Simultaneous dense and nondense orbits for commuting maps
Seminar Room 2, Newton Institute Gatehouse
AbstractWe show that, for two commuting automorphisms of the d-torus, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, the Ledrappier-Young formula, and the Marstrand slicing theorem, we show that the set of points that have dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. This mixed case, dense orbit under one map and nondense orbit under the other, is much more delicate than the other two possible cases. Our technique can also be applied to other settings. For example, we show the analogous result for two elements of the Cartan action on compact higher rank homogeneous spaces. This is joint work with V. Bergelson and M. Einsiedler.
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