Fractal dimension for fractal structures: Applications to the domain of words

How to calculate the Hausdorff dimension using fractal structures

“…Additionally, these new definitions of fractal dimension for a fractal structure can be used in empirical applications and they also designed some algorithms to computationally calculate them. For example, the so-called fractal dimension I was applied in the context of the domain of words, which allows us to calculate the fractal dimension of a language generated by a regular expression and even to estimate the complexity of a board game configuration of Othello [15]. But while the box-counting dimension has been widely used in applications (such as disease diagnosis [30], study of earthquakes [20] and analysis of the human retina [22], to name a few), the Hausdorff dimension, which presents the best analytical properties, still results hard to calculate or to estimate in empirical applications.…”

“…They have interesting analytical properties and can be calculated with easiness. Further, some interdisciplinary applications of these models have been successfully carried out (see [15,32]). A whole study about fractal dimensions for fractal structures including applications can be found in [16].…”

“…Since then, both the study and the identification of fractal patterns have become more and more important due to the large number of applications to diverse scientific fields where fractals have been found, including computation, physics, economics and statistics among them (see [10,11,15,32]). …”

A new topological indicator for chaos in mechanical systems

“…In this way, some definitions of fractal dimension for a fractal structure have been already studied in [7][8][9]11] and applied in [10,23], to quote some of them.…”

Counterexamples in theory of fractal dimension for fractal structures

“…In this way, some theoretical results have been shown to generalize the classical models in the context of fractal structures (see, e.g., [13,Theorem 4.15] and [15,Theorem 3.12]). Moreover, we would like to point out that some fractal dimensions for a fractal structure have been already succesfully applied in non-Euclidean contexts, where the box dimension cannot be applied (see, e.g., [18,19]).…”