Diophantine property of matrices and attractors of projective iterated function systems in \mathbbRP¹

2019/02/28 by Solomyak, Boris, Takahashi, Yuki · 1 citation
#11K60 #15B48 #28A80 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.11059

Abstract

We prove that almost every finite collection of matrices in GL_d(ℝ) and SL_d(ℝ) with positive entries is Diophantine. Next we restrict ourselves to the case d=2. A finite set of SL₂(ℝ) matrices induces a (generalized) iterated function system on the projective line \mathbbRP¹. 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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• On exponential separation of analytic self-conformal sets on the real line

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