Name: Pointwise periodic maps with quantized first integrals
Keywords: Difference equations, Quantized first integrals, Classificació AMS::39 Difference and functional equations::39A Difference equations, Geometria discreta, Regular tessellations., :Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC], Equacions en diferències, Periodic points, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Differentiable dynamical systems

descriptionPublicationkeyboard_double_arrow_right Report 01 Jan 2020 English

Authors: Gasull, Armengol; Cima, Anna; Mañosa Fernández, Víctor; Mañosas, Francesc;

handle: 2117/332940

Pointwise periodic maps with quantized first integrals

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Abstract

In a series of papers, Chang, Cheng and Wang studied the periodic behavior of some piecewise linear maps in the plane. These examples are valuable under the light of the classical result of Montgomery about periodic homeomorphisms, since they are pointwise periodic but not periodic. We revisit them from the point of view of their properties as integrable systems. We describe their global dynamics in terms of the dynamics induced by the maps on the level sets of certain first integrals that we also find. We believe that some of the features that the first integrals exhibit are interesting by themselves, for instance the set of values of the integrals are discrete. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of some prescribed tiles of certain regular tessellations. The existence of these quantized integrals is quite novel in the context of discrete dynamic systems theory.

The third author acknowledges the group’s research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya

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Difference equations, Quantized first integrals, Classificació AMS::39 Difference and functional equations::39A Difference equations, Geometria discreta, Regular tessellations., :Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC], Equacions en diferències, Periodic points, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Differentiable dynamical systems, and nonholonomic systems, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Lagrangian, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals, Classificació AMS::52 Convex and discrete geometry::52C Discrete geometry, Discrete geometry, :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], :52 Convex and discrete geometry::52C Discrete geometry [Classificació AMS], Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Piecewise linear maps, Pointwise periodic maps, Regular tessellations, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, contact

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