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Devil's staircase

How does the rotation function tex2html_wrap_inline1926 depend on the caustic tex2html_wrap_inline1372 . We assume tex2html_wrap_inline1930 and that (0,0) is contained in the convex hull of tex2html_wrap_inline1372 . The set tex2html_wrap_inline1936 of closed convex caustics satisfying these conditions is a compact metric space when equiped with the Hausdorff metric.

Let tex2html_wrap_inline1938 and let tex2html_wrap_inline1940 . Set tex2html_wrap_inline1942 . By Corollary 2.2, tex2html_wrap_inline1944 is a closed connected set. We disregard the case when tex2html_wrap_inline1944 is empty and consider the dichotomy: i) tex2html_wrap_inline1948 has a nonempty interior; ii) tex2html_wrap_inline1944 is a single point tex2html_wrap_inline1952 . By Corollary 2.2, the latter happens if and only if the homeomorphism tex2html_wrap_inline1380 is conjugate to the rotation by tex2html_wrap_inline1956 . In the case ii), we say tex2html_wrap_inline1372 is p/q-exceptional. Let tex2html_wrap_inline1962 be the set of exceptional caustics. We say that tex2html_wrap_inline1964 are p/q-nonexceptional, and call tex2html_wrap_inline1944 a phase locking interval. The set tex2html_wrap_inline1970 consists of exceptional caustics. Its complement tex2html_wrap_inline1972 is the set of nonexceptional caustics.

Remarks.
1) The set tex2html_wrap_inline1974 contains all ellipses. The Birkhoff-Poritsky conjecture is equivalent to the statement that tex2html_wrap_inline1974 is equal the set of ellipses.
2) The set tex2html_wrap_inline1978 is the set of tables of equal width.
3) The set tex2html_wrap_inline1980 is nonempty [14]. The same construction seems sto generalize and lead to nonempty tex2html_wrap_inline1982 .

Recall that a subset, Y, of a compact metric space is a tex2html_wrap_inline1388 , if Y is a countable intersection of open sets. Sets containing dense tex2html_wrap_inline1990 are called residual or Baire generic (see [24]).

  thm310

trivlist1061

A continuous nondecreasing function tex2html_wrap_inline2112 is called a devil's staircase if there exists a family of disjoint open intervals whose union is dense in I, such that tex2html_wrap_inline1578 takes distinct constant values on these intervals [16]. A subset of a compact metric space is residual if it contains a dense tex2html_wrap_inline1388 .

  coro359

trivlist1064

tex2html_wrap2134 tex2html_wrap2136
Next: Periodic orbits and Up: Billiards that share a Previous: Preliminaries

Oliver Knill
Wed Jul 8, 1998