On the law of logarithm of the recurrence time

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July 2004, 10(3): 581-587. doi: 10.3934/dcds.2004.10.581

On the law of logarithm of the recurrence time

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

Received November 2002 Revised May 2003 Published January 2004

Abstract
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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.

Keywords: waiting time., the first return time, Recurrence time.

Mathematics Subject Classification: Primary: 28D05, 37E0.

Citation: Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581

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