TY - JOUR
T1 - MULTIFRACTAL ANALYSIS of the BIRKHOFF SUMS of SAINT-PETERSBURG POTENTIAL
AU - Kim, Dong Han
AU - Liao, Lingmin
AU - Rams, Michal
AU - Wang, Bao Wei
N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.
PY - 2018/6/1
Y1 - 2018/6/1
N2 - Let ((0, 1],T) be the doubling map in the unit interval and φ be the Saint-Petersburg potential, defined by φ(x) = 2n if x (2-n-1, 2-n] for all n ≥ 0. We consider asymptotic properties of the Birkhoff sum Sn(x) = φ(x) + ⋯ + φ(Tn-1(x)). With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that 1 nlog nSn(x) converges to 1 log 2 in probability. We determine the Hausdorff dimension of the level set {x:limn∞Sn(x)/n = α} (α > 0), as well as that of the set {x:limn∞Sn(x)/(n) = α} (α > 0), when (n) = nlog n, na or 2nγ for a > 1,γ > 0. The fast increasing Birkhoff sum of the potential function x1/x is also studied.
AB - Let ((0, 1],T) be the doubling map in the unit interval and φ be the Saint-Petersburg potential, defined by φ(x) = 2n if x (2-n-1, 2-n] for all n ≥ 0. We consider asymptotic properties of the Birkhoff sum Sn(x) = φ(x) + ⋯ + φ(Tn-1(x)). With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that 1 nlog nSn(x) converges to 1 log 2 in probability. We determine the Hausdorff dimension of the level set {x:limn∞Sn(x)/n = α} (α > 0), as well as that of the set {x:limn∞Sn(x)/(n) = α} (α > 0), when (n) = nlog n, na or 2nγ for a > 1,γ > 0. The fast increasing Birkhoff sum of the potential function x1/x is also studied.
KW - Hausdorff Dimension
KW - Multifractal Analysis
KW - Saint-Petersburg Potential
UR - http://www.scopus.com/inward/record.url?scp=85047182114&partnerID=8YFLogxK
U2 - 10.1142/S0218348X18500263
DO - 10.1142/S0218348X18500263
M3 - Article
AN - SCOPUS:85047182114
SN - 0218-348X
VL - 26
JO - Fractals
JF - Fractals
IS - 3
M1 - 1850026
ER -