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

Localized Birkhoff average in beta dynamical systems

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.
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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

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