Ushiki's theorem

The theorem

A biholomorphic mapping $F:\mathbb {C} ^{n}\to \mathbb {C} ^{n}$, where $\mathbb {C} ^{n}$ is the $n$-dimensional complex vector space, does not have a 1-dimensional compact smooth invariant manifold. In particular, such a map cannot have a homoclinic connection or heteroclinic connection.

Comentary

Invariant manifolds typically appear as solutions of certain asymptotic problems in dynamical systems. The most common is the stable manifold or its kin, the unstable manifold.

The publication

Ushiki's theorem was published in 1980. Interestingly, the theorem appeared in print again several years later, in a certain Russian journal, by an author apparently unaware of Ushiki's work.

An application

The standard map cannot have a homoclinic or heteroclinic connection. The practical consequence is that one cannot show the existence of a Smale's horseshoe in this system by a parturbation method, starting from a homoclinic or heteroclinic connection. Nevertheless, one can show that Smale's horseshoe exists in the standard map for many parameter values, based on crude rigorous numerical calculations.

See also

1. Melnikov distance
2. Stephen Smale
3. Rychlik's Theorem