# Diophantine property of matrices and attractors of projective iterated function systems in $\mathbb{RP}^1$

## TL;DR

This paper demonstrates that most finite sets of matrices with positive entries are Diophantine and explores the dimension of attractors of projective iterated function systems in the case of $SL_2(
)$ matrices, linking hyperbolicity and Diophantine properties.

## Contribution

It establishes the Diophantine property for almost all finite matrix collections and relates it to the dimension of attractors in projective iterated function systems for $d=2$.

## Key findings

- Almost every finite collection of matrices in $GL_d(
)$ and $SL_d(
)$ with positive entries is Diophantine.
- Under hyperbolicity and Diophantine assumptions, the attractor's dimension equals the minimum of 1 and the critical exponent.
- The work connects Diophantine properties with fractal dimensions of attractors in projective dynamics.

## Abstract

We prove that almost every finite collection of matrices in $GL_d(\mathbb{R})$ and $SL_d(\mathbb{R})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2(\mathbb{R})$ matrices induces a (generalized) iterated function system on the projective line $\mathbb{RP}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.