On the law of logarithm of the recurrence time

Chihurn Kim; Dong Han Kim

doi:10.3934/dcds.2004.10.581

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2004, Volume 10, Issue 3: 581-587. Doi: 10.3934/dcds.2004.10.581

This issue Previous Article Next Article The limiting distribution and error terms for return times of dynamical systems

On the law of logarithm of the recurrence time

Chihurn Kim^1, and
Dong Han Kim^2,

1.
Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701
2.
School of Mathematics, Korea Institute for Advanced Study, Seoul, 130-722

Received: November 2002

Revised: May 2003

Published: July 2004

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Abstract

Let $T$ be a transformation from $I=[0,1)$ onto itself and let $Q_n(x)$ be the subinterval $[i/2^n,(i+1)/2^n)$, $0 \leq i < 2^n$ containing $x$. Define $K_n (x) =$min{$j\geq 1:T^j (x)\in Q_n(x)$} and $K_n(x,y) =$min{$j\geq 1:T^{j-1} (y) \in Q_n(x)$}. For various transformations defined on $I$, we show that

$ \lim_{n\to\infty}\frac{\log K_n(x)}{n}=1 \quad$and$\quad \lim_{n\to\infty}\frac{\log K_n(x,y)}{n}=1 \quad $a.e.

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Mathematics Subject Classification: Primary: 28D05, 37E05.

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