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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

Chihurn Kim 1, and  Dong Han Kim 2,

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

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