Global periodicity and complete integrability of discrete dynamical systems

Abstract

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.

Keywords:

• Globally periodic discrete dynamical system
• First integrals
• Completely integrable systems
• Difference equations
• Invariants for difference equations

2000 Mathematics Subject Classification:

• 39A11
• 39A20

Acknowledgements

The authors are partially supported by the Government of Catalonia's grant 2001SGR-00173, and CICYT through grants BFM2002-04236-C02-2 (first and second authors) and DPI2005-08-668-C03-01 (third author). This work was partially done while the third author was visiting Barcelona's CRM (Centre de Recerca Matermàtica), having the support of the UPC's “Programa d'ajuts a la mobilitat”; V. Mañosa acknowledges both institutions. We acknowledge F. Mañosas for stimulating and helpful indications.

Notes

^¶Email: [email protected]

Additional information

Notes on contributors

Víctor Mañosa

¶ ¶Email: [email protected]