Invariant distributions for shapes in sequences of randomly-divided rectangles

Interest has been shown in Markovian sequences of geometric shapes. Mostly the equations for invariant probability measures over shape space are extre mely complicated and multidimensional. This paper deals with rectangles whi ch have a simple one-dimensional shape descriptor. We explore the invariant distributions of shape under a variety of randomised rules for splitting t he rectangle into two sub-rectangles, with numerous methods for selecting t he next shape in sequence. Many explicit results emerge. These help to fill a vacant niche in shape theory, whilst contributing at the same time, new distributions on [0,1] and interesting examples of Markov processes or, in the language of another discipline, of stochastic dynamical systems.