# General multifractal analysis of local entropies

Floris Takens; Evgeny Verbitski

Fundamenta Mathematicae (2000)

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

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topTakens, 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] 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] 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] C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems, Cambridge Univ. Press, Cambridge, 1993.
- [4] R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), 193-202. Zbl0299.54031
- [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] K. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
- [7] P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Phys. D 13 (1984), 34-54.
- [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] 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] 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] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, preprint, 1998.
- [12] S. Isola, Dynamical zeta functions and correlation functions for non-uniformly hyperbolic transformations, preprint, 1995.
- [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] C. Liverani, B. Saussol, and S. Vaienti, A probabilistic approach to intermittency, preprint.
- [15] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196. Zbl0841.28012
- [16] Ya. B. Pesin, Dimension Theory in Dynamical Systems, Univ. of Chicago Press, Chicago, IL, 1997.
- [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] G. Pianigiani, First return map and invariant measures, Israel J. Math. 35 (1980), 32-48. Zbl0445.28016
- [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] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973. Zbl0271.26009
- [21] J. Schmeling, On the completeness of multifractal spectra, preprint WIAS, Berlin, 1996.
- [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] M. Urbański, ParabolicCantor sets, Fund. Math. 151 (1996), 241-277.
- [24] T. Ward, Entropy of Compact Group Automorphisms, lecture notes, 1994.
- [25] L.-S. Young, Recurrence times and rates of mixing, preprint, 1997.
- [26] M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, preprint. Zbl0971.37004

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