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July  2019, 39(7): 4157-4185. doi: 10.3934/dcds.2019168

Scaling properties of the Thue–Morse measure

1. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

2. 

Fachbereich Mathematik, Universität Bremen, Postfach 330440, 28359 Bremen, Germany

3. 

Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley St, Acton ACT 2601, Australia

Received  October 2018 Published  April 2019

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.

Citation: M. Baake, P. Gohlke, M. Kesseböhmer, T. Schindler. Scaling properties of the Thue–Morse measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4157-4185. doi: 10.3934/dcds.2019168
References:
[1] J.-P. Allouche and J. Shallit, Automatic 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. Google Scholar

[3]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139025256. Google Scholar

[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.Google Scholar

[5]

R. Bowen, Gibbs Measures and the Ergodic Theory of Anosov Diffiomorphisms, Springer, Berlin, 2008. Google Scholar

[6]

Z. ChengR. Savit and R. Merlin, Structure and electronic properties of Thue–Morse lattices, Phys. Rev. B, 37 (1982), 4375-4382. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A.-H. Fan, Multifractal analysis of infinite products, J. Stat. Phys., 86 (1997), 1313-1336. doi: 10.1007/BF02183625. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

M. Keane, Sur les mesures invariantes d'un recouvrement régulier, C.R. Acad. Sci. Paris, 272 (1971), A585–A587. Google Scholar

[15]

M. Keane, Strongly mixing $g$-measures, Invent. Math., 16 (1972), 309-324. doi: 10.1007/BF01425715. Google Scholar

[16] G. Keller, Equilibrium 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. Google Scholar

[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. Google Scholar

[19]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitsth. Verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471. Google Scholar

[20]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268 pp. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

M. Queffélec, Substitution Dynamical Systems – Spectral Analysis, 2nd ed., LNM 1294, Springer, Berlin, 2010.Google Scholar

[24]

M. Queffélec, Questions around the Thue–Morse sequence, Unif. Distrib. Th., 13 (2018), 1-25. doi: 10.1515/udt-2018-0001. Google Scholar

[25]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar

[26]

P. Walters, Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. doi: 10.2307/1997113. Google Scholar

[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. Google Scholar

show all references

References:
[1] J.-P. Allouche and J. Shallit, Automatic 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. Google Scholar

[3]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139025256. Google Scholar

[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.Google Scholar

[5]

R. Bowen, Gibbs Measures and the Ergodic Theory of Anosov Diffiomorphisms, Springer, Berlin, 2008. Google Scholar

[6]

Z. ChengR. Savit and R. Merlin, Structure and electronic properties of Thue–Morse lattices, Phys. Rev. B, 37 (1982), 4375-4382. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A.-H. Fan, Multifractal analysis of infinite products, J. Stat. Phys., 86 (1997), 1313-1336. doi: 10.1007/BF02183625. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

M. Keane, Sur les mesures invariantes d'un recouvrement régulier, C.R. Acad. Sci. Paris, 272 (1971), A585–A587. Google Scholar

[15]

M. Keane, Strongly mixing $g$-measures, Invent. Math., 16 (1972), 309-324. doi: 10.1007/BF01425715. Google Scholar

[16] G. Keller, Equilibrium 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. Google Scholar

[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. Google Scholar

[19]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitsth. Verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471. Google Scholar

[20]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 268 pp. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

M. Queffélec, Substitution Dynamical Systems – Spectral Analysis, 2nd ed., LNM 1294, Springer, Berlin, 2010.Google Scholar

[24]

M. Queffélec, Questions around the Thue–Morse sequence, Unif. Distrib. Th., 13 (2018), 1-25. doi: 10.1515/udt-2018-0001. Google Scholar

[25]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar

[26]

P. Walters, Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. doi: 10.2307/1997113. Google Scholar

[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. Google Scholar

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 $p(0) = p(1) = \log(2)$
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