Multifractal Measures and a Weak Separation Condition

  title={Multifractal Measures and a Weak Separation Condition},
  author={Ka-Sing Lau and Sze-Man Ngai},
  journal={Advances in Mathematics},
Abstract We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the well-known class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the two-scale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition. 

Figures from this paper

Multifractal Structure of Convolution of the Cantor Measure

The local dimension of the m-time convolution of the standard Cantor measure @m is studied to show that the set E of attainable local dimensions of @m contains an isolated point, implying that the multifractal formalism fails without the open set condition.

Multifractal analysis of infinite products

We construct a family of measures called infinite products which generalize Gibbs measures in the one-dimensional lattice gas model. The multifractal properties of these measures are studied under

Multifractal formalism for self-similar measures with weak separation condition

Multifractal formalism for self-affine measures with overlaps

Abstract.We prove that for a class of self-affine measures defined by an expanding matrix whose eigenvalues have the same modulus, the Lq-spectrum τ(q) is differentiable for all q > 0. Furthermore,

Multifractal Structure of Noncompactly Supported Measures

Many important definitions in the theory of multifractal measures on ℝd, such as the Lq-spectrum, L∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported

Assouad dimensions and self-similar sets satisfying the weak separation condition in R

We discuss the weak separation condition in the context of iterated function systems of similarities in the real line. Then, following [6], we present a discussion and proof of the Assouad dimension


In this paper we study a general separation property for subsystems G, whose attractor KG is a sub-self-similar set. This is a generalization of the Lau-Ngai weak separation property for the bounded

When the weak separation condition implies the generalized finite type condition

We prove that an iterated function system of similarities on R \mathbb {R} that satisfies the weak separation condition and has an interval as its self-similar set is of generalized

Multifractal structure and product of matrices

There is a well established multifractal theory for self-similar measures generated by non-overlapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict

Iterated Function Systems with Overlaps and Self‐Similar Measures

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems



An Improved Multifractal Formalism and Self Similar Measures

Abstract To characterize the geometry of a measure, its generalized dimensions dq have been introduced recently. The mathematically precise definition given by Falconer ["Fractal Geometry," 1990]

The dimension spectrum of some dynamical systems

We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber. Using large-deviation theory, we prove, for some invariant measures of

A dimension formula for Bernoulli convolutions

We present a “dynamical” approach to the study of the singularity of infinitely convolved Bernoulli measuresvβ, for β the golden section. We introducevβ as the transverse measure of the maximum

Multifractal decompositions of digraph recursive fractals

We prove that the multifractal decomposition behaves as expected for a family of sets K known as digraph recursive fractals, using measures μ of Markov type. For each value of a parameter α between a

Dimension of a Family of Singular Bernoulli Convolutions

Abstract Let { X n } ∞ n = 0 be a sequence of i.i.d. Bernoulli random variables (i.e., X n takes values {0, 1} with probability 1 2 each), let X = ∑ ∞ n = 0 ρ n X n and let μ be the corresponding

On a Family of Symmetric Bernoulli Convolutions

1. For any fixed real number a in the interval 0 1. In other words, A (x; a) is the distributioni function whose Fourier-Stieltjes transform is the infinite product

$L^q$-spectrum of the Bernoulli convolution associated with the golden ratio

Based on the higher order self-similarity of the Bernoulli convolution measure for (p 5?1)=2 proposed by Strichartz et al, we derive a formula for the L q-spectrum, q > 0 of the measure. This formula

On the multifractal analysis of measures

The multifractal formalism is shown to hold for a large class of measures.