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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Bibliographic Details
Published in:Journal of Differential Equations Vol. 259; no. 10; pp. 5115 - 5136
Main Authors: Cima, Anna, Gasull, Armengol, Mañosa, Víctor
Format: Journal Article Publication
Language:English
Published: Elsevier Inc 15.11.2015
Subjects:
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
Online Access:Get full text
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Summary: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