Downloads provided by UsageCountsNon-integrability of measure preserving maps via Lie symmetries
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Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor;
Non-integrability of measure preserving maps via Lie symmetries
We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-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.
25 pages
Difference equations, integrable vector field, Dynamical Systems (math.DS), Integrable vector fields, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Classical Analysis and ODEs (math.CA), Equacions diferencials ordinàries, and nonholonomic systems, Mathematics - Dynamical Systems, Lagrangian, 34C14, 37C25, 37J30, 39A05, Isochronous center, Symmetries, Lie group and Lie algebra methods for problems in mechanics, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Cohen map, :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], :39 Difference and functional equations::39A Difference equations [Classificació AMS], difference equations, Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Mathematics - Classical Analysis and ODEs, Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria), contact, Differential equations, integrable vector fields, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, FOS: Physical sciences, Classificació AMS::39 Difference and functional equations::39A Difference equations, measures preserving map, Measure preserving maps, :Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC], Integrability and non-integrability of maps, isochronous centers, FOS: Mathematics, Differentiable dynamical systems, Lie symmetries, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, integrability and non-integrability of maps, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals, Period function, measure preserving maps, Lie symmetry, period function, Dynamical systems involving smooth mappings and diffeomorphisms, Discrete version of topics in analysis, difference equations., Exactly Solvable and Integrable Systems (nlin.SI), Symmetries, invariants of ordinary differential equations, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian
Difference equations, integrable vector field, Dynamical Systems (math.DS), Integrable vector fields, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Classical Analysis and ODEs (math.CA), Equacions diferencials ordinàries, and nonholonomic systems, Mathematics - Dynamical Systems, Lagrangian, 34C14, 37C25, 37J30, 39A05, Isochronous center, Symmetries, Lie group and Lie algebra methods for problems in mechanics, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Cohen map, :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], :39 Difference and functional equations::39A Difference equations [Classificació AMS], difference equations, Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Mathematics - Classical Analysis and ODEs, Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria), contact, Differential equations, integrable vector fields, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, FOS: Physical sciences, Classificació AMS::39 Difference and functional equations::39A Difference equations, measures preserving map, Measure preserving maps, :Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC], Integrability and non-integrability of maps, isochronous centers, FOS: Mathematics, Differentiable dynamical systems, Lie symmetries, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, integrability and non-integrability of maps, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals, Period function, measure preserving maps, Lie symmetry, period function, Dynamical systems involving smooth mappings and diffeomorphisms, Discrete version of topics in analysis, difference equations., Exactly Solvable and Integrable Systems (nlin.SI), Symmetries, invariants of ordinary differential equations, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian
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