Misure di Hausdorff e applicazioni

Obiettivo della tesi è fornire nozioni di teoria della misura tramite cui è possibile l'analisi e la descrizione degli insiemi frattali. A tal fine vengono definite la Misura e la Dimensione di Hausdorff, strumenti matematici che permettono di "misurare" tali oggetti particolari, per i quali la classica Misura di Lebesgue non risulta sufficientemente precisa. Viene introdotto, inoltre, il carattere di autosimilarità, comune a molti di questi insiemi, e sono forniti alcuni tra i più noti esempi di frattali, come l'insieme di Cantor, l'insieme di Mandelbrot e il triangolo di Sierpinski. Infine, viene verificata l'ipotesi dell'esistenza di componenti di natura frattale in serie storiche di indici borsistici e di titoli finanziari (Ipotesi dei Mercati Frattali, Peters, 1990).

Misura di Hausdorff e frattali

In questa tesi sono presentate la misura e la dimensione di Hausdorff, gli strumenti matematici che permettono di descrivere e analizzare alcune delle più importanti proprietà degli insiemi frattali. Inoltre viene introdotto il carattere di autosimilarità, comune a questi insiemi, e vengono mostrati alcuni tra i più noti esempi di frattali, come l'insieme di Cantor, la curva di Koch, l'insieme di Mandelbrot e gli insiemi di Julia. Di quest'ultimi sono presenti immagini ottenute tramite un codice Matlab.

La matematica dei frattali

Questa tesi ha lo scopo di descrivere le proprietà matematiche degli insiemi frattali. Nell'introduzione è spiegato brevemente cosa sono i frattali e vengono fatti alcuni esempi di frattali in natura, per poi passare agli aspetti più matematici nei capitoli. Nel capitolo uno si parla della misura e della dimensione di Hausdorff e viene calcolata, seguendo la definizione, per l'insieme di Cantor. Poi nel secondo capitolo viene descrittà l'autosimilarità e viene enunciato un importante teorema che lega l'autosimilarità e la dimensione di Hausdorff. Nel terzo capitolo vengono descritti degli insiemi frattali molto importanti: quelli di Mandelbrot e di Julia.

Estimating the Hausdorff-Besicovitch dimension of boundary of basin of attraction in helicopter trim

Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geometries. These results led the authors to investigate the fractal dimension of the basin boundary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature. (C) 2012 Elsevier Inc. All rights reserved.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Mandelbrot集与Julia集的图象生成算法

介绍一种在计算机上生成Ｍａｎｄｅｌｂｒｏｔ集和Ｊｕｌｉａ集图象的简易算法。该算法对计算机的软硬件要求均不高，在普通的微机以及工作站上均可实现。

La geometría como matemática aplicada en su evolución histórica: de Euclides a Mandelbrot

En la presente contribución intentamos evidenciar cómo la geometría a lo largo de toda su historia ha desempeñado un papel fundamental interactivo con la ciencia natural, en particular con la física, y más en concreto aún con la mecánica. En la primera parte esbozamos nuestra visión de esta intima interrelación desde el alba de la geometría en China, Mesopotamia y Egipto hasta nuestros días.

La geometría de la naturaleza: Benoit Mandelbrot

Este matemático polaco francés norteamericano gozó siempre de una gran reputación, que se acrecentó con el redescubrimiento de conceptos que condujeron a la dimensión fractal y a los fractales. ¿Existe una matemática que modela de manera acertada a ciertos procesos de la naturaleza? ¿Es sencilla esta matemática?

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a Cr diffeomorphism f of a surface, are not C1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

On Chaos and Fractals in General Topological Spaces

The present study on chaos and fractals in general topological spaces. Chaos theory originated with the work of Edward Lorenz. The phenomenon which changes order into disorder is known as chaos. Theory of fractals has its origin with the frame work of Benoit Mandelbrot in 1977. Fractals are irregular objects. In this study different properties of topological entropy in chaos spaces are studied, which also include hyper spaces. Topological entropy is a measures to determine the complexity of the space, and compare different chaos spaces. The concept of fractals can’t be extended to general topological space fast it involves Hausdorff dimensions. The relations between hausdorff dimension and packing dimension. Regular sets in Metric spaces using packing measures, regular sets were defined in IR” using Hausdorff measures. In this study some properties of self similar sets and partial self similar sets. We can associate a directed graph to each partial selfsimilar set. Dimension properties of partial self similar sets are studied using this graph. Introduce superself similar sets as a generalization of self similar sets and also prove that chaotic self similar self are dense in hyper space. The study concludes some relationships between different kinds of dimension and fractals. By defining regular sets through packing dimension in the same way as regular sets defined by K. Falconer through Hausdorff dimension, and different properties of regular sets also.

Aula de Teatre: El conjunt de Mandelbrot. Preguntes, preguntes i més preguntes

L'Aula de Teatre de la UdG va estrenar el 22 de maig, al Teatre Municipal, l'obra El conjunt de Mandelbrot, una dramatúrgia de Jordi Duran i del matemàtic David Juher que aboca sense complexos la ciència damunt de l'escenari

On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

We study the topology of a set naturally arising from the study of β-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of β-expansions are equal and explicitly calculable.

Sparse block-Jacobi matrices with arbitrarily accurate Hausdorff dimension

We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i.e. nonrandom, block-Jacobi matrices may be determined with any degree of precision, improving a result of Zlatos [Andrej Zlatos,. Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004) 216-252]. (C) 2010 Elsevier Inc. All rights reserved.

WHICH TOPOLOGIES HAVE IMMEDIATE PREDECESSORS IN THE POSET OF HAUSDORFF TOPOLOGIES?

We study which topology have an immediate predecessor in the poset of Sigma(2) of Hausdorff topologies on set X. We show that certain classes of H-closed topologies, do have predecessors. and we give examples of second countable H-closed topologies which are not upper Sigma(2.)