Non-integrability of measure preserving maps via Lie symmetries

We consider the problem of characterizing, for certain natural number m, the local Cm-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with th...

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Vydáno v:	Journal of Differential Equations Ročník 259; číslo 10; s. 5115 - 5136
Hlavní autoři:	Cima, Anna, Gasull, Armengol, Mañosa, Víctor
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	Elsevier Inc 15.11.2015
Témata:	34 Ordinary differential equations 34C Qualitative theory 37 Dynamical systems and ergodic theory 37C Smooth dynamical systems: general theory 37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 39 Difference and functional equations 39A Difference equations Classificació AMS Cohen map difference equations Differentiable dynamical systems Differential equations Equacions diferencials i integrals Equacions diferencials ordinàries Integrability and non-integrability of maps Integrable vector fields isochronous centers Lie symmetries Matemàtiques i estadística Measure preserving maps Period function Sistemes dinàmics diferenciables Àrees temàtiques de la UPC Integrability and non-integrability of maps 34C14 Integrable vector fields Lie symmetries Measure preserving maps 39A05 Period function 37C25 Cohen map 37J30
ISSN:	0022-0396, 1090-2732
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Popis
Shrnutí:	We consider the problem of characterizing, for certain natural number m, the local Cm-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.
ISSN:	0022-0396 1090-2732
DOI:	10.1016/j.jde.2015.06.019