A new approach to metrization

Arenas, F. G.; Sánchez-Granero, M.A.

doi:10.1016/s0166-8641(01)00165-1

Cited by 12 publications

(9 citation statements)

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“…The main tool we provide in order to describe fractals from a rigorous point of view is the concept of fractal structure, which was first sketched in [26], and then used in [27] to characterize non-archimedean quasimetrization, though a more natural use of it is in the study of fractals. A fractal structure is just a countable family of coverings, called levels, of a given set which approaches the space.…”

Section: Exploring the Complexity Of Curves Using Fractal Structuresmentioning

confidence: 99%

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

2012

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Abstract. In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of short length series, the introduced algorithms perform much better than the classical methods. Finally, an empirical application for some stock market indexes as well as some individual stocks is presented.

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Section: Exploring the Complexity Of Curves Using Fractal Structuresmentioning

confidence: 99%

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

2012

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“…Anyway, in this paper we will only use the natural fractal structure on a Euclidean space (which will be defined later) which induces the Euclidean topology. Note that any metric space admits a compatible fractal structure (see [2,Theorem 4.1] or [3,Theorem 3.18]). …”

Section: Definition 21mentioning

confidence: 99%

How to calculate the Hausdorff dimension using fractal structures

Fernández–Martínez

Sánchez-Granero

2015

Applied Mathematics and Computation

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“…It is worth mentioning that if Γ is a fractal structure on X, and St(x, Γ) is a neighborhood base of x, for all x ∈ X, then we will call Γ a starbase fractal structure. Starbase fractal structures are connected to metrizability (see [2,3]). A fractal structure Γ is said to be finite, if all levels Γ n are finite coverings.…”

Section: Preliminariesmentioning

confidence: 99%

Theoretical properties of fractal dimensions for fractal structures

Fernández–Martínez¹

2015

Discrete &Amp; Continuous Dynamical Systems - S

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Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdorff model pretty desirable from a theoretical point of view. On the other hand, it turns out that fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.

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A new approach to metrization

Cited by 12 publications

References 12 publications

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

How to calculate the Hausdorff dimension using fractal structures

Theoretical properties of fractal dimensions for fractal structures

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