Multivariate wavelet leaders Rényi dimension and multifractal formalism in mixed Besov spaces

In this paper, we first establish a general lower bound for the multivariate wavelet leaders Rényi dimension valid for any pair [Formula: see text] of functions on [Formula: see text] where [Formula: see text] belongs to the Besov space [Formula: see text] with [Formula: see text] and [Formula: see text] belongs to [Formula: see text] with [Formula: see text]. We then prove the optimality of this result for quasi all pairs [Formula: see text] in the Baire generic sense. Finally, we compute both iso-mixed and upper-multivariate Hölder spectra for all pairs [Formula: see text] in the same [Formula: see text]-set. This allows to prove (respectively, study) the Baire generic validity of the upper-multivariate (respectively, iso-multivariate) multifractal formalism based on wavelet leaders for such pairs.

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

Very recently bounds for the L q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L q spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions.

IMPLEMENTATION OF A CHUA CIRCUIT TO DEMONSTRATE BIFURCATIONS AND STRANGE ATTRACTORS IN A CLASS

This paper describes the design and implementation of a Chua double-scroll circuit to demonstrate chaos in dynamical systems to students in a graduate course in order to enhance their visualization and understanding of strange attractor and Feigenbaum bifurcation trees.Teaching dynamical systems (i.e., nonlinear systems that can exhibit chaos) is often considered difficult because of the mathematical modeling involved and the inclusion of the fourth strange-attractor state, in addition to the traditional point stability, cyclic stability, and toroidal stability, as found in dynamic systems. A graduate course has been offered at the University of Manitoba for many years to provide both (i) a unified theory of fractal dimensions, together with many practical implementations of algorithms to compute the fractal dimensions, including the Rényi dimension spectrum that is required for characterization of the strange attractors using multifractal analysis.Leon Chua developed a simple nonlinear circuit capable of producing a rich collection of dynamic phenomena, ranging from fixed points to cycle points, standard bifurcations (period doubling), other standard routes to chaos, and chaos itself. The reason for selecting this specific circuit as a class demonstration tool is threefold: (i) the circuit has an analytical model and can be simulated, (ii) the circuit is implementable using available commercial off-the-shelf components, and (iii) the signals in the circuit can be acquired without affecting and altering its operation significantly.This paper describes the architecture, implementation, verification, and testing of the Chua system, as well as an analysis of the data obtained during the current phase of the development. Although there are many possible implementations of Chua’s circuit, our implementation has several innovative design features to make it more applicable to enhance students’ learning in the classroom.

The geometry of conformal measures for parabolic rational maps

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

The singularity spectrumf(α) for cookie-cutters

I use a thermodynamic formalism to study the spectrumf(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, thatf(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimensionDqof the Gibbs state by the formula (1 −q)Dq= τ(q).

Dimension, entropy and Lyapunov exponents

We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.