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.
There are many known results analogous to Theorem for other collections of maps associated with linear isometries of spheres. For example, Crouch and Silva Leite @cite_1 found pairs of one-parameter subgroups of @math which generate all of @math , and produced upper bounds on the number of elements from each subgroup required to generate any element (see also Levitt and Sussmann @cite_13 for related results). An example of this phenomenon is the Euler angles decomposition, a formula for writing any element of @math as a product of three rotations about the @math and @math axes. Rosenthal @cite_5 has established bounds on the rate of convergence to the steady state for random walks generated by conjugacy classes of planar rotations on @math . Porod @cite_9 has obtained analogous results for random walks on @math , @math and @math generated by the conjugacy class of reflections.
@cite_13 @cite_5 @cite_9 @cite_1
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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.