On stable laws in a one-dimensional intermittent map equipped with a uniform invariant measure

Abstract

We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under the mapping. The probability density of the residence time and the correlation function are found to behave polynomially: $f(m) \sim m^{-(\kappa+1)}$ and $C(\tau) \sim \tau^{-(\kappa-1)}$ $(\kappa > 1)$. Using the Doeblin-Feller theorems in probability theory, we derive the conjecture that the rescaled fluctuations of the time average of some observable functions obey the stable distribution with the exponent $1 < \alpha \le 2$. Some exponents of the stable distribution are precisely determined by numerical simulations, and the conjecture is verified numerically. The polynomial decay of large deviations is also discussed, and it is found that the entropy function does not exist, because the moment generating function of the stable distribution can not be defined.