ULTRADISCRETIZATION OF A SOLVABLE TWO-DIMENSIONAL CHAOTIC MAP ASSOCIATED WITH THE HESSE CUBIC CURVE

Kenji KAJIWARA; Masanobu KANEKO; Atsushi NOBE; Teruhisa TSUDA

doi:10.2206/kyushujm.63.315

Kenji KAJIWARA, Masanobu KANEKO, Atsushi NOBE, Teruhisa TSUDA

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Keywords: ultradiscretization, tropical geometry, theta functions, plane cubic curve, discrete dynamical systems

JOURNAL FREE ACCESS

2009 Volume 63 Issue 2 Pages 315-338

DOI https://doi.org/10.2206/kyushujm.63.315

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Abstract

We present a solvable two-dimensional piecewise linear chaotic map that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as ‘the minus-sign problem.’

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