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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The law of exponential decay for expanding transformations of the unit interval into itself


Author: M. Jabłoński
Journal: Trans. Amer. Math. Soc. 284 (1984), 107-119
MSC: Primary 28D05
DOI: https://doi.org/10.1090/S0002-9947-1984-0742414-1
MathSciNet review: 742414
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Abstract: Let $T:[0,1] \to [0,1]$ be an expanding map of the unit interval and let ${\xi _\varepsilon }(x)$ be the smallest integer $n$ for which ${T^n}(x) \in [0,\varepsilon ]$; that is, it is the random variable given by the formula \[ {\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}. \] It is shown that for any $z \geqslant 0$ and for any integrable function $f:[0,1] \to {R^ + }$ the measure ${\mu _f}$ (where $\mu$ is Lebesgue measure and ${\mu _f}$ is defined by $d{\mu _f} = fd\mu$) of the set of points $x$ for which ${\xi _\varepsilon }(x) \leqslant z/\varepsilon$ tends to an exponential function of $z$ as $\varepsilon$ tends to zero.


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Article copyright: © Copyright 1984 American Mathematical Society