Summary: Appeared in volume 44 of the IMA volumes in mathematics and its applications, "Twistmappings and their
applications", edited by R.McGeehee and K.R.Meyer. Springer Verlag, 1992
THE TOPOLOGICAL ENTROPY AND INVARIANT CIRCLES OF AN AREA
Let A be the annulus S1 0; 1 , and let f : A ! A be an area preserving twist
homeomorphism of A. The two boundary components of A, Aj = S1 fjg, are invariant
under fq, and we shall denote the rotation number of fjAj by j.
In this note we wish to point out that the folowing holds:
Theorem A. If the topological entropy htopfof f vanishes, then f must have an invariant
circle of rotation number !, for any ! 2 0; 1.
In fact, we'll show that if "one of the invariant circles of f is missing," there must exist
a compact subset K A which is invariant under fq, for some q 1, and such that fqjK
has a Bernoulli shift as a factor.
If the map f is a C1; diffeomorphism, then a theorem of A. Katok implies that f
must have a "horse shoe" if htopf 0. Thus our theorem says that any C1; twist
diffeomorphism of the annulus either has a transversal homoclinic point, or else it has
invariant circles for any rotation number in its rotation interval 0; 1.
We shall give two proofs of this theorem. The first proof consists of simply combining
two results obtained by Dick Hall and Phil Boyland.