We show that if the Julia set J(f) of a rational function f is invariant under translation by one and infinity is a periodic or preperiodic point for f , then J(f) must either be a line or the Riemann sphere.

The theory of translation-invariant operators on various spaces of functions (or measures or distributions) is a well-trodden field. The problem is to decide, first, whether or not a linear operator between two function spaces on, say, IR or R + which commutes with one or many translations on the two spaces is necessarily continuous, and, second, to give a canonical form for all such continuous...

This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space L(R,H) of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of L(R,H) and its bo...

A set of polynomials is called a submodule $\mathbb C [x_1 ,\ldots , x_n ]$ if it translation invariant linear subspace ]$. We present description the submodules [x,y]$ in terms