Explicit Cross-Sections of Singly Generated Group Actions

We consider two classes of actions on ℝn—one continuous and one discrete. For matrices of the form A = e B with B ∈ M n(ℝ), we consider the action given by γ → γA t. We characterize the matrices A for which there is a cross-section for this action. The discrete action we consider is given by γ → γA k, where A ∈ GL n(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 minimally supported frequency (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.