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Path-dependent shrinking targets in generic affine iterated function systems
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We calculate the Hausdorff dimension of path-dependent shrinking target sets in generic affine iterated function systems. Here, by a path-dependent shrinking target set, we mean a set of points whose orbits infinitely often hit small balls with a fixed generic centre and with radius decreasing and dependent on the point itself. It turns out that the Hausdorff dimension of such a set is given by the zero point of a certain limsup pressure function. The result generalizes the work of Koivusalo and Ramirez, and Barany and Troscheit, as well as that of Hill and Velani.
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Dichotomy laws for the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems
For beta-dynamical systems with integer bases, the Hausdorff f-measure of shrinking target sets is zero or full according to divergence of an explicit series.
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