TY - JOUR
T1 - Large deviation principle for the backward continued fraction expansion
AU - Takahasi, Hiroki
N1 - Funding Information:
I thank anonymous referees for their very useful comments and suggestions. I thank Ryoki Fukushima, Hiroaki Ito, Johannes Jaerisch for fruitful discussions. This research was partially supported by the JSPS, Japan KAKENHI 19K21835 , 20H01811 . Part of this paper was written during the conference “Thermodynamic Formalism: Dynamical Systems, Statistical Properties and their Applications“ at CIRM Marseille December 2019. I thank Mark Pollicott and Sandro Vaienti for their hospitality during the conference.
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2022/2
Y1 - 2022/2
N2 - We investigate stochastic properties of the backward continued fraction expansion of irrational numbers in (0,1). For the mean process associated with a real-valued observable which depends only on the first digit of the expansion, we establish the large deviation principle. For any such observable which is non-negative, we completely determine the set of minimizers of the rate function in terms of a growth rate of the observable. Our method of proof employs the thermodynamic formalism for topological Markov shifts, and a multifractal analysis of pointwise Lyapunov exponents for the Rényi map generating the backward continued fraction expansion.
AB - We investigate stochastic properties of the backward continued fraction expansion of irrational numbers in (0,1). For the mean process associated with a real-valued observable which depends only on the first digit of the expansion, we establish the large deviation principle. For any such observable which is non-negative, we completely determine the set of minimizers of the rate function in terms of a growth rate of the observable. Our method of proof employs the thermodynamic formalism for topological Markov shifts, and a multifractal analysis of pointwise Lyapunov exponents for the Rényi map generating the backward continued fraction expansion.
KW - Backward continued fractions
KW - Large deviation principle
KW - Multifractal analysis
KW - Thermodynamic formalism
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U2 - 10.1016/j.spa.2021.11.002
DO - 10.1016/j.spa.2021.11.002
M3 - Article
AN - SCOPUS:85120443497
SN - 0304-4149
VL - 144
SP - 153
EP - 172
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
ER -