# Multifractal Measures and a Weak Separation Condition

@article{Lau1999MultifractalMA, title={Multifractal Measures and a Weak Separation Condition}, author={Ka-Sing Lau and Sze-Man Ngai}, journal={Advances in Mathematics}, year={1999}, volume={141}, pages={45-96} }

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.

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#### References

SHOWING 1-10 OF 50 REFERENCES

An Improved Multifractal Formalism and Self Similar Measures

- Mathematics
- 1995

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

- Mathematics
- 1987

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… Expand

Multifractal decompositions of Moran fractals

- Mathematics
- 1992

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 nonnegative… Expand

The multifractal spectrum of statistically self-similar measures

- Mathematics
- 1994

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 to… Expand

A dimension formula for Bernoulli convolutions

- Mathematics
- 1994

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… Expand

Multifractal decompositions of digraph recursive fractals

- Mathematics
- 1992

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… Expand

Dimension of a Family of Singular Bernoulli Convolutions

- Mathematics
- 1993

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… Expand

Fractal measures and mean p-variations

- Mathematics
- 1992

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 a… Expand

On a Family of Symmetric Bernoulli Convolutions

- Mathematics
- 1939

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

- Mathematics
- 1998

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… Expand