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Lattès Map
In mathematics, a Lattès map is a rational map ''f'' = Θ''L''Θ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ... to the complex sphere and ''L'' is an affine map ''z'' → ''az'' + ''b'' from the complex torus to itself. Lattès maps are named after French mathematician Samuel Lattès, who wrote about them in 1918. References * * {{DEFAULTSORT:Lattes map Dynamical systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Map
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal definition Formally, a rational map f \colon V \to W between two varieties is an equivalence class of pairs (f_U, U) in which f_U is a morphism of varieties from a non-empty open set U\subset V to W, and two such pairs (f_U, U) and (_, U') are considered equivalent if f_U and _ coincide on the intersection U \cap U' (this is, in particular, vacuously true if the intersection is empty, but since V is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: * If two morphisms of varieties are equal on some non-empty open set, then they are equal. f is said to be birational if there exists a rational map g \colon W \to V which is its inverse, where the composition is taken i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Map
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', where ''n'' is the complex dimension of ''M''. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C''n'' considered as real vector space; then the quotient group V/\Lambda is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For ''n'' = 1 this is the classical period lattice construction of elliptic curves. For ''n'' > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when ''n'' > 1, and are real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Lattès
Samuel Lattès (21 February 1873 (Nice) – 5 July 1918) was a French mathematician. From 1892 to 1895 he studied at the École Normale Superieure. After this he was a teacher in Algiers, Dijon and Nice. After a promotion to Paris in 1906 he moved first to Montpellier in 1908 and then to Besançon, before he took up a professorship at the University of Toulouse in 1911. He died of typhus in 1918. Today Lattès is best known for his work on complex sets, particularly for examples of rational functions including the Riemann sphere in its Julia set. Today these examples are described as Lattès maps or Lattès examples.Für eine moderne Darstellung der Lattèsschen Beispiele und neuere Ergebnisse dazu siehe: John Milnor, ''On Lattès maps''. In ''Dynamics on the Riemann sphere'', European Mathematical Society, Zürich, 2006, S. 9–43 See also *Pierre Fatou *Gaston Julia *Lattès map In mathematics, a Lattès map is a rational map ''f'' = Θ''L''Θ−1 from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
