The Dimension of Subsets of Moran Sets Determined by the Success Run Behaviour of Their Codings

 Let be a Moran set associated with the set . Let Γ be a non-empty subset of with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space is a decomposition of F such that Depending on F this might be a partition of F or almost a partition of F in the sense that . We prove that each F _t is dense in F, and with . For -a.e. and F _t is an s-set when . Moreover, associated with this decomposition of F is a measurable function Y such that each F _t is a level set of Y. The fractal dimensions of the graph of Y are also determined. (Received 17 January 2000; in revised form 13 July 2000)