General Approach to Integrating Invertible Dynamical Systems Defined by Transformations from the Cremona group $\operatorname{Cr}(P^n_k)$ of Birational Transformations

K. V. Rerikh

Matematičeskie zametki, Tome 68 (2000) no. 5, pp. 699-709.

Résumé

A general approach is developed for integrating an invertible dynamical system defined by the composition of two involutions, i.e., a nonlinear one which is a standard Cremona transformation, and a linear one. By the Noether theorem, the integration of these systems is the foundation for integrating a broad class of Cremona dynamical systems. We obtain a functional equation for invariant homogeneous polynomials and sufficient conditions for the algebraic integrability of the systems under consideration. It is proved that Siegel's linearization theorem is applicable if the eigenvalues of the map at a fixed point are algebraic numbers.

Bibliographie

[1] Moser J., “On the theory of quasi-periodic motions”, SIAM Rev., 8 (1966), 145–172 | DOI | MR | Zbl

[2] Moser J. K., “Convergent series expansions for quasi-periodic motions”, Math. Ann., 169 (1967), 136–176 | DOI | MR

[3] Moser J. K., Stable and Random Motion in Dynamical Systems, Princeton Univ. Press, Princeton, 1973 | Zbl

[4] Arnol'd V. I., “Reversible systems”, Nonlinear and Turbulent Processes in Physics, ed. R. Z. Sagdeev, Chur, Harwood, 1984, 1161–1174

[5] Arnol'd V. I., Sevryuk M. B., “Oscillations and bifurcations in reversible systems”, Nonlinear Phenomena in Plasma Physics and Hydrodynamics, ed. R. Z. Sagdeev, Mir, Moscow, 1986, 31–64

[6] Sevryuk M. B., Reversible Systems, Lecture Notes in Math., 1211, Springer, Berlin, 1986 | MR | Zbl

[7] Devaney R. L., “Reversible diffeomorphisms and flows”, Trans. Amer. Math. Soc., 218 (1976), 89–113 | DOI | MR | Zbl

[8] Quispel G. R. W., Roberts J. A. G., Thompson C. J., “Integrable mappings and soliton equations”, Phys. D, 34 (1989), 183–192 | DOI | MR | Zbl

[9] Hudson H., Cremona Transformations in Plane and Space, University Press, Cambridge, 1927

[10] Matematicheskaya entsiklopediya, T. 3, Sovetskaya entsiklopediya, M., 1982

[11] Shirkov D. V., Serebryakov V. V., Mescheryakov V. A., Dispersionnye teorii silnykh vzaimodeistvii pri nizkikh energiyakh, Nauka, M., 1967

[12] Rerikh K. V., “Cremona transformation and general solution of one dynamical system of the static model”, Phys. D, 57 (1992), 337–354 | DOI | MR | Zbl

[13] Moser J., “On the integrability of area-preserving Cremona mappings near an elliptic fixed point”, Bol. Soc. Mat. Mexicana, 1960, 176–180 | Zbl

[14] Veselov A. P., “Gruppa Kremony i dinamicheskie sistemy”, Matem. zametki, 45 (1989), 118–120 | MR | Zbl

[15] Arnold V. I., Ilyashenko Yu. S., “Obyknovennye differentsialnye uravneniya”, Itogi nauki i tekhniki. Sovr. probl. matem. Fundament. napravleniya, 1, VINITI, M., 1985, 7–149

[16] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978

[17] Baker A., “Linear forms in logarithms of algebraic numbers”, Mathematika, 13 (1966), 204–216 | Zbl

[18] Baker A., Transcendental Number Theory, Cambridge University Press, Cambridge, 1990

[19] Feldman N. I., “Improvement of evaluation of the linear form of logarithms of algebraic numbers”, Mat. Sbornik, 77 (119):3 (1968), 423–436 | MR | Zbl

[20] Feldman N. I., Sedmaya problema Gilberta, Izd-vo MGU, M., 1982 | Zbl

[21] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966

[22] Mescheryakov V. A., Metod postroeniya nekotorykh klassov reshenii uravnenii tipa Chu–Lou, Preprint OIYaI R-2369, OIYaI, Dubna, 1965

[23] Fairlie D. B., “Structure of the crossing matrix for arbitrary internal symmetry groups”, J. Math. Phys., 7 (1966), 811–815 | DOI | MR | Zbl

[24] Wanders G., “Analytic structure of the scattering matrix in the neutral pseudoscalar theory”, Nuovo Cimento, 23 (1962), 817–837 | DOI | MR | Zbl

[25] Mescheryakov V. A., Metod resheniya uravnenii tipa uravnenii Chu–Lou, Preprint OIYaI R-1965, OIYaI, Dubna, 1965

[26] Rothelutner T., “Exacte Lösungen von Low–Geichungen”, Z. Phys., 177 (1964), 287–299 | DOI | MR

[27] Zhuravlev V. I., Mescheryakov V. A., Rerikh K. V., “Ob odnom novom vide reshenii uravnenii tipa uravnenii Chu–Lou”, YaF, 10 (1969), 168–175 | MR

[28] Meshcheryakov V. A., Rerikh K. V., “A method of the local construction of invariant subspaces in the space of solutions of the Chew–Low type equations”, Ann. Phys., 59 (1970), 408–425 | DOI | MR

[29] Gerdt V. P., Mescheryakov V. A., “Lokalnaya forma resheniya uravnenii Chu–Lou”, TMF, 24 (1975), 155–163 | MR

[30] Rerikh K. V., “Uravneniya Chu–Lou kak preobrazovaniya Kremona. Struktura obschikh integralov”, TMF, 50 (1982), 251–260 | MR

[31] Rerikh K. V., “Chew–Low equations as the Cremona transformation. Structure of the first integral and general solution”, Proc. of the XIII Int. Conf. on Differential Geometric Methods in Theoretical Physics (Shumen, Bulgaria, 1984), World Scientific, Singapore, 1986, 170–178 | MR

[32] Rerikh K. V., “Non-algebraic integrability of one reversible Cremona dynamical system. The Poincaré $(1.1)$ resonance and the Birkhoff–Moser analytical invariants”, Proc. of Internat. Workshop “Finite-Dimensional Integrable Systems”, JINR, Dubna, 1995, 171–180

[33] Rerikh K. V., “Algebraic addition concerning the Siegel theorem on the linearization of a holomorphic mapping”, Math. Z., 224 (1997), 445–448 | DOI | MR | Zbl

[34] Rerikh K. V., “Non-algebraic integrability of the Chew–Low reversible dynamical system of the Cremona type and the relation with the 7th Hilbert problem (non-resonant case)”, Phys. D, 82 (1995), 60–78 | DOI | MR | Zbl

[35] Baker A., “Effective methods in the theory of numbers”, Proceedings of the International Congress of Mathematicians, V. 1 (Nice, September 1970), Gauthier-Villars, Paris, 1971, 19–26

[36] Baker A., Selected Studies: Physics-Astrophysics, Mathematics, Hystory of Science, Scientific Research and Technology Agency. A Volume Dedicated to the Memory of Albert Einstein (Athens, Greece), eds. T. M. Rassias, G. M. Rassias, North-Holland, Amsterdam–New York–Oxford, 1982

[37] Gekke E., Lektsii po teorii algebraicheskikh chisel, M., 1940

[38] Baker A., Wüstholz G., “Logarithmic forms and group varieties”, J. Reine Angew. Math., 442 (1993), 19–62 | MR | Zbl

[39] Rerikh K. V., “Non-algebraic integrability of one reversible dynamical system of the Cremona type”, J. Math. Phys., 39 (1998), 2821–2832 | DOI | MR | Zbl

[40] Moser J., “On quadratic symplectic mappings”, Math. Z., 216 (1994), 417–430 | DOI | MR | Zbl

[41] Rerikh K. V., “Integrability of functional equations defined by byrational mappings (general theory of integration of birational cascades and cascade-flows), II”, Math. Physics, Analysis and Geometry (to appear)

[42] Rerikh K. V., “Integrability of functional equations defined by plane byrational mappings”, The Electronic Data Base of Abstracts, Section “Dynamical Systems and Differential Equations” of International Congress of Mathematicians (August, 18–27, Berlin, 1998)