Explicit cross-sections of singly generated group actions

Abstract

We consider two classes of actions on ${\mathbb{R}}^n$ - one continuous and one discrete. For matrices of the form $A=e^B$ with $B\in M_n(\mathbb{R})$, we consider the action given by ${\hat{I}}^3\to {\hat{I}}^3{A}^t$. We characterize the matrices $A$ for which there is a cross-section for this action. The discrete action we consider is given by ${\hat{I}}^3\to {\hat{I}}^3{A}^k$, where $A\in G{L}_n(\mathbb{R})$. We characterize the matrices $A$ for which there exists a cross-section for this action as well. We also characterize those $A$ for which there exist special types of cross-sections; namely, bounded cross-sections and finite measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist MSF wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators.