Multifractal Measures and a Weak Separation Condition

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.

[1]  L. Olsen,et al.  A Multifractal Formalism , 1995 .

[2]  O. Lanford.  ENTROPY AND EQUILIBRIUM STATES IN CLASSICAL STATISTICAL MECHANICS , 1973 .

[3]  K. Lau,et al.  $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio , 1998 .

[4]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[5]  A. Garsia.  Arithmetic properties of Bernoulli convolutions , 1962 .

[6]  A. Lopes.  The dimension spectrum of the maximal measure , 1989 .

[7]  The local dimensions of the Bernoulli convolution associated with the golden number , 1997 .

[8]  R. Mauldin,et al.  Multifractal decompositions of digraph recursive fractals , 1992 .

[9]  G. Michon,et al.  On the multifractal analysis of measures , 1992 .

[10]  L. Olsen,et al.  Random Geometrically Graph Directed Self-Similar Multifractals , 1994 .

[11]  I. Daubechies,et al.  ON THE THERMODYNAMIC FORMALISM FOR MULTIFRACTAL FUNCTIONS , 1994 .

[12]  Tong Zhang,et al.  Densities of Self-Similar Measures on the Line , 1995, Exp. Math..

[13]  T. Tél.  Fractals, Multifractals, and Thermodynamics , 1988 .

[14]  P. Erdös.  On a Family of Symmetric Bernoulli Convolutions , 1939 .

[15]  J. Igusa.  ON JACOBI'S DERIVATIVE FORMULA AND ITS GENERALIZATIONS* , 1980 .

[16]  Szépfalusy,et al.  Singularities in Rényi information as phase transitions in chaotic states. , 1989, Physical review. A, General physics.

[17]  F. Ledrappier,et al.  A dimension formula for Bernoulli convolutions , 1994 .

[18]  J. Alexander,et al.  The Entropy of a Certain Infinitely Convolved Bernoulli Measure , 1991 .

[19]  D. Rand.  The singularity spectrum f (α) for cookie-cutters , 1989 .

[20]  K. Lau.  Dimension of a Family of Singular Bernoulli Convolutions , 1993 .

[21]  Kenneth Falconer,et al.  The multifractal spectrum of statistically self-similar measures , 1994 .

[22]  Pierre Collet,et al.  The dimension spectrum of some dynamical systems , 1987 .

[23]  Jensen,et al.  Fractal measures and their singularities: The characterization of strange sets. , 1987, Physical review. A, General physics.

[24]  R. Mauldin,et al.  Multifractal decompositions of Moran fractals , 1992 .

[25]  Lq-spectrum of Bernoulli convolutions associated with P.V. numbers , 1999 .

[26]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[27]  K. Lau.  Fractal measures and mean p-variations , 1992 .

[28]  Rudolf H. Riedi,et al.  An Improved Multifractal Formalism and Self Similar Measures , 1995 .

[29]  I. Daubechies,et al.  Two-scale difference equations I: existence and global regularity of solutions , 1991 .

[30]  James A. Yorke,et al.  Fat baker's transformations , 1984, Ergodic Theory and Dynamical Systems.

[31]  David A. Rand,et al.  The entropy function for characteristic exponents , 1987 .

[32]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[33]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[34]  R. Strichartz.  Self-similar measures and their Fourier transforms. II , 1993 .