A note on the run length function for intermittent maps

Abstract

We study the run length function for intermittent maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. Our results are compensational to the Erdős-Rényi law obtained by Denker and Nicol in [8]. * Corresponding author, and every author contributes equally.

Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables

“…It is interesting to note that the typical growth rate of ξ (1) n depends on α, while that of ξ (0) n does not. This is in contrast to the corresponding run length results in the probabilistic cases [14,Theorem 1], due to the additional scaling contribution arising from the asymptotics of the return time function R. The intuitive reason for this is that when α is larger than 1, then the orbit spends very little time away from the neutral fixed point. When α is less than one, then a typical orbit spends a positive proportion of the time in the right half of the interval (Birkhoff's ergodic theorem), but this is not true when α is larger than one.…”

“…uniformly hyperbolic Gibbs-Markov systems; logistic-like maps satisfying the Collet-Eckmann condition; and families of intermittent maps preserving a.c.i.p. [14,15,16,33,49]), we show that apart from the local dimension, there is an additional scaling contribution, arising from the asymptotics of the return time function associated to the induced transformation, which needs to be taken into account in the growth rate for ξ n of infinite systems. As the reader will realize, our proof is based on a natural link between the dynamical run length function, hitting time, and growth of maximum for the return time functions.…”

Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables

“…uniformly hyperbolic Gibbs-Markov systems; logistic-like maps satisfying the Collet-Eckmann condition; and families of intermittent maps preserving a.c.i.p. [13,14,15,33,48]), we show that apart from the local dimension, there is an additional scaling contribution, arising from the asymptotics of the return time function associated to the induced transformation, which needs to be taken into account in the growth rate for ξ n of infinite systems. As the reader will realize, our proof is based on a natural link between the dynamical run length function, hitting time, and growth of maximum for the return time functions.…”