As we have seen in Chapter 3, the pseudochaotic web (residual set) of aperiodic, discontinuity avoiding orbits in a renormalizable piecewise isometric system, while occupying zero area, nevertheless is far from trivial in its dynamics. In probing more deeply, we are naturally led to ask questions of a probabilistic nature, concerning, for example, what fraction of the web is contained in a particular subset of the plane, or with what probability will an orbit initiated in a particular set return to that set within a designated time? To investigate such questions requires a measure, and so we turn our attention to the task of constructing one. Fortunately, Hausdorff measure, widely used in the study of Cantor sets and other fractals (Falconer, 1990), turns out to be the most natural choice.
KeywordsTransfer Matrix Hausdorff Dimension Incidence Matrix Hausdorff Measure Transfer Matrix Method
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