Dimension Of Inhomogeneous Sub-Self-Similar Sets

In this paper, we introduce the concept of Inhomogeneous sub-self-similar (ISSS) sets, building upon the foundations laid by Falconer (Trans. Amer. Math. Soc. 347 (1995) 3121-3129) in the study of sub-self-similar sets and drawing inspiration from Barnsley’s work on inhomogeneous self-similar sets (Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 24). We explore a range of examples of ISSS sets and elucidate a method to construct ISSS sets. We also investigate the upper and lower box dimensions of ISSS sets and discuss the continuity of the Hausdorff dimension.

💡 Research Summary

The paper introduces a new class of fractal sets called Inhomogeneous Sub‑Self‑Similar (ISSS) sets, which extend both Falconer’s sub‑self‑similar (SSS) sets and Barnsley’s inhomogeneous iterated function systems (IFS). After reviewing the classical theory of self‑similar sets, IFS, and the open set condition, the authors replace the equality in the self‑similar equation with a set inclusion and add a compact “condensation” set C, obtaining the defining relation

 F ⊆ ⋃_{i=1}^N f_i(F) ∪ C.

When C is empty the definition collapses to an SSS set; when the inclusion is an equality it reduces to an inhomogeneous self‑similar set. Thus ISSS unifies several previously studied objects.

Section 3 provides a suite of examples: trivial cases (self‑similar and SSS sets), unions of ISSS sets, augmentations by additional contractions, and boundaries of ISSS sets. Remarks emphasize that ISSS sets are generally non‑unique; any compact set can be turned into an ISSS set by choosing C appropriately, and classical inhomogeneous self‑similar sets are special cases.

The constructive core appears in Section 4. Given an SSS set E with associated code space S⊂I^∞ and a condensation set C, the authors define

 O_S = { f_ω(C) : ω∈S* }

and prove (Theorem 4.1) that E∪O_S is compact and satisfies the ISSS inclusion, hence is an ISSS set for the original IFS together with C. The proof uses the coding map Θ: I^∞→E, left‑shift invariance of S, and the contraction ratios ρ_i<1 to show closure and inclusion. Theorem 4.2 shows that O_S is actually the smallest closed set containing all images f_ω(C), i.e., O_S = E∪O_S.

Section 5 tackles dimension theory. First, the authors recall Falconer’s result for SSS sets: there exists a unique s≥0 solving τ(s)=1, where τ(s)=lim_{k→∞}(∑_{ω∈S_k}ρ_ω^s)^{1/k}. Under the open set condition dim_H E = dim_B E = s. For an ISSS set F = E∪O_S, Corollary 5.2 gives

 dim_H F = max{s, dim_H C},

since the Hausdorff dimension is countably stable and O_S has the same dimension as C.

The upper box dimension is estimated using Fraser’s δ‑covering technique. Lemma 5.3 shows that for any t>s the series Σ_{ω∈S*}ρ_ω^t converges, yielding a finite constant m_t. Lemma 5.4 bounds the number of “δ‑stopping” words S(δ) by |S(δ)| ≤ m_t δ^{-t}. Subsequent lemmas bound the number of words with ρ_ω≥δ and with ρ_ω<δ, and relate the δ‑covers of O_S to those of the whole space X. Theorem 5.7 combines these estimates to obtain

 max{dim_B E, dim_B C} ≤ dim_B F ≤ max{s, dim_B C}.

Thus the box dimension of an ISSS set is also governed by the larger of the SSS dimension s and the dimension of the condensation set.

Section 6 discusses continuity of the Hausdorff dimension with respect to variations in C, noting that the dimension varies continuously as C changes, unlike some pathological self‑similar examples where dimension jumps.

Section 7 briefly mentions product constructions for inhomogeneous IFS, hinting at possible extensions to graph‑directed or random IFS frameworks.

The conclusion summarises the contributions: a unified definition of ISSS sets, a constructive method via code space, precise dimension bounds, and observations on continuity. The paper suggests future work on measure‑theoretic properties, dynamical aspects, and numerical simulations of ISSS attractors.

Overall, the work provides a solid theoretical foundation for a broader class of fractals, blending coding‑space techniques with covering arguments, and opens avenues for further exploration in fractal geometry and dynamical systems.