Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages.
The aim of this paper is to develop a unifying multifractal framework, based on the so-called deformations of empirical measures, which allows significant extensions of already known results in multifractal analysis of both dynamical systems and fractal measures. One considers a finite directed graph \((V,E)\). To each vertex \(i\in V\) one associates the set \(\Sigma_i\) of all infinite paths in \((V,E)\) starting at \(i\), and the set \(\Sigma\) of all infinite paths in \(V\). \(S:\Sigma\to\Sigma\) is the shift map, and \({\mathcal P}(\Sigma)\) the family of probability measures on \(\Sigma\). For \(n\in \mathbb N\), \(L_{n}:\Sigma\to{\mathcal P}(\Sigma)\) is the \(n\)th empirical measure, defined by: \[ L_{n}\omega= {{1}\over{n}} \sum_{k=0}^{n-1} \delta_{S^{k}\omega}, \] where \(\delta_{x}\) is the Dirac measure at \(x\). A continuous deformation of \(L_{n}\) is a pair \((X,\Xi)\), where \(X\) is a metric space, and \[ \Xi:{\mathcal P}(\Sigma)\to X \] is a continuous map with respect to the weak topology. The multifractal spectrum of this deformation is defined by: \[ f_{i}(t)=\text{dim} \left(\pi_{i}\{\omega\in \Sigma_{i}\mid \lim_{n\to\infty}\Xi L_{n}\omega=t\}\right), \] where \(\pi_{i}:\Sigma_{i}\to \mathbb R^{d}\) is the natural projection. For different choices of the deformation one obtains different multifractal spectra. As examples of deformations of empirical measures are given: the spectrum of ergodic averages, the multifractal spectrum of the self-conformal measures \(\mu_i\), associated with a directed graph iterated self-conformal function system with probabilities, the spectrum of local entropies of \(\mu_i\), the spectrum of local Lyapunov exponents. The main result (Theorem 5) gives lower and upper bounds for the multifractal spectra for arbitrary nonlinear continuous deformations. There are also given necessary conditions ensuring the equality in Theorem 5. These results lead to an abstract notion of relative or conditional multifractal spectra. A variational principle for the mixed multifractal spectrum of the local entropy and local Lyapunov exponent is also derived. As an application of the obtained results, the author proposes the study of the so-called set of divergence points of a deformation \((X, \Xi)\): \[ \{\omega\in \Sigma_{i}\mid \text{the limit } \lim_{n}\Xi L_{n}\omega \text{ does not exist}\}, \] as well as the study of the Hausdorff dimension of several classes of sets in number theory.
scientific article; zbMATH DE number 2032770
scientific article