Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

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 SL 2 (R) matrices induces a (generalized) iterated function system on the projective line 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.

The Hausdorff dimension of self-projective sets

“…In principle, K A may be empty or the whole of RP 1 . The attractor will be the main object of study in this paper and our main result is a generalisation of a recent result of Solomyak and Takahashi [19,Theorem 1.7] concerning its Hausdorff dimension. Before stating our main result, we introduce the two key assumptions that we will need to make on our set A.…”

“…As we mentioned earlier, Theorem 1.4 is a generalisation of the following theorem due to Solomyak and Takahashi [19,Theorem 1.7].…”

“…Given a finite set A ⊆ SL(2, R) we call the collection of maps Φ A := {φ A } A∈A a projective iterated function system (projective IFS). Iterated function systems on real projective space have been previously studied in [4,7,8,19] and on complex projective space in [20].…”

“…It is clear that if A is Diophantine then it generates a free semigroup. In [13,19] the authors also consider a weaker version of this definition with the freeness of A * removed, but we do not require this more general form.…”

“…In order to properly describe the contribution of Theorem 1.4, however, we must first review the literature which is most relevant to our problem. To this end, we introduce the notion of uniform hyperbolicity following [4,7,19]. Definition 1.5.…”