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 ${\Lambda }_Q(x)$ as a function of the path determined by $x\in \mathbb{R}{P}^1$ are twice the values of the corresponding exponents for the growth of Markov numbers \cite{SV}, except for the paths corresponding to the Conway rivers, when ${\Lambda }_Q(x)=0.$ The relation with Galois results about continued fraction expansions for quadratic irrationals is explained and interpreted geometrically.

A few typos and the claim in Proposition 5 about semidefinite case are corrected

References in corpus (2)

• The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms
• Lyapunov spectrum of Markov and Euclid trees

Cited by in corpus (2)

• The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms
• Conway river and Arnold sail