Growth of values of binary quadratic forms and Conway rivers

2018-04-06T13:24:31Z (GMT) by Kathryn Spalding Alexander Veselov
We study the growth of the values of integer 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 2 RP1 are twice the values of the corresponding exponents for the growth of Markov numbers [10], except for the paths corresponding to the Conway river, when _Q(x) = 0: The relation with the Galois result about pure periodic continued fractions is explained and interpreted geometrically.