In revising this paper, we noticed yet another surprising and/or confusing aspect of one of our main results (Theorem 13).
In this result, we show that if a single basepoint log-exp scheme S has its underlying linear subdivision scheme SLin with a dual time-symmetry (i.e. SLin commutes with the 'reflection about 1/2' operator), then S is not only C3 equivalent to SLin, but also automatically C4 equivalent. The same result holds when exp is replaced by a retraction f: TM → M and log is replaced the local inverse of f, provided that f satisfies a third order condition Pf = 0, see Section 3.
The first surprise of this result, already noted in the paper, is that this elevation of equivalence order can happen without any fourth order assumption on the retraction f.
[Note: It is, however, not surprising that such an elevation happens in the presence of symmetry, such kind of results is seen already in basic numerical analysis. For example, a forward divided difference approximates a derivative with accuracy O(h), whereas a central divided difference approximates the same derivative with accuracy O(h2). Likewise, the left/right endpoint quadrature approximates an integral with accuracy O(h), whereas a midpoint quadrature approximates the same integral with accuracy O(h2). In both cases, symmetry buys you an extra accuracy order.]
What makes dual time-symmetry so special in this case? What about primal time-symmetry?
When the manifold M and the retraction map f: TM→M have the right kind of symmetry (e.g. when M is a symmetric space and f is the corresponding exponential map), then, together with the dual symmetry of SLin, the nonlinear S inherits the dual time-symmetry. Such an inheritance of time-symmetry does not hold if SLin is assumed to have a primal time-symmetry, even when (M,f) has all the right `space-symmetries'. The distinction between primal and dual symmetry here is simply an artifact of the single basepoint structure in S; see Section 5.1 of our earlier paper.
But all results in this paper are proved without any symmetry assumption on the manifold or on the retraction map f (as opposed to our earlier paper). While Theorem 13 is a result about the nonlinear S, we only need to assume a dual time-symmetry in the underlying linear scheme SLin, no space-symmetry -- and hence also no time-symmetry inheritance from SLin to S -- is needed.
(Thomas Yu, April 27, 2012)