How does the rotation function
depend on the caustic .
We assume and that (0,0) is contained in the
convex hull of . The set of closed
convex caustics satisfying these conditions is a compact
metric space when equiped with the Hausdorff metric.
Let and let .
By Corollary 2.2,
is a closed connected set. We disregard the case when
is empty and consider the dichotomy:
has a nonempty interior; ii) is a single point
. By Corollary 2.2, the latter happens
if and only if the homeomorphism is conjugate to the
rotation by . In the case ii), we say is
be the set of exceptional caustics.
We say that
are p/q-nonexceptional, and
call a phase locking interval. The set
consists of exceptional caustics.
Its complement is the set of
1) The set contains all ellipses. The Birkhoff-Poritsky conjecture is equivalent to the statement that is equal the set of ellipses.
2) The set is the set of tables of equal width.
3) The set is nonempty . The same construction seems sto generalize and lead to nonempty .
Recall that a subset, Y, of a compact metric space is a , if Y is a countable intersection of open sets. Sets containing dense are called residual or Baire generic (see ).
A continuous nondecreasing function is called a devil's staircase if there exists a family of disjoint open intervals whose union is dense in I, such that takes distinct constant values on these intervals . A subset of a compact metric space is residual if it contains a dense .