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Transversal family of non-autonomous conformal iterated function systems (2308.13213v2)

Published 25 Aug 2023 in math.DS

Abstract: We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset $X\subset\mathbb{R}m$ is a sequence $\Phi=({\phi{(j)}{i}}{i\in I{(j)}})_{j=1}{\infty}$ of collections of uniformly contracting maps $\phi{(j)}_{i}: X\rightarrow X$, where $I{(j)}$ is a finite set. In comparison to usual iterated function systems, we allow the contractions $\phi{(j)}_{i}$ applied at each step $j$ to depend on $j$. In this paper, we focus on a family of parameterized NIFSs on $\mathbb{R}m$. 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 ${\Phi_t}{t\in U}$ of parameterized NIFSs such that ${\Phi_t}{t\in U}$ satisfies the transversality condition but $\Phi_t$ does not satisfy the open set condition for any $t\in U$.

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