May  2022, 42(5): 2461-2497. doi: 10.3934/dcds.2021199

On the multifractal spectrum of weighted Birkhoff averages

1. 

Budapest University of Technology and Economics, Department of Stochastics, MTA-BME Stochastics Research Group, P.O.Box 91, 1521 Budapest, Hungary

2. 

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland

3. 

Sorbonne Université, LPSM, 75005 Paris, France

*Corresponding author: Balázs Bárány

Received  September 2020 Revised  September 2021 Published  May 2022 Early access  January 2022

Fund Project: Balázs Bárány acknowledges support from grants OTKA K123782 and OTKA FK134251. Michaƚ Rams was supported by National Science Centre grant 2019/33/B/ST1/00275 (Poland)

In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.

Citation: Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199
References:
[1]

E. H. El AbdalaouiJ. Kuƚaga-PrzymusM. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899-2944.  doi: 10.3934/dcds.2017125.

[2]

L. BarreiraB. Saussol and J. Schmeling, Distribution of frequencies of digits via multifractal analysis, J. Number Theory, 97 (2002), 410-438.  doi: 10.1016/S0022-314X(02)00003-3.

[3]

L. BarreiraB. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, J. Math. Pures Appl., 81 (2002), 67-91.  doi: 10.1016/S0021-7824(01)01228-4.

[4]

A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann., 110 (1935), 321-330.  doi: 10.1007/BF01448030.

[5] C. J. Bishop and Y. Peres, Fractals in Probability and Analysis, Cambridge Studies in Advanced Mathematics, vol. 162, Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781316460238.
[6]

F. Cellarosi and Y. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 1 (2011), 245-275.  doi: 10.1007/s13373-011-0011-6.

[7]

H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math., Oxford Ser, 20 (1949), 31–36. doi: 10.1093/qmath/os-20.1.31.

[8]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272. Springer-Verlag, New York, 2015. doi: 10.1007/978-3-319-16898-2.

[9]

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

[10]

A. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275-1289.  doi: 10.1017/etds.2017.81.

[11]

A. Fan, Multifractal analysis of weighted ergodic averages, Adv. Math., 377 (2021), 107488.  doi: 10.1016/j.aim.2020.107488.

[12]

A. Fan and D. Feng, Analyse multifractale de la recurrence sur l'espace symbolique, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 629-632.  doi: 10.1016/S0764-4442(99)80091-3.

[13]

A. Fan and D. Feng, On the distribution of long-term time averages on symbolic space, J. Statist. Phys., 99 (2000), 813-856.  doi: 10.1023/A:1018643512559.

[14]

A. FanD. Feng and W. Jun, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.

[15]

A. Fan and Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 1709-1744.  doi: 10.1017/etds.2016.121.

[16]

A. FanL. Liao and J. Peyriere, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., 21 (2008), 1103-1128.  doi: 10.3934/dcds.2008.21.1103.

[17]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.

[18]

D. FengK. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.  doi: 10.1006/aima.2001.2054.

[19]

S. FerencziJ. Kuƚaga-Przymus and M. Lemańczyk, Sarnak's conjecture: What's new, Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, 2213 (2018), 163-235. 

[20]

K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergodic Theory Dynam. Systems, 29 (2009), 919-940.  doi: 10.1017/S0143385708080462.

[21]

Y. Heurteaux, Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 309-338.  doi: 10.1016/S0246-0203(98)80014-9.

[22]

H. B. Keynes and J. B. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59.  doi: 10.1007/BF01695625.

[23]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.  doi: 10.1112/jlms/s2-16.3.568.

[24]

E. Olivier, Analyse multifractale de fonctions continues, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1171-1174.  doi: 10.1016/S0764-4442(98)80221-8.

[25]

M. Rams, On some non-conformal fractals, Nonlinearity, 23 (2010), 2423-2428.  doi: 10.1088/0951-7715/23/10/004.

[26]

V. A. Rohlin, Lectures on the entropy theory of measure-preserving transformations, Russian Math. Surveys, 22 (1967), 1-52. 

[27]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

[28]

J. Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems, 19 (1999), 1595-1616.  doi: 10.1017/S0143385799151988.

[29]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.

[30]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

E. H. El AbdalaouiJ. Kuƚaga-PrzymusM. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899-2944.  doi: 10.3934/dcds.2017125.

[2]

L. BarreiraB. Saussol and J. Schmeling, Distribution of frequencies of digits via multifractal analysis, J. Number Theory, 97 (2002), 410-438.  doi: 10.1016/S0022-314X(02)00003-3.

[3]

L. BarreiraB. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, J. Math. Pures Appl., 81 (2002), 67-91.  doi: 10.1016/S0021-7824(01)01228-4.

[4]

A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann., 110 (1935), 321-330.  doi: 10.1007/BF01448030.

[5] C. J. Bishop and Y. Peres, Fractals in Probability and Analysis, Cambridge Studies in Advanced Mathematics, vol. 162, Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781316460238.
[6]

F. Cellarosi and Y. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 1 (2011), 245-275.  doi: 10.1007/s13373-011-0011-6.

[7]

H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math., Oxford Ser, 20 (1949), 31–36. doi: 10.1093/qmath/os-20.1.31.

[8]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272. Springer-Verlag, New York, 2015. doi: 10.1007/978-3-319-16898-2.

[9]

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

[10]

A. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275-1289.  doi: 10.1017/etds.2017.81.

[11]

A. Fan, Multifractal analysis of weighted ergodic averages, Adv. Math., 377 (2021), 107488.  doi: 10.1016/j.aim.2020.107488.

[12]

A. Fan and D. Feng, Analyse multifractale de la recurrence sur l'espace symbolique, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 629-632.  doi: 10.1016/S0764-4442(99)80091-3.

[13]

A. Fan and D. Feng, On the distribution of long-term time averages on symbolic space, J. Statist. Phys., 99 (2000), 813-856.  doi: 10.1023/A:1018643512559.

[14]

A. FanD. Feng and W. Jun, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.

[15]

A. Fan and Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 1709-1744.  doi: 10.1017/etds.2016.121.

[16]

A. FanL. Liao and J. Peyriere, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., 21 (2008), 1103-1128.  doi: 10.3934/dcds.2008.21.1103.

[17]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.

[18]

D. FengK. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.  doi: 10.1006/aima.2001.2054.

[19]

S. FerencziJ. Kuƚaga-Przymus and M. Lemańczyk, Sarnak's conjecture: What's new, Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, 2213 (2018), 163-235. 

[20]

K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergodic Theory Dynam. Systems, 29 (2009), 919-940.  doi: 10.1017/S0143385708080462.

[21]

Y. Heurteaux, Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 309-338.  doi: 10.1016/S0246-0203(98)80014-9.

[22]

H. B. Keynes and J. B. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59.  doi: 10.1007/BF01695625.

[23]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.  doi: 10.1112/jlms/s2-16.3.568.

[24]

E. Olivier, Analyse multifractale de fonctions continues, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1171-1174.  doi: 10.1016/S0764-4442(98)80221-8.

[25]

M. Rams, On some non-conformal fractals, Nonlinearity, 23 (2010), 2423-2428.  doi: 10.1088/0951-7715/23/10/004.

[26]

V. A. Rohlin, Lectures on the entropy theory of measure-preserving transformations, Russian Math. Surveys, 22 (1967), 1-52. 

[27]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

[28]

J. Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems, 19 (1999), 1595-1616.  doi: 10.1017/S0143385799151988.

[29]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.

[30]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

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