Pointwise Dimension Research Articles

Recurrence rate in rapidly mixing dynamical systems

For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point.We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal.This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.

A note on the relationship of pointwise dimensions of an invariant measure and its natural extension

In this note we present an estimate on the relationship of lower pointwise dimensions of an invariant measure of a Lipschitz map on a compact metric space and its natural extension.

Birkhoff Averages for Hyperbolic Flows: Variational Principles and Applications

We establish a higher-dimensional version of multifractal analysis for hyperbolic flows. This means that we consider simultaneously the level sets of several Birkhoff averages. Examples are the Lyapunov exponents as well as the pointwise dimension and the local entropy of a given measure. More precisely, we consider multifractal spectra associated to multi-dimensional parameters, obtained by computing the entropy of the level sets associated to the Birkhoff averages. We also consider the more general class of flows with upper semi-continuous metric entropy. The multifractal analysis is obtained here from a variational principle for the topological entropy of the level sets, showing that their topological entropy can be arbitrarily approximated by the entropy of ergodic measures. This principle unifies many results. An analogous principle holds for the Hausdorff dimension. The applications include the study of the regularity of the spectra, the description of how these vary under small perturbations, and the detailed study of the finer structure. The higher-dimensional spectra also exhibit new nontrivial phenomena absent in the one-dimensional multifractal analysis.

Dimension of measures for suspension flows

We consider hyperbolic flows and more generally suspension flows over two-sided subshifts of finite type, and establish an explicit formula for the pointwise dimension of an arbitrary measure (in particular the measure is not necessarily ergodic and it may not possess a local product structure). The applications of this formula include the description of the Hausdorff dimension of a given measure in terms of an ergodic decomposition, and the proof of the existence of measures of maximal dimension.

The pointwise dimension of self-similar measures in complete metric spaces

Abstract In this paper we investigate the pointwise dimension of self-similar measures in complete metric spaces. Due to the lack of the geometry property of general complete metric spaces, some new techniques will be needed.

Pointwise Dimensions of the Lorenz Attractor

We discuss a connection between two complementary views of the Lorenz attractor: the first is the accepted view where the attractor has a smooth measure on a fractal support. This complex system is alternatively manifest as a self-similar curve for the pointwise dimension alpha. We describe why the latter approach is accessible for the analysis of an experimental signal.

Most probable dimension value and most flat interval methods for automatic estimation of dimension from time series

We present and compare two automatic methods for dimension estimation from time series. Both methods, based on conceptually different approaches, work on the derivative of the bi-logarithmic plot of the correlation integral versus the correlation length (log–log plot). The first method searches for the most probable dimension values (MPDV) and associates to each of them a possible scaling region. The second one searches for the most flat intervals (MFI) in the derivative of the log–log plot. The automatic procedures include the evaluation of the candidate scaling regions using two reliability indices. The data set used to test the methods consists of time series from known model attractors with and without the addition of noise, structured time series, and electrocardiographic signals from the MIT-BIH ECG database. Statistical analysis of results was carried out by means of paired t-test, and no statistically significant differences were found in the large majority of the trials. Consistent results are also obtained dealing with ‘difficult’ time series. In general for a more robust and reliable estimate, the use of both methods may represent a good solution when time series from complex systems are analyzed. Although we present results for the correlation dimension only, the procedures can also be used for the automatic estimation of generalized q-order dimensions and pointwise dimension. We think that the proposed methods, eliminating the need of operator intervention, allow a faster and more objective analysis, thus improving the usefulness of dimension analysis for the characterization of time series obtained from complex dynamical systems.

Pointwise dimensions for Poincaré recurrences associated with maps and special flows

We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.

Weak Gibbs measures and the local product structure

We establish a version of the local product structure ( weak local product structure ) for ergodic measures \overline{\mu} which are the invertible extension of ergodic weak Gibbs measures \mu invariant under piecewise C^0 -invertible (infinite to one) Markov maps T . As a special case, \overline{\mu} possesses asymptotically ‘almost’ local product structure in the sense of Barreira, Pesin and Schmeling. Under piecewise conformality of T and the existence of a piecewise smooth representation of the dual map of T , the weak local product structure allows one to show that the pointwise dimension of \overline{\mu} exists almost everywhere and is the sum of the pointwise dimension of \mu and the pointwise dimension of the dual of \mu . Our results can be applicable to a natural extension of piecewise conformal two-dimensional Markov map which is related to a complex continued fraction and admits indifferent periodic points.

On pointwise dimension of non-hyperbolic measures

We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.

Using the pointwise dimension to filter time series

This paper provides an insight into dimension analysis from time series. In particular, we propose a procedure based on the pointwise dimension in order to extract, for each embedding dimension, the subset of points in the phase space (and the corresponding ones in the time series) which give rise to the scaling behaviour. We may consider the output time series as the result of a filtering process, based on correlations of points in the phase-space domain. Furthermore, the procedure gives the statistics of points which determine the scaling behaviour.

Pointwise dimensions and Rényi dimensions

We prove that the local lower and upper pointwise dimensions of a probability measure μ are bounded from below by the lower generalized dimension D μ (q) for q > 1 and from above by the upper generalized dimension D μ (q) for q < 1.

Product structure of Poincaré recurrence

We provide new non-trivial quantitative information on the behavior of Poincare recurrence. In particular we establish the almost everywhere coincidence of the recurrence rate and of the pointwise dimension for a large class of repellers, including repellers without finite Markov partitions.Using this information, we are able to show that for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures.

Higher-dimensional multifractal analysis

We establish a higher-dimensional version of multifractal analysis for several classes of hyperbolic dynamical systems. This means that we consider multifractal decompositions which are associated to multi-dimensional parameters. In particular, we obtain a conditional variational principle, which shows that the topological entropy of the level sets of pointwise dimensions, local entropies, and Lyapunov exponents can be simultaneously approximated by the entropy of ergodic measures. A similar result holds for the Hausdorff dimension. This study allows us to exhibit new nontrivial phenomena absent in the one-dimensional multifractal analysis. In particular, while the domain of definition of a one-dimensional spectrum is always an interval, we show that for higher-dimensional spectra the domain may not be convex and may even have empty interior, while still containing an uncountable number of points. Furthermore, the interior of the domain of a higher-dimensional spectrum has in general more than one connected component.

On pointwise dimensions and spectra of measures

We give a new definition of the lower pointwise dimension associated with a Borel probability measure with respect to a general Carathéodory–Pesin structure. Then we show that the spectrum of the measure coincides with the essential supremum of the lower pointwise dimension. We provide an example coming from dynamical systems.

Convolution of multifractals and the local magnetization in a random-field Ising chain

The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.

Multifractal Analysis of Conformal Axiom A Flows

We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. Examples include suspensions over axiom A conformal diffeomorphisms, Anosov flows, and in particular, geodesic flows on compact smooth surfaces of negative curvature.

Nonlinear Dynamics Estimation of EEG Signals Accompanying Self-Paced Goal-Directed Movements

The study was aimed at comparison of several nonlinear characteristics (NC) computed in time for one and the same EEG record accompanying voluntary goal-directed movements. This allowed us to reveal different aspects of the nonlinear behavior of the process underlying the self-paced movement organization. Parallel alterations of these characteristics during the task performance supported the hypothesis that cognitive task performance is reflected by the changes in the nonlinear dynamics of the EEG activity. Four NCs of scalar EEG time series were estimated: Point-wise Correlation Dimension (PD2), Kolmogorov Entropy (K2), and Largest Lyapunov Exponent (LLE) as a function of time, and Nonlinear Prediction (NP) for successive EEG segments. The time evolution of these characteristics exhibited several transients between chaos-like states to almost periodic states during the task performance: the gradual increase to higher values indicating chaos are probably related to the onset of the successive phases of brain organization of the movement. The results suggest that even in short periods an EEG signal changes its dynamic structure and these periods could be determined precisely enough using different methods of nonlinear dynamics.

Orbits and phase transitions in the multifractal spectrum

We consider the one dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q < 0 at some critical strength of the random field. We introduce the concept of orbits which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of random iterated function systems in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role.

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.