Lekha Priya Patil

San Francisco State University

“Hausdorff Dimension of Totally Disconnected Thin Fractals with Very Thick Shadows”

The projections of fractals is a central study in fractal geometry. In this work, we study whether we can have fractals in the 2D plane with as small of a dimension as possible whose projections onto the real line in every direction have as large of dimensions as possible. We provide a classification of self-affine fractals whose projections have as large of a dimension as possible.

ABSTRACT

The projection of fractals is a central study in fractal geometry and geometric measure theory. We say that a fractal has a very thick shadow on a one-dimensional subspace if its projection onto the subspace is an interval. We are interested in finding a totally disconnected fractal in the 2D plane with very thick shadows in every direction. Applying the Marstrand projection theorem, we realized that the fractal can not have Hausdorff dimension less than 1. In 2008, Mendivil and Taylor discovered a totally disconnected self-affine fractal with very thick shadows in every direction. We show that the Mendivil-Taylor fractal can have Hausdorff dimension arbitrarily close to 1. Additionally, we generalize their work and present a classification to determine which self-affine fractals project very thick shadows in every direction. In doing so, we demonstrate that every polygon P contains a totally disconnected fractal K with very thick shadows in every direction, given that P is the convex hull of K. Moreover, such a fractal has Hausdorff dimension arbitrarily close to 1.
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