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Journal ArticleDOI

A New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

13 Jul 2021-Symmetry Integrability and Geometry-methods and Applications (SIGMA. Symmetry, Integrability and Geometry: Methods and Applications)-Vol. 17, pp 067
TL;DR: In this article, the notion of an involution curve was introduced for cubic pencils, which is a curve which intersects each curve of the pencil in exactly one non-base point on the pencil and is used to construct integrable maps of the plane which leave invariant a cubic pencil.
Abstract: For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.

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Citations
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Journal ArticleDOI
TL;DR: In this paper , the boundary consistency around a half of a rhombic dodecahedron was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary.
Abstract: In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary. In this paper, we formalize the notions of boundary equations as boundary conditions for quad-graph systems, and provide a systematic method for solving the boundary consistency, which results in a classification of integrable boundary equations for quad-graph equations in the Adler–Bobenko–Suris classification. This relies on factorizing, first the quad-graph equations into pairs of dual boundary equations, and then the consistency on a rhombic dodecahedron into two equivalent boundary consistencies. Generalization of the method to rhombic-symmetric equations is also considered.
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TL;DR: In this paper, the boundary consistency around a half of a rhombic dodecahedron was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary.
Abstract: In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary. In this paper, we formalize the notions of boundary equations as boundary conditions for quad-graph systems, and provide a systematic method for solving the boundary consistency, which results in a classification of integrable boundary equations for quad-graph equations in the Adler-Bobenko-Suris classification. This relies on factorizing, first the quad-graph equations into pairs of dual boundary equations, and then the consistency on a rhombic dodecahedron into two equivalent boundary consistencies. Generalizations of the method to rhombic-symmetric equations are also considered.
References
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Journal ArticleDOI
TL;DR: It is shown that the sequence ║fn*║ can be bounded, grow linearly, grow quadratically, or grow exponentially, and that after conjugating, f is an automorphism virtually isotopic to the identity, f preserves a rational fibration, or f preserves an elliptic fibration.
Abstract: We classify bimeromorphic self-maps f : X [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of compact Kahler surfaces X in terms of their actions f *: H 1,1 ( X ) [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i "/] on cohomology. We observe that the growth rate of ║ f n *║ is invariant under bimeromorphic conjugacy, and that by conjugating one can always arrange that f n * = f * n . We show that the sequence ║ f n *║ can be bounded, grow linearly, grow quadratically, or grow exponentially. In the first three cases, we show that after conjugating, f is an automorphism virtually isotopic to the identity, f preserves a rational fibration, or f preserves an elliptic fibration, respectively. In the last case, we show that there is a unique (up to scaling) expanding eigenvector θ+ for f *, that θ+ is nef, and that f is bimeromorphically conjugate to an automorphism if and only if θ 2 + = 0. We go on in this case to construct a dynamically natural positive current representing θ+, and we study the growth rate of periodic orbits of f . We conclude by illustrating our results with a particular family of examples.

432 citations

Journal ArticleDOI
TL;DR: In this paper, an 18-parameter family of integrable reversible mappings of the plane is presented, which are shown to occur in soliton theory and in statistical mechanics.

311 citations


"A New Class of Integrable Maps of t..." refers background or methods in this paper

  • ...to a QRT map [14, 15] through a projective collineation....

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  • ...A rather large (18-parameter) family of integrable maps of the plane was obtained by Quispel, Roberts and Thompson (QRT) [14, 15]....

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  • ...Recall that the QRT map preserves a special N = 4 pencil, namely a biquadratic pencil....

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  • ...The corresponding involutions are the horizontal switch ι1, and the vertical switch ι2, and the QRT map is the composition τ = ι2 ◦ ι1 [5, p. viii]....

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  • ...In [18] it was shown that this construction does not generalise to maps which preserve a pencil of degree > 4 and that all generalised Manin transformations obtained in this way are equivalent to a QRT map [14, 15] through a projective collineation....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors introduce reversible dynamical systems, which generalise classical mechanical systems possessing time-reversal symmetry and are found in ordinary differential equations, partial differential equations and diffeomorphisms (mappings) modelling many physical problems.

279 citations


"A New Class of Integrable Maps of t..." refers background in this paper

  • ...For planar maps, integrability is equivalent to the preservation of a pencil of curves and measure-preservation [8]....

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Journal ArticleDOI
TL;DR: In this article, a systematic derivation of polynomial invariants of integrable partial differential equations is given using the associated linear spectral problems (Lax pairs), where the level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.

200 citations


"A New Class of Integrable Maps of t..." refers methods in this paper

  • ...One way to obtain such maps is by reduction from integrable lattice equations [12, 19]....

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Book
09 Jan 2016
TL;DR: This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant.
Abstract: This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Backlund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Pade approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painleve equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.

182 citations


"A New Class of Integrable Maps of t..." refers background in this paper

  • ...One clear definition of an integrable map is the notion of Liouville integrability, which requires the existence of sufficiently many invariant functions in involution with each other [6, 20]....

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