Mathematics Colloquia and Seminars

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The dilatation of a pseudo-Anosov map is a measure of the complexity of its dynamics. The minimum dilatation problem asks for the minimum dilatation among all pseudo-Anosov maps defined on a fixed surface, which can be thought of as the smallest amount of mixing one can perform while still doing something topologically interesting. In this talk, we present some recent work on this problem with Eriko Hironaka, which shows a sharp lower bound for dilatations of fully-punctured pseudo-Anosov maps with at least two puncture orbits. We will explain some ideas in the proof, including standardly embedded train tracks and Perron-Frobenius matrices.