Localized Birkhoff average in beta dynamical systems

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 3191-3215. 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), 717-751. 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), 67-91. doi: 10.1016/S0021-7824(01)01228-4.
[4]	F. Blanchard, $\beta$-expansion and symbolic dynamics, Theor. Comp. Sci., 65 (1989), 131-141. 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), 775-790. doi: 10.1007/BF01055700.
[6]	G. Brown and Q. Yin, $\beta$-expansions and frequency of zero, Acta Math. Hungar., 84 (1999), 275-291. doi: 10.1023/A:1006625032066.
[7]	K. J. Falconer, "Fractal Geometry - Mathematical Foundations and Application," Wiley, New York, 1990.
[8]	A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimension and entropy, J. Lond. Math. Soc., 64 (2001), 229-244. 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), 1103-1128. 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), 1329-1343. doi: 10.1088/0951-7715/25/5/1329.
[11]	D. Färm and T. Persson, Non-typical points for $\beta$-shift, arXiv:1004.4812.
[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), 157-176. 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), 58-91. doi: 10.1006/aima.2001.2054.
[14]	F. Hofbauer, $\beta$-shifts have unique maximal measure, Monatsh. Math., 85 (1978), 189-198.
[15]	W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416.
[16]	T. Persson and J. Schmeling, Dyadic Diophantine approximation and Katok's horseshoe approximation, Acta Arith., 132 (2008), 205-230. 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), 237-261. 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), 929-956. doi: 10.1017/S0143385706000824.
[19]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hunger., 8 (1957), 477-493.
[20]	J. Schmeling, Symbolic dynamics for $\beta$-shfits and self-normal numbers, Ergod. Th. Dynam. Sys., 17 (1997), 675-694. 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), 317-348. doi: 10.1017/S0143385702000913.
[22]	B. Tan and B. W. Wang, Quantitive recurrence properties of beta dynamical systems, Adv. Math., 228 (2011), 2071-2097. 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), 5395-5414. 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), 2565-2579.
[25]	P. Walters, "An Introduction to Ergodic Theory," Grad. Texts in Math., 79, Springer-Verlag, New York/Berlin, 1982.

show all references

References:

[1]	J. Barral and S. Seuret, A localized Jarník-Besicovitch Theorem, Adv. Math., 226 (2011), 3191-3215. 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), 717-751. 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), 67-91. doi: 10.1016/S0021-7824(01)01228-4.
[4]	F. Blanchard, $\beta$-expansion and symbolic dynamics, Theor. Comp. Sci., 65 (1989), 131-141. 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), 775-790. doi: 10.1007/BF01055700.
[6]	G. Brown and Q. Yin, $\beta$-expansions and frequency of zero, Acta Math. Hungar., 84 (1999), 275-291. doi: 10.1023/A:1006625032066.
[7]	K. J. Falconer, "Fractal Geometry - Mathematical Foundations and Application," Wiley, New York, 1990.
[8]	A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimension and entropy, J. Lond. Math. Soc., 64 (2001), 229-244. 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), 1103-1128. 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), 1329-1343. doi: 10.1088/0951-7715/25/5/1329.
[11]	D. Färm and T. Persson, Non-typical points for $\beta$-shift, arXiv:1004.4812.
[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), 157-176. 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), 58-91. doi: 10.1006/aima.2001.2054.
[14]	F. Hofbauer, $\beta$-shifts have unique maximal measure, Monatsh. Math., 85 (1978), 189-198.
[15]	W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416.
[16]	T. Persson and J. Schmeling, Dyadic Diophantine approximation and Katok's horseshoe approximation, Acta Arith., 132 (2008), 205-230. 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), 237-261. 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), 929-956. doi: 10.1017/S0143385706000824.
[19]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hunger., 8 (1957), 477-493.
[20]	J. Schmeling, Symbolic dynamics for $\beta$-shfits and self-normal numbers, Ergod. Th. Dynam. Sys., 17 (1997), 675-694. 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), 317-348. doi: 10.1017/S0143385702000913.
[22]	B. Tan and B. W. Wang, Quantitive recurrence properties of beta dynamical systems, Adv. Math., 228 (2011), 2071-2097. 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), 5395-5414. 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), 2565-2579.
[25]	P. Walters, "An Introduction to Ergodic Theory," Grad. Texts in Math., 79, Springer-Verlag, New York/Berlin, 1982.

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