Transversal family of non-autonomous conformal iterated function systems

2023/08/25 by Nakajima, Yuto
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.13213

Abstract

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset X⊂ℝ^m is a sequence Φ=(\ϕ^(j)_i\_{i∈ I^(j)})_j=1^∞ of collections of uniformly contracting maps ϕ^(j)_i: X→ X, where I^(j) is a finite set. In comparison to usual iterated function systems, we allow the contractions ϕ^(j)_i applied at each step j to depend on j. In this paper, we focus on a family of parameterized NIFSs on ℝ^m. Here, we do not assume the open set condition. We show that if a d-parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of m and the Bowen dimension. Moreover, we give an example of a family \Φ_t\_{t∈ U} of parameterized NIFSs such that \Φ_t\_{t∈ U} satisfies the transversality condition but Φ_t does not satisfy the open set condition for any t∈ U.

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