814 research outputs found
Goldfishing by gauge theory
A new solvable many-body problem of goldfish type is identified and used to
revisit the connection among two different approaches to solvable dynamical
systems. An isochronous variant of this model is identified and investigated.
Alternative versions of these models are presented. The behavior of the
alternative isochronous model near its equilibrium configurations is
investigated, and a remarkable Diophantine result, as well as related
Diophantine conjectures, are thereby obtained.Comment: 22 page
Yang-Baxter maps: dynamical point of view
A review of some recent results on the dynamical theory of the Yang-Baxter
maps (also known as set-theoretical solutions to the quantum Yang-Baxter
equation) is given. The central question is the integrability of the transfer
dynamics. The relations with matrix factorisations, matrix KdV solitons,
Poisson Lie groups, geometric crystals and tropical combinatorics are discussed
and demonstrated on several concrete examples.Comment: 24 pages. Extended version of lectures given at the meeting
"Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004
Dunkl operators at infinity and Calogero-Moser systems
We define the Dunkl and Dunkl-Heckman operators in infinite number of
variables and use them to construct the quantum integrals of the
Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple
proof of integrability of the deformed quantum CMS systems related to classical
Lie superalgebras. We show how this naturally leads to a quantum version of the
Moser matrix, which in the deformed case was not known before.Comment: 22 pages. Corrected version with minor change
Tropical Markov dynamics and Cayley cubic
We study the tropical version of Markov dynamics on the Cayley cubic,
introduced by V.E. Adler and one of the authors. We show that this action is
semi-conjugated to the standard action of on a torus, and
thus is ergodic with the Lyapunov exponent and entropy given by the logarithm
of the spectral radius of the corresponding matrix.Comment: Extended version, accepted for publication in "Integrable Systems and
Algebraic Geometry" (Editors: R. Donagi, T. Shaska), Cambridge Univ. Press:
LMS Lecture Notes Series, 201
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