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Transactions of the American Mathematical Society

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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

$\displaystyle {\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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DOI: https://doi.org/10.1090/S0002-9947-1984-0742414-1
Article copyright: © Copyright 1984 American Mathematical Society

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