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Search Results (4)

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Keywords = Billingsley dimension

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17 pages, 290 KB  
Article
Billingsley-Type Theorem of Weighted Bowen Topological Entropy for Amenable Group Actions
by Yuan Lian and Hongjun Liu
Mathematics 2025, 13(23), 3776; https://doi.org/10.3390/math13233776 - 25 Nov 2025
Viewed by 511
Abstract
Let (Xi,di) be a compact metric space with metric di, i=1,2…,k, and G be a discrete infinitely countable amenable group. This paper is based on continuous [...] Read more.
Let (Xi,di) be a compact metric space with metric di, i=1,2…,k, and G be a discrete infinitely countable amenable group. This paper is based on continuous actions G↷Xi on compact metric spaces (Xi,di). Firstly, we introduce the concept of weighted Bowen balls, and then use the concept of weighted Bowen balls to introduce the corresponding lower (upper) weighted local entropy, as well as propose the concept of weighted Bowen topological entropy defined in terms of Hausdorff dimension by weighted Bowen balls, and prove Billingsley-type theorem between these two types of entropies by using the equivalent definition of weighted Bowen topological entropy. Full article
16 pages, 319 KB  
Article
A φ-Contractivity and Associated Fractal Dimensions
by Nifeen H. Altaweel, Olayan Albalawi and Razan Albalawi
Fractal Fract. 2025, 9(10), 628; https://doi.org/10.3390/fractalfract9100628 - 26 Sep 2025
Viewed by 778
Abstract
In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function φ, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power [...] Read more.
In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function φ, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power relative to the associated measure power, and where the gauge is a set-valued function or a measure in the majority of cases, we no longer assume this hypothesis. The introduced variant generalizes many existing cases, such as Haudorff, packing, Carathéodory, and Billingsley original variants. Many characteristics of the dimensions are investigated, such as bijectivity, convexity, monotony, asymptotic behavior, and fixed points. Full article
(This article belongs to the Section General Mathematics, Analysis)
13 pages, 261 KB  
Article
Bowen Lemma in the Countable Symbolic Space
by Mingtian Li and Jihua Ma
Entropy 2017, 19(10), 532; https://doi.org/10.3390/e19100532 - 11 Oct 2017
Cited by 2 | Viewed by 3770
Abstract
We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent [...] Read more.
We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent of the Gibbs measure, are given by a variational principle, which generalizes Li and Ma’s result and Bowen’s result. Full article
(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications)
30 pages, 328 KB  
Article
The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
by Victor Bakhtin
Entropy 2014, 16(12), 6624-6653; https://doi.org/10.3390/e16126624 - 19 Dec 2014
Cited by 2 | Viewed by 5469
Abstract
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of [...] Read more.
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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