Lq-spectrum of the Bernoulli convolution associated with the golden ratio
Lau, Ka-Sing ; Ngai, Sze-Man
Studia Mathematica, Tome 129 (1998), p. 225-251 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Based on a set of higher order self-similar identities for the Bernoulli convolution measure for (√5-1)/2 given by Strichartz et al., we derive a formula for the Lq-spectrum, q >0, of the measure. This formula is the first obtained in the case where the open set condition does not hold.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216578 
@article{bwmeta1.element.bwnjournal-article-smv131i3p225bwm,
     author = {Ka-Sing Lau and Sze-Man Ngai},
     title = {$L^q$-spectrum of the Bernoulli convolution associated with the golden ratio},
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {225-251},
     zbl = {0929.28005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv131i3p225bwm}
}

Lau, Ka-Sing; Ngai, Sze-Man. $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio. Studia Mathematica, Tome 129 (1998) pp. 225-251. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv131i3p225bwm/

[00000] [AY] J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory Dynam. Systems 4 (1984), 1-23. | Zbl 0553.58020

[00001] [AZ] J. C. Alexander and D. Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc. 44 (1991), 121-134. | Zbl 0691.58024

[00002] [CM] R. Cawley and R. D. Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), 196-236. | Zbl 0763.58018

[00003] [CLP] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys. 47 (1987), 609-644. | Zbl 0683.58023

[00004] [DL] I. Daubechies and J. Lagarias, On the thermodynamic formalism for multifractal functions, Rev. Math. Phys. 6 (1994), 1033-1070. | Zbl 0843.58091

[00005] [EM] G. A. Edgar and R. D. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. London Math. Soc. 65 (1992), 604-628. | Zbl 0764.28007

[00006] [E] P. Erdős, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974-976. | Zbl 0022.35402

[00007] [F] K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications, Wiley, New York, 1990. | Zbl 0689.28003

[00008] [Fe] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.

[00009] [FP] U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, in: Proc. Internat. School Phys., "Enrico Fermi" Course LXXXVIII, North-Holland, Amsterdam, 1985, 84-88.

[00010] [G] A. M. Garsia, Entropy and singularity of infinite convolutions, Pacific J. Math. 13 (1963), 1159-1169. | Zbl 0126.14901

[00011] [H] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986), 1141-1151. | Zbl 1184.37028

[00012] [Hu] T. Y. Hu, The local dimensions of the Bernoulli convolution associated with the golden number, Trans. Amer. Math. Soc. 349 (1997), 2917-2940. | Zbl 0876.28013

[00013] [Hut] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. | Zbl 0598.28011

[00014] [La] S. P. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, J. London Math. Soc., to appear.

[00015] [L1] K.-S. Lau, Fractal measures and mean p-variations, J. Funct. Anal. 108 (1992), 427-457. | Zbl 0767.28007

[00016] [L2] K.-S. Lau, Dimension of a family of singular Bernoulli convolutions, ibid. 116 (1993), 335-358. | Zbl 0788.60055

[00017] [LN1] K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math., to appear.

[00018] [LN2] K.-S. Lau and S.-M. Ngai, Lq-spectrum of Bernoulli convolutions associated with P.V. numbers, preprint.

[00019] [LW] K.-S. Lau and J. Wang, Mean quadratic variations and Fourier asymptotics of self-similar measures, Monatsh. Math. 115 (1993), 99-132. | Zbl 0778.28005

[00020] [LP] F. Ledrappier and A. Porzio, A dimension formula for Bernoulli convolution, J. Statist. Phys. 76 (1994), 1307-1327. | Zbl 0840.58026

[00021] [Lo] A. O. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal. 20 (1989), 1243-1254. | Zbl 0683.58029

[00022] [N] S.-M. Ngai, A dimension result arising from the Lq-spectrum of a measure, Proc. Amer. Math. Soc. 125 (1997), 2943-2951. | Zbl 0886.28006

[00023] [O] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-195. | Zbl 0841.28012

[00024] [PU] F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math. 93 (1989), 155-186. | Zbl 0691.58029

[00025] [R] D. Rand, The singularity spectrum f(α) for cookie-cutters, Ergodic Theory Dynam. Systems 9 (1989), 527-541. | Zbl 0664.58022

[00026] [Ré] A. Rényi, Probability Theory, North-Holland, 1970.

[00027] [Ri] R. Riedi, An improved multifractal formalism and self-similar measures, J. Math. Anal. Appl. 189 (1995), 462-490. | Zbl 0819.28008

[00028] [S] R. Salem, Algebraic Numbers and Fourier Transformations, Heath Math. Monographs, Boston, 1962.

[00029] [Se] E. Seneta, Non-negative Matrices, Wiley, New York, 1973.

[00030] [So] B. Solomyak, On the random series ∑±λn (an Erdős problem), Ann. of Math. 142 (1995), 611-625. | Zbl 0837.28007

[00031] [St] R. S. Strichartz, Self-similar measures and their Fourier transforms III, Indiana Univ. Math. J. 42 (1993), 367-411. | Zbl 0790.28003

[00032] [STZ] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math. 4 (1995), 101-128. | Zbl 0860.28005