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Anosov Diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all). Dmitri V. Anosov, ''Geodesic flows on closed Riemannian manifolds with negative curvature'', (1967) Proc. Steklov Inst. Mathematics. 90. Overview Three closely related definitions must be distinguished: * If a differentiable map ''f'' on ''M'' has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map,_and_Arnold's_cat_map.html" ;"title=", 1)^\infty : x \mapsto (x_0, x_1, x_2, ..., and Arnold's cat map">, 1)^\infty : x \mapsto (x_0, x_1, x_2, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
