A note on the run length function for intermittency maps

TL;DR

This paper investigates the run length function in intermittency maps, demonstrating specific growth rates for consecutive zero and one digits using elementary probabilistic methods, complementing previous Erdős-Rényi law results.

Contribution

It provides new elementary proofs for the growth rates of run lengths in intermittency maps, expanding understanding of their digit patterns.

Findings

Longest zero digits grow polynomially

Longest one digits grow logarithmically

Results complement existing Erdős-Rényi law

Abstract

We study the run length function for intermittency 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 intermittency maps. Our results are compensational to the Erd\H{o}s-R\'{e}nyi law obtained by Denker and Nicol in \cite{dennic13}.