Some Dimensional Results of a Class of Homogeneous Moran Sets

Jingru Zhang,Yanzhe Li,Manli Lou

doi:10.1155/2021/6965671

Abstract

In this paper, we construct a class of special homogeneous Moran sets: m k -quasi-homogeneous perfect sets, and obtain the Hausdorff dimension of the sets under some conditions. We also prove that the upper box dimension and the packing dimension of the sets can get the maximum value of the homogeneous Moran sets under the condition sup k ≥ 1 m k < ∞ and estimate the upper box dimension of the sets in two cases.

Highlights

Introduction e fractal dimensions of theMoran sets have been studied by many authors and close connected with many subjects, such as the multifractals, the quasiconformal mappings, the ergodic theory, and the number theory
Let E ∈ M(I, 􏼈nk􏼉, 􏼈ck􏼉), and we say that J 􏼈Iσ: σ ∈ D􏼉 satisfies homogeneous perfect structure if it satisfies the following: there exists a sequence of real numbers 􏽮ηk,j: k ≥ 1, 0 ≤ j ≤ nk􏽯 such that, for any k ≥ 1, 0 ≤ j ≤ nk, we have ηk,j ≥ 0, and for any k ≥ 0, σ ∈ Dk, and 1 ≤ i ≤ nk+1 − 1, we have min(Iσ∗1) − min(Iσ) ηk+1,0, min(Iσ∗(i+1))− max(Iσ∗i) ηk+1,i, and max(Iσ ) − max(Iσ∗nk+1) ηk+1,nk+1
If there exist positive constants c􏽥1, c􏽥2(c􏽥2 ≥ 1) such that at least one of the following two conditions is satisfied for any k ≥ 1: (A) max1≤l≤mk− 1η∗k,l ≤ c􏽥1 · c1c2, . . . , ck. (B) max1≤l≤mk− 1η∗k,l ≤ c􏽥2 · min1≤l≤mk− 1η∗k,l; dimHE

Summary

Introduction

Introduction e fractal dimensions of theMoran sets have been studied by many authors and close connected with many subjects, such as the multifractals (see [1,2,3]), the quasiconformal mappings (see [4, 5]), the ergodic theory (see [6]), and the number theory (see [7]). Let E ∈ M(I, 􏼈nk􏼉, 􏼈ck􏼉), and we say that J 􏼈Iσ: σ ∈ D􏼉 satisfies homogeneous perfect structure if it satisfies the following: there exists a sequence of real numbers 􏽮ηk,j: k ≥ 1, 0 ≤ j ≤ nk􏽯 such that, for any k ≥ 1, 0 ≤ j ≤ nk, we have ηk,j ≥ 0, and for any k ≥ 0, σ ∈ Dk, and 1 ≤ i ≤ nk+1 − 1, we have min(Iσ∗1) − min(Iσ) ηk+1,0, min(Iσ∗(i+1))− max(Iσ∗i) ηk+1,i, and max(Iσ ) − max(Iσ∗nk+1) ηk+1,nk+1.

Results

Conclusion