Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets

Abstract

Abstract. We construct covering maps from infinite blowups of the n-dimensional Sierpinski gasket SGn to certain compact fractafolds based on SGn. These maps are fractal analogs of the usual covering maps from the line to the circle. The construction extends work of the second author in the case n = 2, but a different method of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

A Spectral Triple for a Solenoid Based on the Sierpinski Gasket

“…Periodic and almost periodic functions on the infinite fractafold are considered, and a Fourier series description for the periodic functions is given, based on periodic eigenfunctions of the Laplacian (cf. also [46] for higher-dimensional examples). Let us remark that such coverings, as the ones considered in this paper, are not associated with groups of deck transformations.…”

Good labeling property of simple nested fractals

“…It can be also used to define periodic functions on nested fractals. Such objects on the Sierpiński gasket and its higher dimension analogues were examined by Ruan and Strichartz [20,23].…”

The mathieu differential equation and generalizations to infinite fractafolds

“…Strichartz [24]. Existing research on the infinite Sierpinski gasket and on other 'fractafolds' be found in [12,20,23,25,26,27]. In particular, we will use the important result concerning the spectrum of the Laplacian on the infinite Sierpinski gasket by A. Teplyaev [27].…”

“…Normalized solutions corresponding to p(cos 2t, ε), with ε = 0, 1, 2, 3, 4, 5, 6 in the left, and ε = 6, 7,8,9,10,20, 30, 40, 60, 80, 100, 160 in the right.…”