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arXiv · 1902.11059

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

Boris Solomyak·Yuki Takahashi
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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.

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Boris Solomyak, Yuki Takahashi. 2019-02-28. Diophantine property of matrices and attractors of projective iterated function systems in $\mathbb{RP}^1$. https://arxiv.org/abs/1902.11059

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