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# Devil's staircase

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/q-exceptional. Let 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 nonexceptional caustics.

Remarks.
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 [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 .

Next: Periodic orbits and Up: Billiards that share a Previous: Preliminaries

Oliver Knill
Wed Jul 8, 1998