Growth of values of binary quadratic forms and Conway rivers
Abstract
We study the growth of the values of binary quadratic forms Q on a binary planar tree as it was described by Conway. We show that the corresponding Lyapunov exponents Q(x) as a function of the path determined by x∈ RP1 are twice the values of the corresponding exponents for the growth of Markov numbers SV, except for the paths corresponding to the Conway rivers, when Q(x)=0. The relation with Galois results about continued fraction expansions for quadratic irrationals is explained and interpreted geometrically.
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