In mathematics, particularly in the study of functions of
several complex variable
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
s, Ushiki's theorem, named after S. Ushiki, states that certain
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
functions cannot have certain kinds of well-behaved invariant manifolds.
The theorem
A
biholomorphic
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definiti ...
mapping cannot have a 1-dimensional
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
invariant manifold
In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. Examples include the slow manifold, center manifold, stable manifold, stable manifold, unsta ...
. In particular, such a map cannot have a
homoclinic connection or
heteroclinic connection.
Commentary
Invariant manifolds typically appear as solutions of certain asymptotic problems in
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. The most common is the
stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repello ...
or its kin, the unstable manifold.
The publication
Ushiki's theorem was published in 1980.
[S. Ushiki. Sur les liaisons-cols des systèmes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447–449, 1980] 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
The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side 2\pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rota ...
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 perturbation 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
*
Melnikov distance
*
Equichordal point problem
In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points. The problem was originally posed in 1916 by Fujiwara and in 1917 by Wilhelm Blaschke, Hermann Rothe, a ...
References
{{DEFAULTSORT:Ushiki's Theorem
Dynamical systems
Theorems in complex analysis
Several complex variables