General multifractal analysis of local entropies

Floris Takens; Evgeny Verbitski

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 3, page 203-237
  • ISSN: 0016-2736

Abstract

top
We address the problem of the multifractal analysis of local entropies for arbitrary invariant measures. We obtain an upper estimate on the multifractal spectrum of local entropies, which is similar to the estimate for local dimensions. We show that in the case of Gibbs measures the above estimate becomes an exact equality. In this case the multifractal spectrum of local entropies is a smooth concave function. We discuss possible singularities in the multifractal spectrum and their relation to phase transitions.

How to cite

top

Takens, Floris, and Verbitski, Evgeny. "General multifractal analysis of local entropies." Fundamenta Mathematicae 165.3 (2000): 203-237. <http://eudml.org/doc/212467>.

@article{Takens2000,
abstract = {We address the problem of the multifractal analysis of local entropies for arbitrary invariant measures. We obtain an upper estimate on the multifractal spectrum of local entropies, which is similar to the estimate for local dimensions. We show that in the case of Gibbs measures the above estimate becomes an exact equality. In this case the multifractal spectrum of local entropies is a smooth concave function. We discuss possible singularities in the multifractal spectrum and their relation to phase transitions.},
author = {Takens, Floris, Verbitski, Evgeny},
journal = {Fundamenta Mathematicae},
keywords = {multifractal analysis; local entropy; multifractal spectrum},
language = {eng},
number = {3},
pages = {203-237},
title = {General multifractal analysis of local entropies},
url = {http://eudml.org/doc/212467},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Takens, Floris
AU - Verbitski, Evgeny
TI - General multifractal analysis of local entropies
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 3
SP - 203
EP - 237
AB - We address the problem of the multifractal analysis of local entropies for arbitrary invariant measures. We obtain an upper estimate on the multifractal spectrum of local entropies, which is similar to the estimate for local dimensions. We show that in the case of Gibbs measures the above estimate becomes an exact equality. In this case the multifractal spectrum of local entropies is a smooth concave function. We discuss possible singularities in the multifractal spectrum and their relation to phase transitions.
LA - eng
KW - multifractal analysis; local entropy; multifractal spectrum
UR - http://eudml.org/doc/212467
ER -

References

top
  1. [1] L. Barreira, Ya. Pesin, and J. Schmeling, Multifractal spectra and multifractal rigidity for horseshoes, J. Dynam. Control Systems 3 (1997), 33-49. Zbl0949.37017
  2. [2] L. Barreira, Ya. Pesin, and J. Schmeling, On a general concept of multifractality: multifractal spectra for dimensions, entropies, andLyapunov exponents. Multifractal rigidity, Chaos 7 (1997), 27-38. Zbl0933.37002
  3. [3] C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems, Cambridge Univ. Press, Cambridge, 1993. 
  4. [4] R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), 193-202. Zbl0299.54031
  5. [5] M. Brin and A. Katok, On local entropy, in: Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math. 1007, Springer, Berlin, 1983, 30-38. 
  6. [6] K. Falconer, Fractal Geometry, Wiley, Chichester, 1990. 
  7. [7] P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Phys. D 13 (1984), 34-54. 
  8. [8] M. Guysinsky and S. Yaskolko, Coincidence of various dimensions associated with metrics and measures on metric spaces, Discrete Contin. Dynam. Systems 3 (1997), 591-603. Zbl0948.37014
  9. [9] 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 (3), 33 (1986), 1141-1151. Zbl1184.37028
  10. [10] N. T. A. Haydn and D. Ruelle, Equivalence ofGibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Comm. Math. Phys. 148 (1992), 155-167. Zbl0763.54028
  11. [11] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, preprint, 1998. 
  12. [12] S. Isola, Dynamical zeta functions and correlation functions for non-uniformly hyperbolic transformations, preprint, 1995. 
  13. [13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995. Zbl0878.58020
  14. [14] C. Liverani, B. Saussol, and S. Vaienti, A probabilistic approach to intermittency, preprint. 
  15. [15] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196. Zbl0841.28012
  16. [16] Ya. B. Pesin, Dimension Theory in Dynamical Systems, Univ. of Chicago Press, Chicago, IL, 1997. 
  17. [17] Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 50-63 (in Russian). 
  18. [18] G. Pianigiani, First return map and invariant measures, Israel J. Math. 35 (1980), 32-48. Zbl0445.28016
  19. [19] T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions, J. Statist. Phys. 66 (1992), 503-514. Zbl0892.58024
  20. [20] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973. Zbl0271.26009
  21. [21] J. Schmeling, On the completeness of multifractal spectra, preprint WIAS, Berlin, 1996. 
  22. [22] F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys. 203 (1999), 593-612. Zbl0955.37002
  23. [23] M. Urbański, ParabolicCantor sets, Fund. Math. 151 (1996), 241-277. 
  24. [24] T. Ward, Entropy of Compact Group Automorphisms, lecture notes, 1994. 
  25. [25] L.-S. Young, Recurrence times and rates of mixing, preprint, 1997. 
  26. [26] M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, preprint. Zbl0971.37004

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.