# Growth of values of binary quadratic forms and Conway rivers

## TL;DR

This paper investigates the growth rates of binary quadratic form values along Conway's tree, revealing a connection to Markov numbers and identifying special paths where growth halts, with implications for Galois theory and continued fractions.

## Contribution

It establishes a relationship between Lyapunov exponents of quadratic forms and Markov number growth, highlighting the unique behavior along Conway rivers.

## Key findings

- Lyapunov exponents are twice those for Markov number growth.
- Along Conway rivers, the Lyapunov exponent is zero.
- The results connect quadratic form growth to Galois theory and continued fractions.

## 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 $x\in \mathbb RP^1$ 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.