Expanding Polynomials and Connectedness of Self-Affine Tiles

Little is known about the connectedness of self-affine tiles in ${Bbb R}^n$. In this note we consider this property on the self-affine tilesthat are generated by consecutive collinear digit sets. By using an algebraiccriterion, we call it the {it height reducing property}, on expanding polynomials(i.e., all the roots have moduli $ > 1$), we show that all such tiles in ${BbbR}^n, n leq 3$, are connected. The problem is still unsolved for higherdimensions. For this we make another investigation on this algebraic criterion.We improve a result of Garsia concerning the heights of expanding polynomials.The new result has its own interest from an algebraic point of view and alsogives further insight to the connectedness problem.