Recherches autour de la theorie de Markoff

Details Recherches autour de la theorie de Markoff Recherches autour de la theorie de Markoff

2003-07-16

arxiv.org/abs/math-ph/0307032v1

Physics

Mathematical Physics

155 pages. Compact version of a text published as La theorie de Markoff et ses developpements, by S. Perrine, Tessier&Ashpool

Scientific paper

The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic geometry. This algebraic approach is completed with an analytical development concerning fuchsian groups. The link with the Teichmuller theory for punctured toruses is completely described, giving their classification with a reduction theory. More general considerations about Riemann surfaces, geodesics and their hamiltonian study are quoted, together with applications in physics, 1/f noise and zeta function. Ideas about important conjectures are presented. Reasons why the Markoff theory appears in different geometrical contexts are given, thanks to decomposition results in the group GL(2,Z).

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