International Mathematics Research Notices ( IF 1.452 ) Pub Date : 2020-02-10 , DOI: 10.1093/imrn/rnz309
Solomyak B, Takahashi Y.

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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