# American Institute of Mathematical Sciences

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 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.
 [1] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 [2] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [3] Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055 [4] Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103 [5] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [6] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [7] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [8] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [9] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [10] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [11] Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098 [12] Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125 [13] Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 [14] Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417 [15] Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 [16] Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006 [17] Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699 [18] Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077 [19] Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 [20] David Färm, Tomas Persson. Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3525-3537. doi: 10.3934/dcds.2012.32.3525

2021 Impact Factor: 1.588