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 S_{Lin} with a dual
time-symmetry (i.e. S_{Lin} commutes with the 'reflection about
1/2' operator), then S is not only C3
equivalent to S_{Lin}, 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 S_{Lin},
* the nonlinear S inherits the dual time-symmetry*. Such an
inheritance of time-symmetry does not hold if S

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 S_{Lin},
no space-symmetry -- and hence also no time-symmetry inheritance from S_{Lin}
to S -- is needed.

(Thomas Yu, April 27, 2012)