Phase transitions for the multifractal analysis of self-similar measures

Nonlinearity

Volumn 19, Issue 5, 2006, Pages 1201-1217

  Testud, B ^a  

^a UNIVERSITÉ D'ORLÉANS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33645987925     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/19/5/009     Document Type: Article

References (37)

1. Ben Nasr F 1994 Analyse multifractale de mesures C. R Acad. Sci. Paris Sér. I Math. 319 807-10
2. Ben Nasr F, Bhouri I and Heurteaux Y 2002 The validity of the multifractal formalism: results and examples Adv. Math. 165 264-84
3. Benzi R, Paladin G, Parisi G and Vulpiani A 1984 On the multifractal nature of fully developed turbulence and chaotic system J. Phys. A: Math. Gen. 17 3521-31
4. Bowen R 1975 Equilibrium States and the Ergodic Theory of Anasov Diffeomorphisms (Lecture Notes in Math vol 470) (Berlin: Springer)
5. Brown G, Michon G and Peyrière J 1992 On the multifractal analysis of measures J. Stat. Phys. 66 775-90
6. Cawley R and Mauldin R D 1992 Multifractal decompositions of Moran fractals Adv. Math. 92 196-236
7. Edgar G A and Mauldin R D 1992 Multifractal decompositions of digraph recursive fractals Proc. London Math. Soc. 65 604-28
8. Falconer K 1990 Fractal Geometry, Mathematical Foundations and Applications (New York: John Wiley & Sons Ltd)
9. Falconer K 1997 Techniques in Fractal Geometry (New York: John Wiley & Sons Ltd)
10. q-spectrum of self-similar measures with overlaps Proc. London Math. Soc. 68 102-18
11. Feng D J 2005 The limit Rademacher functions and Bernoulli convolutions associated with Pisot numbers Adv. Math. 195 24-101
12. Feng D J and Olivier E 2003 Multifractal analysis of weak Gibbs measures and phase transition-application to some Bernoulli convolution Ergod. Theory Dynam. Syst. 23 1751-84
13. Frisch U and Parisi G 1985 On the singularity structure of fully developed turbulence Appendix to Fully Developed Turbulence and Intermittency (Proc. Int. School Phys. Enrico Fermi) ed U Frisch (Amsterdam: North-Holland) pp 84-8
14. Grassberger P 1983 Generalized dimension of strange attractors Phys. Lett. A 97 227-30
15. Grassberger P and Procaccia I 1983 Characterization of strange sets Phys. Rev. Lett. 50 346-9
16. Halsey T C, Jensen M H, Kadanoff L P, Procaccia I and Shraiman B I 1986 Fractal measures and their singularities: the characterization of strange sets Phys. Rev. A 33 1141-51
17. Hentschel H and Procaccia I 1983 The infinite number of generalized dimensions of fractals and strange attractors Physica D 8 435-44
18. Heurteaux Y 1998 Estimations de la dimension inférieure et de la dimension supérieure des mesures Ann. Inst. H Poincaré Probab. Statist. 34 309-38
19. Hu T Y and Lau K S 2001 Multifractal structure of convolution of the Cantor measure Adv. Appl. Math. 27 1-16
20. Hutchinson J E 1981 Fractals and self similarity Indiana Univ. Math. J. 30 713-47
21. q-spectrum of the Bernoulli convolution associated with the golden ratio Studia Math. 131 225-51
22. Lau K S and Ngai S M 1999 Multifractal measures and a weak separation condition Adv. Math. 141 45-96
23. Mandelbrot B B 1974 Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier J. Fluid Mech. 62 331-58
24. Meakin P, Conoglio A, Stanley H and Witten T 1986 Scaling properties for the surfaces of fractal and nonfractal objects: an infinite hierarchy of critical exponents Phys. Rev. A 34 3325-40
25. McMullen C 1984 The Hausdorff dimension of general Sierpiński carpets Nagoya Math. J. 96 1-9
26. Michon G 1983 Mesures de Gibbs sur les Cantor réguliers Ann. Inst. H Poincaré Phys. Théor. 58 267-85
27. Olivier E, Sidorov N and Thomas A 2005 On the Gibbs properties of Bernoulli convolutions related to β-numeration in multinacci bases Monatsh. Math. 145 145-74
28. Olsen L 1995 A multifractal formalism Adv. Math. 116 82-196
29. Peyrière J 1992 Multifractal measures Proc. NATO Adv. Study Inst. II Ciocco (NATO ASI Series C) vol 372 (Dordrecht: Kluwer Academic) pp 175-86
30. Przytycki F and Urbański M 1989 On Hausdorff dimension of some fractal sets Studia Math. 93 155-86
31. Renyi A 1970 Probability Theory (Amsterdam: North-Holland)
32. Riedi R 1995 An improved multifractal formalism and self-similar measures J. Math. Anal. Appl. 189 462-90
33. Ruelle D 1978 Thermodynamic Formalism (Reading, MA: Addison-Wesley)
34. Testud B 2004 Etude d'une large classe de mesures autosimilaires: calculs de dimensions et analyse multifractale PhD Thesis Université Blaise Pascal Clermont-Ferrand
35. Testud B 2005 Transitions de phase dans l'analyse multifractale de mesures auto-similaires C. R. Acad. Sci. Paris Sér. I Math. 340 653-8
36. Testud B 2006 Mesures quasi-Bernoulli au sens faible: résultats et exemples Ann. Inst. H Poincaré Probab. Statist. 42 1-35
37. Urbański M 1990 The Hausdorff dimension of the graphs of continuous self-affine functions Proc. Am. Math. Soc. 108 921-30