Random Iteration of Unimodal Linear Transformations
Shunji Ito, Shigeru Tanaka · Tokyo Journal of Mathematics · 1982
In this paper we are concerned with a random family $\{f_{\alpha};a\leqq\alpha\leqq b\}$ of transformations of an interval $I$ into itself.On one hand, the random family of transformations may serve as more realistic models, e.g., models of population dynamics, if one takes into account of the random- ness of the environment.On the other hand, there may appear some interesting situations.For example, the random system $\{f_{\alpha}\}$ may be mixing (a fortiori, exact ([10])), although each transformation $f_{\alpha}$ is not mixing.We formulate the problem in the following manner.Let $\{f_{\alpha};a\leqq\alpha\leqq b\}$ be a one-parameter family of transformations of an interval $I$ into itself, and let $X_{1},$ $X_{2},$ $\cdots,$ $X_{n},$ $\cdots$ be a sequence of independent and identically distributed random variables defined on a probability space $(\Omega, P)$ with $a\leqq X_{n}\leqq b$ .Then, for each $\omega\in\Omega,$ $x_{0}\in I$ is transformed to $x_{1}=f_{X_{1}(\omega)}(x_{0}),$ $x_{2}=$ $f_{X_{2}(\omega)}(x_{1}),$ $\cdots,$ $x_{n}=f_{X,(\omega)}(x_{n-1}),$ $\cdots$ .Our aim is to investigate the behaviour of the orbit $\{x_{n};n\geqq 0\}$ for almost all $\omega\in\Omega$ .In this paper we only treat the following simplest case where the one-parameter family of transformations is that of unimodal linear transformations, that is,