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

Publisher Website
Google Scholar
Abstract
Let T : [ 0 , 1 ] → [ 0 , 1 ] T:[0,1] \to [0,1] be an expanding map of the unit interval and let ξ ε ( x ) {\xi _\varepsilon }(x) be the smallest integer n n for which T n ( x ) ∈ [ 0 , ε ] {T^n}(x) \in [0,\varepsilon ] ; that is, it is the random variable given by the formula \[ ξ ε ( x ) = min { n : T n ( x ) ⩽ ε } . {\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}. \] It is shown that for any z ⩾ 0 z \geqslant 0 and for any integrable function f : [ 0 , 1 ] → R + f:[0,1] \to {R^ + } the measure μ f {\mu _f} (where μ \mu is Lebesgue measure and μ f {\mu _f} is defined by d μ f = f d μ d{\mu _f} = fd\mu ) of the set of points x x for which ξ ε ( x ) ⩽ z / ε {\xi _\varepsilon }(x) \leqslant z/\varepsilon tends to an exponential function of z z as ε \varepsilon tends to zero.

This publication has 12 references indexed in Scilit: