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## Studia Mathematica

1998 | 131 | 3 | 225-251
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $L^q$-spectrum, q >0, of the measure. This formula is the first obtained in the case where the open set condition does not hold.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
225-251
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-07-16
poprawiono
1998-04-15
Twórcy
autor
• Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
• Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.
autor
• Department of Mathematics, Cornell University, Ithaca, New York 14853 U.S.A.
Bibliografia
• [AY] J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory Dynam. Systems 4 (1984), 1-23.
• [AZ] J. C. Alexander and D. Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc. 44 (1991), 121-134.
• [CM] R. Cawley and R. D. Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), 196-236.
• [CLP] P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys. 47 (1987), 609-644.
• [DL] I. Daubechies and J. Lagarias, On the thermodynamic formalism for multifractal functions, Rev. Math. Phys. 6 (1994), 1033-1070.
• [EM] G. A. Edgar and R. D. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. London Math. Soc. 65 (1992), 604-628.
• [E] P. Erdős, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974-976.
• [F] K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications, Wiley, New York, 1990.
• [Fe] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
• [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.
• [G] A. M. Garsia, Entropy and singularity of infinite convolutions, Pacific J. Math. 13 (1963), 1159-1169.
• [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.
• [Hu] T. Y. Hu, The local dimensions of the Bernoulli convolution associated with the golden number, Trans. Amer. Math. Soc. 349 (1997), 2917-2940.
• [Hut] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
• [La] S. P. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, J. London Math. Soc., to appear.
• [L1] K.-S. Lau, Fractal measures and mean p-variations, J. Funct. Anal. 108 (1992), 427-457.
• [L2] K.-S. Lau, Dimension of a family of singular Bernoulli convolutions, ibid. 116 (1993), 335-358.
• [LN1] K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math., to appear.
• [LN2] K.-S. Lau and S.-M. Ngai, $L^q$-spectrum of Bernoulli convolutions associated with P.V. numbers, preprint.
• [LW] K.-S. Lau and J. Wang, Mean quadratic variations and Fourier asymptotics of self-similar measures, Monatsh. Math. 115 (1993), 99-132.
• [LP] F. Ledrappier and A. Porzio, A dimension formula for Bernoulli convolution, J. Statist. Phys. 76 (1994), 1307-1327.
• [Lo] A. O. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal. 20 (1989), 1243-1254.
• [N] S.-M. Ngai, A dimension result arising from the $L^q$-spectrum of a measure, Proc. Amer. Math. Soc. 125 (1997), 2943-2951.
• [O] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-195.
• [PU] F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math. 93 (1989), 155-186.
• [R] D. Rand, The singularity spectrum f(α) for cookie-cutters, Ergodic Theory Dynam. Systems 9 (1989), 527-541.
• [Ré] A. Rényi, Probability Theory, North-Holland, 1970.
• [Ri] R. Riedi, An improved multifractal formalism and self-similar measures, J. Math. Anal. Appl. 189 (1995), 462-490.
• [S] R. Salem, Algebraic Numbers and Fourier Transformations, Heath Math. Monographs, Boston, 1962.
• [Se] E. Seneta, Non-negative Matrices, Wiley, New York, 1973.
• [So] B. Solomyak, On the random series $∑ ± λ ^n$ (an Erdős problem), Ann. of Math. 142 (1995), 611-625.
• [St] R. S. Strichartz, Self-similar measures and their Fourier transforms III, Indiana Univ. Math. J. 42 (1993), 367-411.
• [STZ] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math. 4 (1995), 101-128.
Typ dokumentu
Bibliografia
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