Lower dimensions of some fractal sets

Chen, Haipeng; Wu, Min; Wei, Chun

doi:10.1016/j.jmaa.2017.06.009

Cited by 5 publications

(4 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It follows from the dimension formulae of Cantor sets [2,4] that for any θ ∈ (0, 1), we get . Let δ k be the maximal length of k-th basic interval, then we have…”

Section: 3mentioning

confidence: 97%

“…To avoid some 'strange' sets whose lower Assouad type dimensions are 0 or ∞, we are mainly concerned with the uniformly perfect sets in doubling metric spaces. 2 In this paper, we study the behaviours of lower Assouad spectrum as θ ∈ (0, 1), and discuss the relationship among the lower Assouad type dimensions. Our first result is stated as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the lower Assouad type dimensions can give an insight into the fractal sets having a certain degree of homogeneity, like self-similar sets, self-affine sets, etc. More discussion of lower Assouad type dimensions of fractal sets can be found in [2,5,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Chen

Chang

2020

Fractals

Self Cite

View full text Add to dashboard Cite

In this paper, we are concerned with the relationship among the lower Assouad type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as θ tends to 1 equals to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as θ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi lower Assouad dimension.

show abstract

“…It follows from the dimension formulae of Cantor sets [2,4] that for any θ ∈ (0, 1), we get . Let δ k be the maximal length of k-th basic interval, then we have…”

Section: 3mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Chen

Chang

2020

Fractals

Self Cite

View full text Add to dashboard Cite

show abstract

“…In general, since the lower dimension does not have the monotone property and the finitely stable property, it is alway difficult to do the study of the lower dimensions for fractal sets. Chen, Wu, Wei [15] discussed the lower dimensions for some Moran sets. Chen [10] , Yang, Li and Hu [16] obtained some results of the lower dimension formulas for some Moran cut-out sets and some homogeneous Moran sets.…”

Section: Introductionmentioning

confidence: 99%

Assouad dimensions of Moran sets and Cantor-like sets

Miao

et al. 2016

Front. Math. China

View full text Add to dashboard Cite

We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set introduced in [15], we also study the Assouad dimensions of Cantor-like sets.where B(x, r) is the closed ball centered at x with radius r and | · | denotes the diameter of set. Olsen [20] proved that for a class of fractals with some flexible graphdirected construction, their Assouad dimensions coincide with their Hausdorff and box dimensions. He also pointed out that the fractals in [20] are Ahlfors-David regular. It is well known that self-similar sets and self-conformal sets satisfying 2000 Mathematics Subject Classification. 28A80.

show abstract

Assouad dimensions and spectra of Moran cut-out sets

Chen

2019

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

Lower dimensions of some fractal sets

Cited by 5 publications

References 8 publications

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

Assouad dimensions of Moran sets and Cantor-like sets

Assouad dimensions and spectra of Moran cut-out sets

Contact Info

Product

Resources

About