Related papers: On multifractal formalism for self-similar measure…

We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\mu$, we show that there is a finite set of concave…

It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this…

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of…

We study the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the L^q spectrum tau(q) as well as the singularity spectrum f(alpha), is sufficiently large to point…

For any self-similar measure $\mu$ in $\mathbb{R}$, we show that the distribution of $\mu$ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the IFS. This generalizes the net…

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also…

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also…

In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures…

In this work, we investigate the H\"older spectrum of typical measures (in the Baire category sense) in a general compact set and we compute the multifractal spectrum of a typical measures supported by a self-similar set. Such mesures…

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.…

Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $\mu$…

For a Borel probability measure $\mu$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(\mu)$ admits an orthogonal basis of exponential functions. In this paper, we study the spectrality of fractal measures…

We refine the multifractal formalism for the local dimension of a Gibbs measure $\mu$ supported on the attractor $\Lambda$ of a conformal iterated functions system on the real line. Namely, for given $\alpha\in \mathbb{R}$, we establish the…

Let $\mu$ be a self-similar measure satisfying the finite type condition. It is known that the set of attainable local dimensions for such a measure is a union of disjoint intervals, where some intervals may be degenerate points. Despite…

We establish matrix representations for self-similar measures on $\mathbb{R}^d$ generated by equicontractive IFSs satisfying the finite type condition. As an application, we prove that the $L^q$-spectrum of every such self-similar measure…

We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…

We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode…

This paper introduces Fourier duality for a class of affine iterated function systems (IFS) T_i. These systems are determined by a finite family of contractive affine maps in R^d. Our Fourier duality applies to the resulting probability…

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $\Phi$ and a positive probability vector $p$. Suppose that the maps in $\Phi$ do not have a common fixed point, and that standard irreducibility and…

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$…