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

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.