Pointwise Dimension Research Articles

Dimension spectrum of asymptotically additive potentials for C1 average conformal repellers

In this paper, the dimension spectrum of asymptotically additive potentials for C1 average conformal repellers is considered. First, we give dimension estimates of arbitrary subsets for C1 average conformal repellers and we also consider the pointwise dimension of invariant measures. Furthermore, we consider the dimension spectrum of asymptotically additive potentials and its properties. As applications, we give an affirmative answer to the problem posed by Olsen (2003 J. Math. Pures Appl. 82 1591–649) for general types of level sets and obtain the dimension spectrum of weak Gibbs measures for continuous potentials on C1 average conformal repellers.

Dimensions of Attractors in Pinched Skew Products

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.

Equilibrium measures on saddle sets of holomorphic maps on $$\mathbb{P }^2$$

We consider the case of hyperbolic basic sets \(\Lambda \) of saddle type for holomorphic maps \(f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}\). We study equilibrium measures \(\mu _\phi \) associated to a class of Holder potentials \(\phi \) on \(\Lambda \), and find the measures \(\mu _\phi \) of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension \(\delta _{\mu _\phi }\) of \(\mu _\phi \) that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of \(\mu _\phi \) in the case when the preimage counting function is constant on \(\Lambda \). For terminal/minimal saddle sets we prove that an invariant measure \(\nu \) obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction \(f|_\Lambda \). This allows then to obtain formulas for the measure \(\nu \) of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of \(\nu \).

The geometry of chaotic dynamics — a complex network perspective

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently $\epsilon$-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using $\epsilon$-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that $\epsilon$-recurrence networks exhibit an important link between dynamical systems and graph theory.

Pointwise dimensions of general Moran measures with open set condition

Recently Lou and Wu obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in ℝd under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different from that used by Lou and Wu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.

Dimension Theory for Multimodal Maps

This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum, generalising previous results by Todd. We also study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra.

Nice Inducing Schemes and the Thermodynamics of Rational Maps

We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential \({-t {\rm ln} \left|f^{\prime} \right|}\) , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in \({\mathbb{R}}\) , with at most two exceptions.

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ . We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.

Poincaré recurrence for observations

A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan’s work, for a measure preserving system we study Poincaré recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relative to the push-forward are equal.

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \({\mathbb{C}\mathbb{P}^k}\) and ν be an f-invariant measure with positive Lyapunov exponents (λ1, . . . , λk). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by \({{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}\), which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of fn in \({\mathbb{C}\mathbb{P}^k}\) . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \({\mathbb{C}\mathbb{P}^k}\) and a normalization theorem for the ν-generic inverse branches of fn.

The pointwise dimensions of Moran measures

In this paper, we get the formulas of upper (lower) pointwise dimensions of some Moran measures on Moran sets in ℝ d under the strong separation condition. We also obtain formulas for the dimension of the Moran measures. Our results extend the known results of some self-similar measures and Moran measures studied by Cawley and Mauldin.

Dimensional characteristics of invariant measures for circle diffeomorphisms

AbstractWe consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

Estimating Effective Degrees of Freedom in Motor Systems

Studies of the degrees of freedom and "synergies" in musculoskeletal systems rely critically on algorithms to estimate the "dimension" of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are the most popular. However, many biological data (or realistic experimental data) may be better represented by nonlinear sets than linear subspaces. We evaluate the performance of PCA and compare it to two nonlinear algorithms [Isomap and our novel pointwise dimension estimation (PD-E)] using synthetic and motion capture data from a robotic arm with known kinematic dimensions, as well as motion capture data from human hands. We find that PCA can lead to more accurate dimension estimates when considering additional properties of the PCA residuals, instead of the dominant method of using a threshold of variance captured. In contrast to the single integer dimension estimates of PCA and Isomap, PD-E provides a distribution and range of estimates of fractal dimension that identify the heterogeneous geometric structure in the experimental data. A strength of the PD-E method is that it associates a distribution of dimensions to the data. Since there is no a priori reason to assume that the sets of interest have a single dimension, these distributions incorporate more information than a single summary statistic. Our preliminary findings suggest that fewer than ten DOFs are involved in some hand motion tasks. Contrary to common opinion regarding fractal dimension methods, PD-E yielded reasonable results with reasonable amounts of data. Given the complex nature of experimental and biological data, we conclude that it is necessary and feasible to complement PCA with methods that take into consideration the nonlinear properties of biological systems for a more robust estimation of their DOFs.

Recurrence rates for loosely Markov dynamical systems

AbstractThe concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.

Dimension and ergodic decompositions for hyperbolic flows

For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.

Pointwise dimension and ergodic decompositions

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approach, pertaining to the fundamentals of ergodic theory, which in particular does not require Pesin theory. This allows us to show, for those systems, that the Hausdorff dimension of a (non-ergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition. We also establish corresponding results for non-uniformly hyperbolic diffeomorphisms on surfaces, now using Pesin theory. To the best of our knowledge, the formula for the Hausdorff dimension of an invariant measure along an ergodic decomposition is new even in the case of uniformly hyperbolic dynamics.

Multifractal Structure of Two-Dimensional Horseshoes

We give a complete description of the dimension spectra of Birkhoff averages on a hyperbolic set of a surface diffeomorphism. The main novelty is that we are able to consider simultaneously Birkhoff averages into the future and into the past, i.e., both for positive and negative time. We emphasize that the description of these spectra is not a consequence of the available results in the case of Birkhoff averages simply into the future (or into the past). The main difficulty is that although the local product structure provided by the intersection of stable and unstable manifolds is bi-Lipschitz equivalent to a product, the level sets of the Birkhoff averages are never compact (this causes their box dimension to be strictly larger than their Hausdorff dimension), and thus the product of level sets may have a dimension that need not be the sum of the dimensions of these sets. Instead we construct explicitly noninvariant measures concentrated on each product of level sets with the appropriate pointwise dimension. We also consider the higher-dimensional case of more than one Birkhoff average, as well as the case of ratios of Birkhoff averages.

Pointwise and Renyi Dimensions of an Invariant Measure of Random Dynamical Systems with Jumps

We estimate the lower pointwise dimension and the generalized Renyi dimension of an invariant measure of random dynamical systems with jumps. It is worthwhile to note that the dimensions are a useful tool in studying the Hausdorff dimension of measures and sets. Our model generalizes Markov processes corresponding to iterated function systems and Poisson driven stochastic differential equations. It can be used as a description of many physical and biological phenomena.

Genetic influences on dynamic complexity of brain oscillations

Human electroencephalogram (EEG) consists of complex aperiodic oscillations that are assumed to indicate underlying neural dynamics such as the number and degree of independence of oscillating neuronal networks. EEG complexity can be estimated using measures derived from nonlinear dynamic systems theory. Variations in such measures have been shown to be associated with normal individual differences in cognition and some neuropsychiatric disorders. Despite the increasing use of EEG complexity measures for the study of normal and abnormal brain functioning, little is known about genetic and environmental influences on these measures. Using the pointwise dimension (PD2) algorithm, this study assessed heritability of EEG complexity at rest in a sample of 214 young female twins consisting of 51 monozygotic (MZ) and 56 dizygotic (DZ) pairs. In MZ twins, intrapair correlations were high and statistically significant; in DZ twins, correlations were substantially smaller. Genetic analyses using linear structural equation modeling revealed high and significant heritability of EEG complexity: 62–68% in the eyes-closed condition, and 46–60% in the eyes-open condition. Results suggest that individual differences in the complexity of resting electrocortical dynamics are largely determined by genetic factors. Neurophysiological mechanisms mediating genetic variation in EEG complexity may include the degree of structural connectivity and functional differentiation among cortical neuronal assemblies.

Multifractal analysis for countable Markov shifts

We study the multifractal analysis of the pointwise dimension for equilibrium measures on countable Markov shifts. The main difficulty is that the space is not compact. In order to overcome this, we use an approximation argument based on the theory of convergence of Fenchel pairs developed by Wijsman. The results of Pesin and Weiss on multifractal analysis for compact spaces are used as well. We also prove a Bowen formula for countable Markov shifts. It turns out that, in this setting, this formula provides the Hausdorff dimension of the set of recurrent points.