
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 .
Set .
By Corollary 2.2,
is a closed connected set. We disregard the case when
is empty and consider the dichotomy:
i)
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
p/qexceptional. Let
be the set of exceptional caustics.
We say that
are p/qnonexceptional, and
call a phase locking interval. The set
consists of exceptional caustics.
Its complement is the set of
nonexceptional caustics.
Remarks.
1) The set contains all ellipses.
The BirkhoffPoritsky 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 [14]. 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 [24]).
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 [16]. A subset of a compact metric space is residual if it contains a dense .