Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$

Jian-Lin Lia1 p1

a1 Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong (jllimath@yahoo.com.cn)

Abstract

The aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

(Online publication February 09 2007)

(Received April 24 2003)

Keywords

  • Primary 28A80;
  • Secondary 42C05;
  • 46E30;
  • spectral measure;
  • iterated function system (IFS);
  • compatible pair;
  • transfer operator

Correspondence:

p1 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China