electronic Computers, Dissertationes Math. (Rozprawy Mat.), 39, Polish Acad. Sci., Warsaw, 1964 | MR | Zbl

[11] R. N. Ganikhodzhaev, A. I. Eshniyazov, “Bistokhasticheskie kvadratichnye operatory”, Uzbek. matem. zhurn., 2004, no. 3, 29–34 | MR

[12] N. N. Ganikhodzhaev, R. T. Mukhitdinov, “Ob odnom klasse kvazivolterrovskikh operatorov”, Uzbek. matem. zhurn., 2003, no. 3–4, 9–12 | MR

[13] R. N. Ganikhodzhaev, A. M. Zhuraboev, “Mnozhestvo ravnovestnykh sostoyanii kvadratichnykh stokhasticheskikh operatorov tipa $V_\pi$”, Uzbek. matem. zhurn., 1998, no. 3, 23–27 | MR

[14] R. N. Ganikhodzhaev, A. Z. Karimov, “O chisle vershin mnozhestva bistokhasticheskikh operatorov”, Uzbek. matem. zhurn., 1999, no. 6, 29–35 | MR

[15] R. N. Ganikhodzhaev, R. E. Abdurakhmanova, “Opisanie kvadratichnykh avtomorfizmov konechno-mernogo simpleksa”, Uzbek. matem. zhurn., 2002, no. 1, 7–16 | MR

[16] M. I. Zakharevich, “O povedenii traektorii i ergodicheskoi gipoteze dlya kvadratichnykh otobrazhenii simpleksa”, UMN, 33:6 (1978), 207–208 | MR | Zbl

[17] N. N. Ganikhodzhaev, “Primenenie teorii gibbsovskikh mer k matematicheskoi genetike”, Dokl. RAN, 372:1 (2000), 13–16 | MR | Zbl

[18] N. N. Ganikhodjaev, U. A. Rozikov, “On quadratic stochastic operators generated by Gibbs distributions”, Regul. Chaotic Dyn., 11:4 (2006), 467–473 | DOI | MR