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(2004 / vol.7 / no.2)
The Waiting Time For Some Interval Transformations
Dong Han Kim
Pages. 177-186     



We discuss the asymptotic behaviour of the waiting time for some transformation on the interval including irrational rotations. Like the fist return time formula, logarithm of the waiting time to a ball divided by logarithm of the radius of the ball goes to 1 as the radius goes to 0 for many transformations.
Let $T$ be a transformation from $I = [0,1)$ onto itself and let $Q_n(x)$ be the subinterval $[i/2^n), (i+1)/2^2)$, $0 leq i < 2^n$ 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
$limlimts_{n
ightarrow infty} frac{log K_n(x)}{n}=1$ and $limlimts_{n
ightarrow infty} frac{log K_n(x,y)}{n}=1$ a.e.
let $Tx=x+ heta (mod 1)$. Then for irrational $ heta$ of type $eta$.
$lim inflimts_{n
ightarrow infty} frac{log K_n(x,y)}{n}=1$ and $lim suplimts_{n
ightarrow infty} frac{log K_n(x,y)}{n}=eta$ a.e.
Since the set of irrational numbers of type 1 has measure 1, for almost every $ heta$ the limit exists and is 1.



1. Introduction
2. The piecewise monotone transformation on I
3. the waiting time for irrational rotations
References



37E05, 11K50