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Dynamics, Geometry and Groups Seminar

Friday, September 30th, 2022

Time: 10:00 a.m.  Place: Jeffery Hall, Room 422

Speaker: Neil MacVicar (¾ÅÐãÖ±²¥)

Title: Multiplicative Invariance in the Complex Plane

Abstract: In the late 60s, Hillel Furstenberg proved results and made conjectures concerning the fractal nature of subsets of the unit interval that are invariant under the map that scales a real number by a positive integer r and then kills the integer part. Since then, these sets have been deeply studied. Considerably less has been said about analogous sets in the complex plane. I will review what is known for the case when r is a positive integer that scales real numbers and then introduce a notion of times b invariance in the plane where b is a Gaussian integer. The fractal dimension of these times b invariant sets will be discussed.