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Scaling properties of the Thue–Morse measure

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  • The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

    Mathematics Subject Classification: 37D35, 37C45, 52C23.


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  • Figure 1.  The graph of the Birkhoff spectrum $b$ from Eq. (3)

    Figure 2.  Illustration of the graphs of $\psi(x)$ (solid line), $\psi(2x)$ (dashed line) and $\psi(4x)$ (dotted line)

    Figure 3.  Illustration of $\psi^{}_{3} (x)$ (dotted line) and $\psi^{}_{5} (x)$ (solid line)

    Figure 4.  The graph of the pressure function $p$ (solid line) with the two asymptotes $x\mapsto \log(3/2) x$ and $x\mapsto\left(1-x\right)\log(2)$ (dashed lines). The dotted lines are added to illustrate $p(0) = p(1) = \log(2)$

  • [1] J.-P. Allouche and  J. ShallitAutomatic Sequences, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511546563.
    [2] M. Baake and M. Coons, A natural probability measure derived from Stern's diatomic sequence, Acta Arithm., 183 (2018), 87–99, arXiv: 1706.00187. doi: 10.4064/aa170709-22-1.
    [3] M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139025256.
    [4] M. Baake, U. Grimm and J. Nilsson, Scaling of the Thue–Morse diffraction measure, Acta Phys. Pol. A, 126 (2014), 431–434, arXiv: 1311.4371.
    [5] R. Bowen, Gibbs Measures and the Ergodic Theory of Anosov Diffiomorphisms, Springer, Berlin, 2008.
    [6] Z. ChengR. Savit and R. Merlin, Structure and electronic properties of Thue–Morse lattices, Phys. Rev. B, 37 (1982), 4375-4382. 
    [7] M. Denker and M. Kesseböhmer, Thermodynamic formalism, large deviation, and multifractals, in Stochastic Climate Models, P. Imkeller and J.-S. von Storch (eds.), Birkhäuser, Basel, 49 (2001), 159–169.
    [8] A.-H. Fan, On uniqueness of $G$-measures and $g$-measures, Studia Math., 119 (1996), 255-269.  doi: 10.4064/sm-119-3-255-269.
    [9] A.-H. Fan, Multifractal analysis of infinite products, J. Stat. Phys., 86 (1997), 1313-1336.  doi: 10.1007/BF02183625.
    [10] A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arithm., 13 (1967/68), 259-265.  doi: 10.4064/aa-13-3-259-265.
    [11] C. Godrèche and J. M. Luck, Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures, J. Phys. A: Math. Gen., 23 (1990), 3769-3797. 
    [12] J. Jaerisch and M. Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc., 363 (2011), 313–330; arXiv: 0902.2473. doi: 10.1090/S0002-9947-2010-05326-7.
    [13] S. Kakutani, Strictly ergodic symbolic dynamical systems, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, L.M. Le Cam, J. Neyman and E.L. Scott (eds.), University of California Press, Berkeley, CA (1972), 319–326.
    [14] M. Keane, Sur les mesures invariantes d'un recouvrement régulier, C.R. Acad. Sci. Paris, 272 (1971), A585–A587.
    [15] M. Keane, Strongly mixing $g$-measures, Invent. Math., 16 (1972), 309-324.  doi: 10.1007/BF01425715.
    [16] G. KellerEquilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781107359987.
    [17] M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math. (Crelle), 605 (2007), 133–163; arXiv: 0509603. doi: 10.1515/CRELLE.2007.029.
    [18] D.H. Kim, L. Liao, M. Rams and B. Wang, Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential, Fractals, 26 (2019), 1850026, 13pp, arXiv: 1707.06059. doi: 10.1142/S0218348X18500263.
    [19] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitsth. Verw. Gebiete, 30 (1974), 185-202.  doi: 10.1007/BF00533471.
    [20] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268 pp.
    [21] L. Peng and T. Kamae, Spectral measure of the Thue–Morse sequence and the dynamical system and random walk related to it, Ergodic Th. & Dynam. Syst., 36 (2016), 1246-1259.  doi: 10.1017/etds.2014.121.
    [22] Y. Pesin and H. Weiss, The multifractal analysis of Birkhoff averages and large deviations, in, Global Analysis of Dynamical Systems, H.W. Broer, B. Krauskopf and G. Vegter (eds.), IoP Publishing, Bristol and Philadelphia, (2001), 419–431.
    [23] M. Queffélec, Substitution Dynamical Systems – Spectral Analysis, 2nd ed., LNM 1294, Springer, Berlin, 2010.
    [24] M. Queffélec, Questions around the Thue–Morse sequence, Unif. Distrib. Th., 13 (2018), 1-25.  doi: 10.1515/udt-2018-0001.
    [25] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278.  doi: 10.1007/BF01654281.
    [26] P. Walters, Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387.  doi: 10.2307/1997113.
    [27] M. A. ZaksA. S. Pikovsky and J. Kurths, On the generalized dimensions for the Fourier spectrum of the Thue–Morse sequence, J. Phys. A: Math. Gen., 32 (1999), 1523-1530.  doi: 10.1088/0305-4470/32/8/018.
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