On statistical convergence in fractal analysis
AIMS Mathematics, cilt.10, sa.8, ss.18197-18215, 2025 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 10 Sayı: 8
- Basım Tarihi: 2025
- Doi Numarası: 10.3934/math.2025812
- Dergi Adı: AIMS Mathematics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Directory of Open Access Journals
- Sayfa Sayıları: ss.18197-18215
- Anahtar Kelimeler: Banach fixed point theorem, box dimension, fractal analysis, Hausdorff distance, statistical Cauchy sequences, statistical convergence of sets
- Süleyman Demirel Üniversitesi Adresli: Evet
Özet
This study investigated the statistical convergence of fractal-generating set sequences, motivated by the observation that natural fractals, influenced by external biological, chemical, or physical factors, rarely exhibit strict classical convergence. Instead, their limiting behavior often aligns with statistical patterns. We formalized the concept of statistical convergence for compact subsets of Rn, introduced the notion of statistical Cauchy sequences, and established their sufficiency for statistical convergence—mirroring the classical relationship. Several illustrative examples and graphical simulations, including variants of the Sierpiński triangle and Koch snowflake, highlight the distinction between classical and statistical convergence. The proposed framework provides a more realistic and robust approach to understanding fractal structures in both theoretical and applied contexts.