Hereditarily non uniformly perfect sets

Abstract

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give a detailed construction of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

Hereditarily non uniformly perfect non-autonomous Julia sets

“…Selecting a subsequence a kn → 0, the reader can quickly check that inf n {b kn } > 0, inf n {δ kn } > 0, and η kn = a kn · diam(X) → 0, and thus Corollary 1.1 applies (since Φ clearly satisfies the Strong Separation Condition). We also note that when lim inf a k = 0, Corollary 1.1 shows J is pointwise thin even when the strict setup above is considerably relaxed (e.g., the sets ϕ Lastly, note that Theorem 4.1(2) of [8] shows that J is uniformly perfect when lim inf a k > 0, which also follows from the more general Theorem 2.1 of [3]…”

“…The following well-known lemma (see, e.g., Theorem 2.1 of [5]) often allows one to replace a conformal annulus with an easier to work with round annulus. The concept of hereditarily non uniformly perfect was introduced in [8] and can be thought of as a thinness criterion for sets which is a strong version of failing to be uniformly perfect. Often a compact set is shown to be HNUP by showing it satisfies the following stronger property of pointwise thinness.…”

“…Often a compact set is shown to be HNUP by showing it satisfies the following stronger property of pointwise thinness. This is done in several examples in [8,2,3], and will be done in the proof of Corollary 1.1.…”

“…In particular, The following result follows from Lemma 1.4 and the fact that locally non-constant analytic maps are either conformal or behave like z → z k for some k ∈ N, which can distort the modulus of an annulus by at most a factor of k. We leave the details to the reader noting, however, that one may follow the style of argument used to prove Proposition 3.1 of [3]. [8] shows that J pointwise thin (and thus HNUP) when lim inf a k = 0. We now show that this also follows from Corollary 1.1.…”

Uniformly perfect analytic and conformal non-autonomous attractor sets

“…We also note that [7] includes related results for similar systems (which require an open set condition). Certain constructions in [15] are non-autonomous iterated function systems shown to have uniformly perfect attractors (though those examples were not presented as attractors, but rather as Cantor-like constructions -see Example 4.1 in this paper), while other examples there are not uniformly perfect. We look to generalize those results here, and we begin by following [12] to introduce the main framework and definitions (with some key differences) of non-autonomous iterated function systems (NIFS's).…”