Topic
Mandelbox
About: Mandelbox is a research topic. Over the lifetime, 308 publications have been published within this topic receiving 4471 citations.
Papers published on a yearly basis
Papers
Posted Content•
TL;DR: The multifractal model of asset returns (MMARCH) as mentioned in this paper is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years.
Abstract: This paper presents the multifractal model of asset returns ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second. the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multi-scaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.
574 citations
Posted Content•
TL;DR: In this paper, the authors provide a proof of Douady and Hubbard's Mandelbrot set theorem, which relies as much as possible on elementary combinatorics, rather than on more difficult analysis.
Abstract: A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c
e 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard proofs with a much more compact argument, and is highly recommended. The proofs given here are rather different from those of Schleicher, and are based on a combinatorial study of the angles of external rays for the Julia set which land on periodic orbits. The results in this paper are mostly well known; there is a particularly strong overlap with the work of Douady and Hubbard. The only claim to originality is in emphasis, and the organization of the proofs.
149 citations
1 Jan 2000
TL;DR: Tan as discussed by the authors showed that the Mandelbrot set at parabolic points can be modelled as an ensemble of non-expanding Julia sets, which is called le ensemble de Julia sets.
Abstract: Introduction L.Tan Preface J. Hubbard 1. The Mandelbrot set is universal C. McMullen 2. Baby Mandelbrot sets are born in cauliflowers A. Douady, X. Buff, R. Devaney and P. Sentenac 3. Modulation dans l'ensemble de Mandelbrot P. Haissinsky 4. Local connectivity of Julia sets: expository lectures J. Milnor 5. Holomorphic motions and puzzles (following M. Shishikura) P. Roesch 6. Local properties of the Mandelbrot set at parabolic points L.Tan 7. Convergence of rational rays in parameter spaces C. Petersen and G. Ryd 8. Bounded recurrence of critical points and Jakobson's Theorem S. Luzzatto 9. The Herman-Swiatek theorems with applications C. Petersen 10. Perturbations d'une fonction linearisable H. Jellouli 11. Indice holomorphe et multiplicateur H. Jellouli 12. An alternative proof of Mane's theorem on non-expanding Julia sets M. Shishikura and L.Tan 13. Geometry and dimension of Julia sets Y. -C. Yin 14. On a theorem of Mary Rees for the matings of polynomials M. Shishikura 15. Le theoreme d'integrabilite des structures presque complexes A. Douady and X. Buff 16. Bifurcation of parabolic fixed points M. Shishikura.
118 citations
Posted Content•
21 Nov 1994
TL;DR: In this paper, an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials is described.
Abstract: We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sections 3 and 4). A simple extension, \emph{angled internal addresses}, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Section~6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Section~7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.
108 citations
TL;DR: In this paper, the authors present and prove Boll's result and show that the number π in the Mandelbrot set is π-approximation of π.
Abstract: The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set.1 Boll's finding is easy to describe and understand, and yet it is not widely known — possibly because the result has not been rigorously shown. The purpose of this paper is to present and prove Boll's result.
92 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 7 |
| 2016 | 8 |
| 2015 | 9 |
| 2014 | 10 |
| 2013 | 17 |
| 2012 | 9 |