Abstract: In this talk, we study the Ahlfors regularity of planar self-affine sets under natural conditions: strong separation condition, strong irreducibility and proximality. Not surprisingly, if the dimension is strictly larger than 1, the set is never Ahlfors regular. In case if the dimension is less than or equal to 1 under the extra condition of dominated splitting, we show that the Ahfors regularity is equivalent to the positive proper dimensional Hausdorff measure and to positive proper dimensional Hausdorff measure of the projections in every Furstenberg direction. Moreover, we introduce a condition called "projective separation", which is equivalent to having positive measure, and which is applicable to show that both positive and zero Hausdorff measure happens on reasonably large sets of translation parameters. This is a joint work with Antti Käenmäki and Han Yu.
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Meeting ID: 868 3323 5350
Tue, 05/01/2021 - 14:00 to 15:00