Relativistic similarity parameter: Difference between revisions

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In mathematics, particularly in the study of functions of [[several complex variable]]s, '''Ushiki's theorem''', named after S. Ushiki, states that certain [[well-behaved]] functions cannot have certain kinds of well-behaved invariant manifolds.
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== The theorem ==
A [[biholomorphic]] [[mapping (mathematics)|mapping]] <math>F:\mathbb{C}^n\to\mathbb{C}^n</math> can not have a 1-dimensional [[compact space|compact]] [[smooth function|smooth]] [[invariant manifold]]. 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]]. The most common is the [[stable manifold]] or its kin, the unstable manifold.
 
== The publication ==
Ushiki's theorem was published in 1980.<ref name="Ushiki_Standard_Map_Theorem">S. Ushiki. Sur les liaisons-cols des systèmes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447–449, 1980</ref> 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 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]]
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Ushiki's Theorem}}
[[Category:Dynamical systems]]
[[Category:Theorems in complex analysis]]
[[Category:Several complex variables]]

Latest revision as of 17:23, 10 July 2014

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