Hausdorff dimension of the limit sets of Tree Iterated Function Systems

We investigate Tree Iterated Function Systems (TIFSs), which we introduce in this paper. TIFSs are the generalizations of Iterated Function Systems in which we take the maps independently at each step and each block. In this paper, we give the definition of TIFSs and the limit sets of them. We show a formula for the Hausdorff dimension of the limit sets of TIFSs, which is a generalization of Bowen’s formula. Moreover, we give an example which emphasizes the difference between TIFSs and non-autonomous IFSs.

💡 Research Summary

The paper introduces Tree Iterated Function Systems (TIFS), a broad generalization of classical Iterated Function Systems (IFS). In a TIFS, a rooted, finitely‑splitting, pruned tree T indexes a family of maps {ϕ_τ : X→X} with τ ranging over the non‑root vertices. Each map is uniformly contractive with a common factor s∈(0,1), and the limit set J is defined as the set of points obtained by infinite compositions along any infinite path in T. This construction removes the usual self‑similarity, allowing the modeling of fractal‑like objects that lack exact scaling invariance, such as natural coastlines or random clouds.

The authors focus on Conformal TIFS (CTIFS), imposing additional regularity: the ambient space X lies in ℝ^d, each ϕ_τ extends to a C¹ diffeomorphism on a common open set, the maps are conformal, satisfy an open set condition, a cone condition at the boundary, and a bounded distortion property. These hypotheses guarantee that the derivatives ‖Φ′_ω‖ (where Φ_ω denotes the composition along a finite word ω) behave in a controlled way, which is essential for dimension estimates.

A central novelty is the use of maximal antichains in the tree. For a fixed depth n, a maximal antichain A ⊂ T(≤n) is a set of vertices none of which are comparable, yet every vertex at level n lies below some element of A. The authors define a pressure‑type quantity \