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
  • T. Hu, K. Lau
  • Mathematics, Computer Science
  • Adv. Appl. Math.
  • 2001
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. Expand
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 underExpand
Multifractal formalism for self-similar measures with weak separation condition
For any self-similar measure μ on Rd satisfying the weak separation condition, we show that there exists an open ball U0 with μ(U0)>0 such that the distribution of μ, restricted on U0, is controlledExpand
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,Expand
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 supportedExpand
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 dimensionExpand
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 boundedExpand
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 systemsExpand
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 restrictExpand
Differentiability of L q -spectrum and multifractal decomposition by using infinite graph-directed IFSs
Abstract By constructing an infinite graph-directed iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the L qExpand


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]Expand
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 ofExpand
Multifractal decompositions of Moran fractals
We present a rigorous construction and generalization of the multifractal decomposition for Moran fractals with infinite product measure. The generalization is specified by a system of nonnegativeExpand
The multifractal spectrum of statistically self-similar measures
We calculate the multifractal spectrum of a random measure constructed using a statistically self-similar process. We show that with probability one there is a multifractal decomposition analogous toExpand
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 maximumExpand
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 aExpand
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 correspondingExpand
Fractal measures and mean p-variations
Abstract Recently Strichartz proved that if μ is locally uniformly α-dimensional on R d, then , where 0 ⩽ α ⩽ d, and BT denotes the ball of radius T center at 0; if μ is self-similar and satisfies aExpand
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 formulaExpand