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Discrete Integrable Systems: QRT Maps and Elliptic Surfaces
TLDR
The QRT Map as discussed by the authors is the pencil of biquadratic curves in the projective plane of the QRT surface and is used to measure the distance between two points.Abstract:
The QRT Map.- The Pencil of Biquadratic Curves in .- The QRT surface.- Cubic Curves in the Projective Plane.- The Action of the QRT Map on Homology.- Elliptic Surfaces.- Automorphisms of Elliptic Surfaces.- Elliptic Fibrations with a Real Structure.- Rational elliptic surfaces.- Symmetric QRT Maps.- Examples from the Literature.- Appendices.read more
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Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks
Thomas Dreyfus,Kilian Raschel +1 more
TL;DR: In this article, the authors consider weighted small step walks in the positive quadrant and provide algebraicity and differential transcendence results for the underlying generating functions, which involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves).
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Integrability and non-integrability of periodic non-autonomous Lyness recurrences
TL;DR: In this paper, the authors studied non-autonomous Lyness-type recurrences of the form x n+2 = (a n + X n+1)/x n, where {a n } is a k-periodic sequence of positive numbers with primitive period k. They showed that for the cases k ∈ {1, 2, 3, 6, 6}, the behaviour of the sequence {x n} is simple (integrable), while for the remaining cases satisfying this behaviour can be much more complicated (chaotic).
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A note on minimization of rational surfaces obtained from birational dynamical systems
TL;DR: In this article, the authors proposed a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure, and applied this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.
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Non-integrability of measure preserving maps via Lie symmetries
TL;DR: In this article, the authors considered the problem of characterizing the local C^m-non-integrability near elliptic fixed points of smooth planar preserving maps, and derived a criterion that relates this non-Integrability with the existence of some Lie Symmetries associated to the maps, together with the finiteness of its periodic points.
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Length derivative of the generating series of walks confined in the quarter plane
TL;DR: In this article, the authors use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane, where the variables $x and $y$ are associated to the ending point of the path, while the variable $t$ encodes its length.