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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 |
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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