arXiv subjects
Publications and source records attributed to Eva Curry.
Given any m-dimensional dilation matrix A with rational eigenvalues, we demonstrate the existence of a digit set D such that the attractor T(A,D) of the iterated function system generated by A and D is connected. We give an easily verified sufficient condition on A for a specific digit set, which we call the centered canonical digit set for A, to give rise to a connected attractor T(A,D).
We define radix representations for vectors in Z^n analogously with radix representations in Z, and give a sufficient condition for a matrix A:Z^n -> Z^n to yield a radix representation with a given canonical digit set. We relate our results to a sufficient condition given recently by Jeong. We also show that any expanding matrix A:Z^n -> Z^n will not be too far from yielding a radix representation, in that we can partition Z^n into a finite number of sets such that A yields a radix representation on each set up to translation by (A^N)s for some vector s (N >= 0 will vary). We call the vectors s "pseudodigits", and call this decomposition of Z^n a "pseudodigit representation".
We investigate the connection between radix representations for Z^n and self-affine tilings of R^n. We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficie