We consider the trajectories of points on under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is d+1.
A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair @math of points of M there exists a trajectory of S from m to @math . Here a trajectory of S is a curve which is an integral curve of some @math or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class @math , @math , or @math , there exists a completely controllable set S consisting of two vector fields of class @math . We analyze a random walk on the orthogonal group SO(N) given by repeatedly rotating by a fixed angle through randomly chosen planes of R . We derive estimates of the rate at which this random walk will converge to Haar measure on SO(N), using character theory and the Upper Bound Lemma of Diaconis and Shashahani. In some cases we are able to establish the existence of a “cut-off phenomenon” for the random walk. This is the first interesting such result on a non-finite group. For many random walks on sufficiently large finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number k 0 , depending on the size of the group, such that k 0 steps are necessary and sufficient for the random walk to closely approximate uniformity. As a first example on a continuous group, Rosenthal recently proved the occurrence of this cut-off phenomenon for a specific random walk on SO(N). Here we present and [for the case of O(N)] prove results for random walks on O(N), U(N) and Sp(N), where the one-step distribution is a suitable probability measure concentrated on reflections. In all three cases the cut-off phenomenon occurs at k 0 = 1 2N log N. We present closed forms for the exponential of some infinitesimal generators of Lie groups which play an important role in physics and engineering applications. These explicit forms are based on the Putzer’s method. We also compare this methodology and results with related work by other authors.