For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $ -transformations and random Lüroth maps with a hole.

中文翻译：

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区间随机系统的不变密度

对于随机分段线性系统吨在平均扩展的区间中，我们明确地构造了绝对连续的密度函数吨- 不变的措施。如果随机系统仅使用扩展映射，我们的过程会产生系统的所有不变密度。示例包括随机帐篷地图、随机W形状的地图，随机的$\beta $- 带有孔的变换和随机 Lüroth 贴图。