Abstract
We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c ∗ > 0 .
Highlights
Let us begin with the definition of the Assouad dimension and the lower dimension
Definition 1. e Assouad dimension of a nonempty set F ⊆ Rd is defined by dimAF inf{α ≥ 0. ere exists a constant c > 0 such that, for any 0 < r < R, and x ∈ F, Nr(B(x, R) ∩ F) ≤ c(R/r)α}
If the Hausdorff dimension provides fine, but global, geometric information, the Assouad dimension which was introduced by Assouad [1] provides coarse, but local, geometric information. e Assouad dimension is a fundamental notion of dimension used to study fractal objects in a wide variety of contexts
Summary
Received 17 July 2020; Revised 22 August 2020; Accepted 12 September 2020; Published 23 September 2020. We consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c∗ > 0
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