Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T03:24:09.810Z Has data issue: false hasContentIssue false
  • English
  • Français

The thermodynamic approach to multifractal analysis

Published online by Cambridge University Press:  04 August 2014

VAUGHN CLIMENHAGA
VAUGHN CLIMENHAGA*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Most results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

Type
Survey
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 5 , October 2014 , pp. 1409 - 1450
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barral, J. and Qu, Y.-H.. Localized asymptotic behavior for almost additive potentials. Discrete Contin. Dyn. Syst. 32(3) (2012), 717–751.Google Scholar
Barral, J. and Qu, Y.-H.. On the higher-dimensional multifractal analysis. Discrete Contin. Dyn. Syst. 32(6) (2012), 1977–1995.Google Scholar
Barreira, L.. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 16 (1996), 871–928.Google Scholar
Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics, 272). Birkhäuser, Basel, 2008.Google Scholar
Barreira, L. and Doutor, P.. Birkhoff averages for hyperbolic flows: variational principles and applications. J. Statist. Phys. 115(5-6) (2004), 1567–1603.CrossRefGoogle Scholar
Barreira, L. and Doutor, P.. Almost additive multifractal analysis. J. Math. Pures Appl. (9) 92(1) (2009), 1–17.Google Scholar
Barreira, L. and Gelfert, K.. Multifractal analysis for Lyapunov exponents on nonconformal repellers. Comm. Math. Phys. 267(2) (2006), 393–418.CrossRefGoogle Scholar
Barreira, L. and Iommi, G.. Multifractal analysis and phase transitions for hyperbolic and parabolic horseshoes. Israel J. Math. 181 (2011), 347–379.Google Scholar
Barreira, L., Pesin, Y. and Schmeling, J.. On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos 7(1) (1997), 27–38.Google Scholar
Barreira, L., Pesin, Y. and Schmeling, J.. Multifractal spectra and multifractal rigidity for horseshoes. J. Dyn. Control Syst. 3(1) (1997), 33–49.Google Scholar
Barreira, L. and Radu, L.. Multifractal analysis of non-conformal repellers: a model case. Dyn. Syst. 22(2) (2007), 147–168.Google Scholar
Barreira, L. and Saraiva, V.. Multifractal nonrigidity of topological Markov chains. J. Stat. Phys. 130(2) (2008), 387–412.Google Scholar
Barreira, L. and Saussol, B.. Variational principles and mixed multifractal spectra. Trans. Amer. Math. Soc. 353(10) (2001), 3919–3944 (electronic).Google Scholar
Barreira, L., Saussol, B. and Schmeling, J.. Higher-dimensional multifractal analysis. J. Math. Pures Appl. (9) 81(1) (2002), 67–91.Google Scholar
Barreira, L. and Schmeling, J.. Sets of ‘non-typical’ points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 29–70.Google Scholar
Benzi, R., Paladin, G., Parisi, G. and Vulpiani, A.. On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17(18) (1984), 3521–3531.Google Scholar
Blokh, A. M.. Decomposition of dynamical systems on an interval. Uspekhi Mat. Nauk. 38(5/233) (1983), 179–180.Google Scholar
Bohr, T. and Rand, D.. The entropy function for characteristic exponents. Phys. D 25(1–3) (1987), 387–398.Google Scholar
Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math Soc. 184 (1973), 125–136.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory 8(3) (1974/75), 193–202.Google Scholar
Brown, G., Michon, G. and Peyrière, J.. On the multifractal analysis of measures. J. Stat. Phys. 66(3–4) (1992), 775–790.Google Scholar
Bruin, H. and Keller, G.. Equilibrium states for S-unimodal maps. Ergod. Th. & Dynam. Sys. 18(4) (1998), 765–789.Google Scholar
Bruin, H. and Todd, M.. Equilibrium states for interval maps: the potential − t log|D f|. Ann. Sci. Éc. Norm. Supér. (4) 42(4) (2009), 559–600.CrossRefGoogle Scholar
Buzzi, J.. Specification on the interval. Trans. Amer. Math. Soc. 349(7) (1997), 2737–2754.Google Scholar
Cao, Y.. Dimension spectrum of asymptotically additive potentials for C 1 average conformal repellers. Nonlinearity 26(9) (2013), 2441–2468.Google Scholar
Cawley, R. and Mauldin, R. D.. Multifractal decompositions of Moran fractals. Adv. Math. 92(2) (1992), 196–236.Google Scholar
Chen, E., Küpper, T. and Shu, L.. Topological entropy for divergence points. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1173–1208.Google Scholar
Chung, Y. M.. Birkhoff spectra for one-dimensional maps with some hyperbolicity. Stoch. Dyn. 10(1) (2010), 53–75.Google Scholar
Climenhaga, V.. Multifractal formalism derived from thermodynamics. Preprint, 2010, arXiv:1002.0789.CrossRefGoogle Scholar
Climenhaga, V.. Multifractal formalism derived from thermodynamics for general dynamical systems. Electron. Res. Announc. Math. Sci. 17 (2010), 1–11.Google Scholar
Climenhaga, V.. Bowen’s equation in the non-uniform setting. Ergod. Th. & Dynam. Sys. 31 (2011), 1163–1182.Google Scholar
Climenhaga, V.. Topological pressure of simultaneous level sets. Nonlinearity 26 (2013), 241–268.CrossRefGoogle Scholar
Collet, P., Lebowitz, J. L. and Porzio, A.. The dimension spectrum of some dynamical systems. Proceedings of the symposium on statistical mechanics of phase transitions—mathematical and physical aspects (Trebon 1986). J. Stat. Phys. 47(5–6) (1986), 609–644.Google Scholar
Edgar, G. A. and Mauldin, R. D.. Multifractal decompositions of digraph recursive fractals. Proc. Lond. Math. Soc. (3) 65(3) (1992), 604–628.CrossRefGoogle Scholar
Falconer, K. J.. Bounded distortion and dimension for nonconformal repellers. Math. Proc. Cambridge Philos. Soc. 115(2) (1994), 315–334.Google Scholar
Falconer, K. J.. The multifractal spectrum of statistically self-similar measures. J. Theoret. Probab. 7(3) (1994), 681–702.Google Scholar
Falconer, K. J.. Fractal Geometry (Mathematical Foundations and Applications), 2nd edn. John Wiley & Sons, Hoboken, NJ, 2003.Google Scholar
Fan, A.-H. and Feng, D.-J.. On the distribution of long-term time averages on symbolic space. J. Stat. Phys. 99(3–4) (2000), 813–856.CrossRefGoogle Scholar
Fan, A.-H., Feng, D.-J. and Wu, J.. Recurrence, dimension and entropy. J. Lond. Math. Soc. (2) 64(1) (2001), 229–244.Google Scholar
Fan, A.-H., Liao, L. and Ma, J.-H.. Level sets of multiple ergodic averages. Monatsh. Math. 168(1) (2012), 17–26.Google Scholar
Fan, A., Liao, L. and Peyrière, J.. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete Contin. Dyn. Syst. 21(4) (2008), 1103–1128.Google Scholar
Fan, A., Schmeling, J. and Wu, M.. Multifractal analysis of multiple ergodic averages. C. R. Math. Acad. Sci. Paris 349(17–18) (2011), 961–964.Google Scholar
Feng, D.-J.. Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices. Israel J. Math. 138 (2003), 353–376.Google Scholar
Feng, D.-J.. Lyapunov exponents for products of matrices and multifractal analysis. II. General matrices. Israel J. Math. 170 (2009), 355–394.Google Scholar
Feng, D.-J. and Huang, W.. Lyapunov spectrum of asymptotically sub-additive potentials. Commun. Math. Phys. 297 (2010), 1–43.Google Scholar
Feng, D.-J. and Olivier, E.. Multifractal analysis of weak Gibbs measures and phase transition—application to some Bernoulli convolutions. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1751–1784.CrossRefGoogle Scholar
Gelfert, K., Przytycki, F. and Rams, M.. On the Lyapunov spectrum for rational maps. Math. Ann. 348(4) (2010), 965–1004.Google Scholar
Gelfert, K., Przytycki, F., Rams, M. and Rivera-Letelier, J.. Lyapunov spectrum for exceptional rational maps. Ann. Acad. Sci. Fenn. Math. 38(2) (2013), 631–656.Google Scholar
Gelfert, K. and Rams, M.. The Lyapunov spectrum of some parabolic systems. Ergod. Th. & Dynam. Sys. 29 (2009), 919–940.Google Scholar
Hadyn, N., Luevano, J., Mantica, G. and Vaienti, S.. Multifractal properties of return time statistics. Phys. Rev. Lett. 88(22) (2002), 224502.Google Scholar
Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I. and Shraiman, B.. Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A (3) 33(2) (1986), 1141–1151.Google Scholar
Hanus, P., Mauldin, R. D. and Urbański, M.. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungar. 96(1–2) (2002), 27–98.Google Scholar
Hofbauer, F.. Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228 (1977), 223–241.Google Scholar
Hofbauer, F.. Local dimension for piecewise monotonic maps on the interval. Ergod. Th. & Dynam. Sys. 15(6) (1995), 1119–1142.Google Scholar
Hofbauer, F.. Multifractal spectra of Birkhoff averages for a piecewise monotone interval map. Fund. Math. 208(2) (2010), 95–121.Google Scholar
Hu, H.. Equilibriums of some non-Hölder potentials. Trans. Amer. Math. Soc. 360(4) (2008), 2153–2190.Google Scholar
Iommi, G.. Multifractal analysis for countable Markov shifts. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1881–1907.CrossRefGoogle Scholar
Iommi, G.. Multifractal analysis of the Lyapunov exponent for the backward continued fraction map. Ergod. Th. & Dynam. Sys. 30(1) (2010), 211–232.Google Scholar
Iommi, G. and Jordan, T.. Multifractal analysis of Birkhoff averages for countable Markov maps. Preprint, 2011, arXiv:1003.2979.Google Scholar
Iommi, G. and Kiwi, J.. The Lyapunov spectrum is not always concave. J. Stat. Phys. 135(3) (2009), 535–546.Google Scholar
Iommi, G. and Todd, M.. Natural equilibrium states for multimodal maps. Comm. Math. Phys. 300(1) (2010), 65–94.Google Scholar
Iommi, G. and Todd, M.. Dimension theory for multimodal maps. Ann. Henri Poincaré 12(3) (2011), 591–620.CrossRefGoogle Scholar
Johansson, A., Jordan, T. M., Öberg, A. and Pollicott, M.. Multifractal analysis of non-uniformly hyperbolic systems. Israel J. Math. 177 (2010), 125–144.Google Scholar
Jordan, T. and Pollicott, M.. Multifractal analysis and the variance of Gibbs measures. J. Lond. Math. Soc. (2) 76(1) (2007), 57–72.Google Scholar
Jordan, T. and Rams, M.. Multifractal analysis for Bedford-McMullen carpets. Math. Proc. Cambridge Philos. Soc. 150(1) (2011), 147–156.Google Scholar
Jordan, T. and Rams, M.. Multifractal analysis of weak Gibbs measures for non-uniformly expanding C 1 maps. Ergod. Th. & Dynam. Sys. 31(1) (2011), 143–164.Google Scholar
Jordan, T. and Simon, K.. Multifractal analysis of Birkhoff averages for some self-affine IFS. Dyn. Syst. 22(4) (2007), 469–483.Google Scholar
Käenmäki, A., Rajala, T. and Suomala, V.. Local multifractal analysis in metric spaces. Nonlinearity 26(8) (2013), 2157–2173.Google Scholar
Keller, G.. Equilibrium States in Ergodic Theory. London Mathematical Society, London, 1998.Google Scholar
Kesseböhmer, M.. Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14(2) (2001), 395–409.CrossRefGoogle Scholar
Kesseböhmer, M. and Stratmann, B. O.. Homology at infinity; fractal geometry of limiting symbols for modular subgroups. Topology 46(5) (2007), 469–491.CrossRefGoogle Scholar
Kesseböhmer, M. and Stratmann, B. O.. Stern-Brocot pressure and multifractal spectra in ergodic theory of numbers. Stoch. Dyn. 4(1) (2004), 77–84.Google Scholar
Kesseböhmer, M. and Urbański, M.. Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems. Nonlinearity 20(8) (2007), 1969–1985.Google Scholar
King, J. F.. The singularity spectrum for general Sierpiński carpets. Adv. Math. 116(1) (1995), 1–11.Google Scholar
Kucherenko, T. and Wolf, C.. Geometry and entropy of generalized rotation sets. Israel J. Math. in press, doi:10.1007/s11856-013-0053-4.Google Scholar
Lau, K.-S. and Ngai, S.-M.. L q-spectrum of the Bernoulli convolution associated with the golden ratio. Studia Math. 131(3) (1998), 225–251.Google Scholar
Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math. 141(1) (1999), 45–96.Google Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials. Commun. Math. Phys. 328 (2014), 397–419.Google Scholar
Lopes, A. O.. The dimension spectrum of the maximal measure. SIAM J. Math. Anal. 20(5) (1989), 1243–1254.Google Scholar
Mauldin, R. D. and Urbański, M.. Graph directed Markov systems. Geometry and Dynamics of Limit Sets (Cambridge Tracts in Mathematics, 148). Cambridge University Press, Cambridge, 2003.Google Scholar
Meson, A. and Vericat, F.. Higher multifractal analysis for simultaneous actions of Fuchsian groups. J. Dyn. Syst. Geom. Theor. 5(2) (2007), 125–140.Google Scholar
Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20(3) (2000), 843–857.Google Scholar
Oliveira, K. and Viana, M.. Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps. Ergod. Th. & Dynam. Sys. 28(2) (2008), 501–533.Google Scholar
Olivier, E.. Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures. Nonlinearity 12(6) (1999), 1571–1585.Google Scholar
Olivier, E.. Structure multifractale d’une dynamique non expansive définie sur un ensemble de Cantor. C. R. Acad. Sci. Paris Sér. I Math. 331(8) (2000), 605–610.Google Scholar
Olsen, L.. Geometric constructions in multifractal geometry. International Conference on Dimension and Dynamics (Miskolc, 1998). Period. Math. Hungar. 37(1–3) (1998), 81–99.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. (9) 82(12) (2003), 1591–1649.Google Scholar
Olsen, L.. Slow and fast convergence to local dimensions of self-similar measures. Math. Nachr. 266 (2004), 68–80.Google Scholar
Olsen, L.. Mixed generalized dimensions of self-similar measures. J. Math. Anal. Appl. 306(2) (2005), 516–539.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV. Divergence points and packing dimension. Bull. Sci. Math. 132(8) (2008), 650–678.Google Scholar
Olsen, L.. On the inverse multifractal formalism. Glasg. Math. J. 52(1) (2010), 179–194.Google Scholar
Olsen, L.. First return times: multifractal spectra and divergence points. Discrete Contin. Dyn. Syst. 10(3) (2004), 635–656.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of the deformed measure theoretical Birkhoff averages. III. Aequationes Math. 71(1–2) (2006), 29–53.Google Scholar
Olsen, L. and Winter, S.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II. Non-linearity, divergence points and Banach space valued spectra. Bull. Sci. Math. 131(6) (2007), 518–558.CrossRefGoogle Scholar
Pesin, Y. B. and Pitskel’, B. S.. Topological pressure and the variational principle for noncompact sets. Funktsional. Anal. i Prilozhen. 18(4) (1984), 50–63 96.Google Scholar
Pesin, Y. B. and Sadovskaya, V.. Multifractal analysis of conformal Axiom A flows. Comm. Math. Phys. 216(2) (2001), 277–312.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, IL, 1998.Google Scholar
Pesin, Y. and Senti, S.. Equilibrium measures for maps with inducing schemes. J. Modern Dyn. 2(3) (2008), 1–31.Google Scholar
Pesin, Y. and Weiss, H.. A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86(1–2) (1997), 233–275.Google Scholar
Pesin, Y. and Weiss, H.. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1) (1997), 89–106.Google Scholar
Pesin, Y. and Weiss, H.. The multifractal analysis of Birkhoff averages and large deviations. Global Analysis of Dynamical Systems. Eds. Broer, H., Krauskopf, B. and Vegter, G.. IoP Publishing, Bristol, UK, 2001.Google Scholar
Pesin, Y. and Zhang, K.. Phase transitions for uniformly expanding maps. J. Stat. Phys. 122(6) (2006), 1095–1110.Google Scholar
Pfister, C.-E. and Sullivan, W. G.. On the topological entropy of saturated sets. Ergod. Th. & Dynam. Sys. 27(3) (2007), 929–956.Google Scholar
Pollicott, M. and Weiss, H.. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville–Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys. 207(1) (1999), 145–171.Google Scholar
Rand, D. A.. The singularity spectrum f (α) for cookie-cutters. Ergod. Th. & Dynam. Sys. 9(3) (1989), 527–541.Google Scholar
Reeve, H. W. J.. Multifractal analysis for Birkhoff averages on Lalley–Gatzouras repellers. Fund. Math. 212(1) (2011), 71–93.CrossRefGoogle Scholar
Reeve, H. W. J.. The packing spectrum for Birkhoff averages on a self-affine repeller. Ergod. Th. & Dynam. Sys. 32(4) (2012), 1444–1470.Google Scholar
Riedi, R. H. and Mandelbrot, B. B.. Inversion formula for continuous multifractals. Adv. Appl. Math. 19(3) (1997), 332–354.Google Scholar
Riedi, R. H. and Mandelbrot, B. B.. Exceptions to the multifractal formalism for discontinuous measures. Math. Proc. Cambridge Philos. Soc. 123(1) (1998), 133–157.Google Scholar
Rockafellar, R. T.. Convex Analysis (Princeton Mathematical Series, 28). Princeton University Press, Princeton, NJ, 1970.Google Scholar
Rudolph, O.. Thermodynamic and multifractal formalism and the Bowen–Series map. Fortschr. Phys. 43(5) (1995), 349–450.Google Scholar
Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.Google Scholar
Shereshevsky, M. A.. A complement to Young’s theorem on measure dimension: the difference between lower and upper pointwise dimensions. Nonlinearity 4(1) (1991), 15–25.Google Scholar
Simpelaere, D.. Dimension spectrum of Axiom A diffeomorphisms. I. The Bowen–Margulis measure. J. Statist. Phys. 76(5–6) (1994), 1329–1358.Google Scholar
Simpelaere, D.. Dimension spectrum of Axiom A diffeomorphisms. II. Gibbs measures. J. Stat. Phys. 76(5–6) (1994), 1359–1375.Google Scholar
Takens, F. and Verbitski, E.. Multifractal analysis of local entropies for expansive homeomorphisms with specification. Comm. Math. Phys. 203(3) (1999), 593–612.Google Scholar
Takens, F. and Verbitski, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dynam. Sys. 23(1) (2003), 317–348.Google Scholar
Tan, B., Wang, B.-W., Wu, J. and Xu, J.. Localized Birkhoff average in beta dynamical systems. Discrete Contin. Dyn. Syst. 33(6) (2013), 2547–2564.Google Scholar
Testud, B.. Phase transitions for the multifractal analysis of self-similar measures. Nonlinearity 19(5) (2006), 1201–1217.Google Scholar
Thompson, D.. A variational principle for topological pressure for certain non-compact sets. J. Lond. Math. Soc. (2) 80(3) (2009), 585–602.Google Scholar
Thompson, D.. The irregular set for maps with the specification property has full topological pressure. Dyn. Syst. 25(1) (2010), 25–51.Google Scholar
Todd, M.. Multifractal analysis for multimodal maps. Preprint, 2010.Google Scholar
Touchette, H. and Beck, C.. Nonconcave entropies in multifractals and the thermodynamic formalism. J. Stat. Phys. 125(2) (2006), 459–475.Google Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2) (2010), 555–593.Google Scholar
Weiss, H.. The Lyapunov spectrum for conformal expanding maps and Axiom-A surface diffeomorphisms. J. Stat. Phys. 95(3–4) (1999), 615–632.Google Scholar
Yuri, M.. Multifractal analysis of weak Gibbs measures for intermittent systems. Comm. Math. Phys. 230(2) (2002), 365–388.Google Scholar
Zindulka, O.. Hentschel–Procaccia spectra in separable metric spaces. Real Anal. Exchange 26 (2002), 115–120.Google Scholar