Mendès France and Thermodynamical Spectra: A Comparative Study of Contractive and Expansive Fractal Processes (original) (raw)

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

Contemporary Mathematics, 2013

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Thermodynamics of fractal spectra: Cantor sets and quasiperiodic sequences

Physical Review E, 2000

We study the properties of the specific heat derived from fractal spectra, for which we extend and generalize some previous known results concerning the log-periodic oscillations of the specific heat C(T). For the monoscale case, we obtain analytically the behavior of C(T) for a two-branch general spectrum, and we show that the oscillatory regime becomes nonharmonic if there exist different gap sizes. In the multiscale case, we connect the role of the spectral dimension as the average value of C(T) with the multifractal properties of the sets, and we give a condition for which the oscillatory regime disappears. Finally, we study the thermodynamics of tight-binding Fibonacci spectra, which are not strictly invariant under changes of scale, and then many of the properties found in Cantor sets become in this case just approximated.

Fractal geometry and dynamical systems in pure and applied mathematics. 1. Fractals in pure mathematics

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Assouad-type spectra for some fractal families

Indiana University Mathematics Journal, 2018

In a previous paper we introduced a new 'dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers.

Comparison of multifractal and thermodynamical properties of fractal and natural spectra

Physica A: Statistical Mechanics and its Applications, 2000

In this paper, we explain the multifractal and thermodynamical properties of deterministic fractal spectra, and compare the most relevant behaviors with those obtained for some natural spectra with multifractal properties. In particular, we compare the heat capacity, C(T ), derived from Cantor-set-type spectra with the ones obtained from several atomic spectra. We ÿnd that, although many of the properties of deterministic fractals, previously reported, are also reproduced in natural fractals (oscillations of C(T ) in log scale), new features appear in the natural case: the mean value of C(T ) is not always related to the multifractal properties, and the oscillations are present in cases for which they disappear in deterministic fractals.

Dynamical characterization of mixed fractal structures

Journal of Mechanics of Materials and Structures, 2011

It is because of people like Marie-Louise and Charles that it is worth fighting for a better world. We present a new technique to determine the fractal or self-similarity dimension of a sequence of curves. The geometric characterization of the sequence is obtained from the mechanical properties of harmonic oscillators with the same shape of the terms composing the given sequence of curves. The definition of "dynamical dimension" is briefly introduced with the help of simple examples. The theory is proved to be valid for a particular type of curves as those of the Koch family. The method is applied to more complex plane curves obtained by superposing two generators of the Koch family with different fractal dimensions. It is shown that this structure is composed by two series of objects one of which is fractal and the other which is not rigorously a fractal sequence but approaches asymptotically a fractal object. The notion of quasifractal structures is introduced. The results are shown to provide good information about the structure formation. It is shown that the dynamical dimension can identify randomness for certain fractal curves. Research project partially funded by CNPq (Brazil) and FAPERJ..

The exact Hausdorff dimension for a class of fractal functions

Journal of Mathematical Analysis and Applications, 1992

  1. have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran's original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].

Fractals, Multifractals, and Thermodynamics

Zeitschrift für Naturforschung A, 1988

The basic concept of fractals and multifractals are introduced for pedagogical purposes, and the present status is reviewed. The emphasis is put on illustrative examples with simple mathematical structures rather than on numerical methods or experimental techniques. As a general characteriza­tion of fractals and multifractals a thermodynamical formalism is introduced, establishing a connec­tion between fractal properties and the statistical mechanics of spin chains.

A new fractal dimension for curves based on fractal structures

In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct space-filling curves. We explore and analyze the behavior of this new fractal dimension compared to classical models for fractal dimension, namely, both the Hausdorff dimension and the box-counting dimension. This analytical study is illustrated through some examples of space-filling curves, including the classical Hilbert's curve. Finally, we contribute some results linking this fractal dimension approach with the self-similarity exponent for random processes.

Fractal sets associated with functions: The spectral lines of hydrogen

Physical review, 1988

There is a strong feeling among many researchers that certain physical phenomena are fractal in nature. The difficulty, in any given case, is to make precise what one means when one says that something is fractal. An example might help. The Cantor middle third set, discussed in detail here, is a fractal in the sense that its Hausdor6'(now often called the fractal) dimension is between zero and 1; is it ln2/ln3, in fact. Now the spectral lines of the hydrogen atom seem to be fractal also, just from a casual observation of the self-similar nature of the various series. They certainly do not have all the properties of the Cantor set and so, we may well ask, in what sense are they fractal? The purpose of this paper is to give one answer to this question.