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June  2013, 33(6): 2547-2564. doi: 10.3934/dcds.2013.33.2547

Localized Birkhoff average in beta dynamical systems

Bo Tan 1, , Bao-Wei Wang 1, , Jun Wu 1, and  Jian Xu 1,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China, China, China, China

Received  January 2012 Revised  October 2012 Published  December 2012

In this note, we investigate the localized multifractal spectrum of Birkhoff average in the beta-dynamical system $([0,1], T_{\beta})$ for general $\beta>1$, namely the dimension of the following level sets $$ \Big\{x\in [0,1]: \lim_{n\to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(T^jx)=f(x)\Big\}, $$ where $f$ and $\psi$ are two continuous functions defined on the unit interval $[0,1]$. Instead of a constant function in the classical multifractal cases, the function $f$ here varies with $x$. The method adopted in the proof indicates that the multifractal analysis of Birkhoff average in a general $\beta$-dynamical system can be achieved by approximating the system by its subsystems.
Keywords: $\beta$-expansion, Hausdorff dimension., localized Birkhoff average.
Mathematics Subject Classification: Primary: 11K55; Secondary: 28A8.
Citation: Bo Tan, Bao-Wei Wang, Jun Wu, Jian Xu. Localized Birkhoff average in beta dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2547-2564. doi: 10.3934/dcds.2013.33.2547
References:
[1]

J. Barral and S. Seuret, A localized Jarník-Besicovitch Theorem,, Adv. Math., 226 (2011), 3191.  doi: 10.1016/j.aim.2010.10.011.  Google Scholar

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F. Blanchard, $\beta$-expansion and symbolic dynamics,, Theor. Comp. Sci., 65 (1989), 131.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

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G. Brown and Q. Yin, $\beta$-expansions and frequency of zero,, Acta Math. Hungar., 84 (1999), 275.  doi: 10.1023/A:1006625032066.  Google Scholar

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A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimension and entropy,, J. Lond. Math. Soc., 64 (2001), 229.  doi: 10.1017/S0024610701002137.  Google Scholar

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A. H. Fan, L. M. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average,, Discrete Contin. Dyn. Syst., 21 (2008), 1103.  doi: 10.3934/dcds.2008.21.1103.  Google Scholar

[10]

A. H. Fan and B. W. Wang, On the lengths of basic intervals in beta expansion,, Nonlinearity, 25 (2012), 1329.  doi: 10.1088/0951-7715/25/5/1329.  Google Scholar

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D. Färm and T. Persson, Non-typical points for $\beta$-shift,, , ().   Google Scholar

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D. Färm, T. Persson and J. Schmeling, Dimenion of countable intersections of some sets arising in expansions in non-integer bases,, Fundamenta Math., 209 (2010), 157.  doi: 10.4064/fm209-2-4.  Google Scholar

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D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar

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F. Hofbauer, $\beta$-shifts have unique maximal measure,, Monatsh. Math., 85 (1978), 189.   Google Scholar

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W. Parry, On the $\beta$-expansions of real numbers,, Acta Math. Acad. Sci. Hunger., 11 (1960), 401.   Google Scholar

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T. Persson and J. Schmeling, Dyadic Diophantine approximation and Katok's horseshoe approximation,, Acta Arith., 132 (2008), 205.  doi: 10.4064/aa132-3-2.  Google Scholar

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C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar

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C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. Dynam. Sys., 27 (2007), 929.  doi: 10.1017/S0143385706000824.  Google Scholar

[19]

A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hunger., 8 (1957), 477.   Google Scholar

[20]

J. Schmeling, Symbolic dynamics for $\beta$-shfits and self-normal numbers,, Ergod. Th. Dynam. Sys., 17 (1997), 675.  doi: 10.1017/S0143385797079182.  Google Scholar

[21]

F. Takens and E. Verbitzkiy, On the variational principle for the topological entropy of certain non-compact sets,, Ergod. Theory Dyn. Syst., 23 (2003), 317.  doi: 10.1017/S0143385702000913.  Google Scholar

[22]

B. Tan and B. W. Wang, Quantitive recurrence properties of beta dynamical systems,, Adv. Math., 228 (2011), 2071.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

[23]

D. Thompson, Irregular sets, the $\beta$-transformation and the almost specification property,, Trans. Amer. Math. Soc., 364 (2012), 5395.  doi: 10.1090/S0002-9947-2012-05540-1.  Google Scholar

[24]

J. Verger-Gaugry, On gaps in Rényi $\beta$-expansions of unity for $\beta>1$ an algebraic number. Numeration, pavages, substitutions,, Ann. Inst. Fourier (Grenoble), 56 (2006), 2565.   Google Scholar

[25]

P. Walters, "An Introduction to Ergodic Theory,", Grad. Texts in Math., 79 (1982).   Google Scholar

show all references

References:
[1]

J. Barral and S. Seuret, A localized Jarník-Besicovitch Theorem,, Adv. Math., 226 (2011), 3191.  doi: 10.1016/j.aim.2010.10.011.  Google Scholar

[2]

J. Barral and Y. H. Qu, Loalized asymptotic behavior for almost additive potentials,, Discrete Contin. Dyn. Syst., 32 (2012), 717.  doi: 10.3934/dcds.2012.32.717.  Google Scholar

[3]

L. Barreira, B. Saussol and J. Schmeling, Higher dimensional multifractal analysis,, J. Math. Pure. Appl., 81 (2002), 67.  doi: 10.1016/S0021-7824(01)01228-4.  Google Scholar

[4]

F. Blanchard, $\beta$-expansion and symbolic dynamics,, Theor. Comp. Sci., 65 (1989), 131.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

[5]

G. Brown, G. Michon and J. Peyriére, On the multifractal analysis of measures,, J. Stat. Phys., 66 (1992), 775.  doi: 10.1007/BF01055700.  Google Scholar

[6]

G. Brown and Q. Yin, $\beta$-expansions and frequency of zero,, Acta Math. Hungar., 84 (1999), 275.  doi: 10.1023/A:1006625032066.  Google Scholar

[7]

K. J. Falconer, "Fractal Geometry - Mathematical Foundations and Application,", Wiley, (1990).   Google Scholar

[8]

A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimension and entropy,, J. Lond. Math. Soc., 64 (2001), 229.  doi: 10.1017/S0024610701002137.  Google Scholar

[9]

A. H. Fan, L. M. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average,, Discrete Contin. Dyn. Syst., 21 (2008), 1103.  doi: 10.3934/dcds.2008.21.1103.  Google Scholar

[10]

A. H. Fan and B. W. Wang, On the lengths of basic intervals in beta expansion,, Nonlinearity, 25 (2012), 1329.  doi: 10.1088/0951-7715/25/5/1329.  Google Scholar

[11]

D. Färm and T. Persson, Non-typical points for $\beta$-shift,, , ().   Google Scholar

[12]

D. Färm, T. Persson and J. Schmeling, Dimenion of countable intersections of some sets arising in expansions in non-integer bases,, Fundamenta Math., 209 (2010), 157.  doi: 10.4064/fm209-2-4.  Google Scholar

[13]

D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar

[14]

F. Hofbauer, $\beta$-shifts have unique maximal measure,, Monatsh. Math., 85 (1978), 189.   Google Scholar

[15]

W. Parry, On the $\beta$-expansions of real numbers,, Acta Math. Acad. Sci. Hunger., 11 (1960), 401.   Google Scholar

[16]

T. Persson and J. Schmeling, Dyadic Diophantine approximation and Katok's horseshoe approximation,, Acta Arith., 132 (2008), 205.  doi: 10.4064/aa132-3-2.  Google Scholar

[17]

C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar

[18]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. Dynam. Sys., 27 (2007), 929.  doi: 10.1017/S0143385706000824.  Google Scholar

[19]

A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hunger., 8 (1957), 477.   Google Scholar

[20]

J. Schmeling, Symbolic dynamics for $\beta$-shfits and self-normal numbers,, Ergod. Th. Dynam. Sys., 17 (1997), 675.  doi: 10.1017/S0143385797079182.  Google Scholar

[21]

F. Takens and E. Verbitzkiy, On the variational principle for the topological entropy of certain non-compact sets,, Ergod. Theory Dyn. Syst., 23 (2003), 317.  doi: 10.1017/S0143385702000913.  Google Scholar

[22]

B. Tan and B. W. Wang, Quantitive recurrence properties of beta dynamical systems,, Adv. Math., 228 (2011), 2071.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

[23]

D. Thompson, Irregular sets, the $\beta$-transformation and the almost specification property,, Trans. Amer. Math. Soc., 364 (2012), 5395.  doi: 10.1090/S0002-9947-2012-05540-1.  Google Scholar

[24]

J. Verger-Gaugry, On gaps in Rényi $\beta$-expansions of unity for $\beta>1$ an algebraic number. Numeration, pavages, substitutions,, Ann. Inst. Fourier (Grenoble), 56 (2006), 2565.   Google Scholar

[25]

P. Walters, "An Introduction to Ergodic Theory,", Grad. Texts in Math., 79 (1982).   Google Scholar

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