Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions on a metric space X, multifractal analysis refers to the study of the Hausdorff and/or packing dimension of the level sets(1) of the limit function limnφn. However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit limnφn(x) exists, has emerged and attracted considerable interest. Namely, for a sequence n(xn) in a metric space X, we let A(xn) denote the set of accumulation points of the sequence n(xn). The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence (φnn(x)) equals a given set C, i.e. computing the Hausdorff dimension of the set(2){x∈X|A(φn(x))=C} has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φn, including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions φn:X→M on a metric space X, multifractal analysis refers to the study of the Hausdorff dimension of the level sets

     

Multifractal analysis of the divergence points of Birkhoff averages for $$\beta $$-transformations

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of \(\beta \)-expansions. More precisely, let \(T_{\beta }\) be the \(\beta \)-transformation on [0, 1) for a general \(\beta >1\) and \(\psi :[0,1]\mapsto \mathbb {R}\) be a continuous function. Denote by \(\textsf {A}(\psi ,x)\) all the accumulation points of \(\{\frac{1}{n}\sum _{j=0}^{n-1}\psi (T^jx): n\ge 1\}\). The Hausdorff dimensions of the sets

$$\begin{aligned} \{x:\textsf {A}(\psi ,x)\supset [a,b]\},\quad \{x:\textsf {A}(\psi ,x)=[a,b]\},\quad \{x:\textsf {A}(\psi ,x)\subset [a,b]\} \end{aligned}$$

i.e., the points for which the Birkhoff averages of \(\psi \) do not exist but behave in a certain prescribed way, are determined completely for any continuous function \(\psi \).

     

A note on the divergence of geodesic rays in manifolds without conjugate points

Let (M, g) be a compact Riemannian manifold without conjugate points and let be its universal covering endowed with the pullback of the metric g by the covering map. We show that geodesic rays in which meet an axis of a covering isometry diverge from this axis. This result generalizes well known results by Morse and Hedlund in the context of globally minimizing geodesics in the universal covering of compact surfaces.      

Nonlinear data analysis of experimental (EEG) data and comparison with theoretical (ANN) data

In this article, nonlinear dynamical tools such as largest Lyapunov exponents (LE), fractal dimension, correlation dimension, pointwise correlation dimension will be used to analyze electroencephalogram (EEG) data obtained from healthy young subjects with eyes open and eyes closed condition with the view to compare brain complexity under this two condition. Results of similar calculations from some earlier works will be produced for comparison with present results. Also, a brief report on difference of opinion among coworkers regarding such tools will be reported; particularly applicability of LE will be reviewed. The issue of nonlinearity will be addressed by using surrogate data technique. We have extracted another data set that represented chaotic state of the system considered in our earlier work of mathematical modeling of artificial neural network. We further attempt to compare results to find nature of chaos arising from such theoretical models. © 2002 Wiley Periodicals, Inc.