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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The law of exponential decay for expanding transformations of the unit interval into itself
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by M. Jabłoński PDF
Trans. Amer. Math. Soc. 284 (1984), 107-119 Request permission

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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Additional Information
  • © Copyright 1984 American Mathematical Society
  • 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