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Anosov diffeomorphism

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	Topics Anosov diffeomorphism Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , more particularly in the fields of dynamical systems and geometric topologyGeometric topology In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot the... , an Anosov map on a manifoldManifold A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in... M is a certain type of mapping, from M to itself, with rather clearly marked local directions of 'expansion' and 'contraction'. Anosov diffeomorphisms were introduced by D. V. Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all). Overview Three closely related definitions must be distinguished: If a differentiable mapMap (mathematics) Summary In mathematics and related technical fields, the term map or mapping is often a synonym for function; see function... f on M has a hyperbolic structure on the tangent bundleTangent bundle In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unio... , then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's cat mapArnold's cat map In dynamical systems theory, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who ... . If the map is a diffeomorphismDiffeomorphism Summary In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds.... , then it is called an Anosov diffeomorphism. If a flowFlow (mathematics) In mathematics, a flow formalizes, in mathematical terms, the general idea of "a variable that depends on time" that occurs ... on a manifold splits the tangent bundle into three invariant subbundleSubbundle In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces ... s, with one subbundle that is exponentially contracting, and one that is exponentially expanding, and a third, non-expanding, non-contracting one-dimensional sub-bundle, then the flow is called an Anosov flow. Anosov proved that Anosov diffeomorphisms are structurally stable and form an open subset of mappings (flows) with the C¹ topology. Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphereSphere A sphere is a perfectly symmetrical geometrical object.... . The simplest examples of compact manifolds admitting them are the tori: they admit the so-called linear Anosov diffeomorphisms, which are isomorphisms having no eigenvalue of modulus 1. It was proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind. The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still as of 2005 has no answer. The only known examples are infranil manifolds, and it is conjectured that they are the only ones. Another famous problem is to determine whether or not the nonwandering set of an Anosov diffeomorphism must be the whole manifold. This is known to be true for linear Anosov diffeomorphisms (and hence for any Anosov diffeomorphism in a torus). As of December 2007, it is believed to be proved for all Anosov diffeomorphisms (Xia 2007). Anosov flow on (tangent bundles of) Riemann surfaces As an example, this section develops the case of the Anosov flow on the tangent bundleFacts About Tangent bundle In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unio... of a Riemann surfaceRiemann surface In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional comp... of negative curvatureCurvature Curvature refers to a number of loosely related concepts in different areas of geometry.... . This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane modelFacts About Poincaré half-plane model In non-Euclidean geometry, the Poincar model, named after Henri Poincar, is a model of two-dimensional hyperbolic geometry a... of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian modelFuchsian model In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R.... s, that is, as the quotients of the upper half-planeUpper half-plane In mathematics, the upper half-plane H is the set of complex numbers... and a Fuchsian groupFuchsian group In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane.... . For the following, let H be the upper half-plane; let G be a Fuchsian group; let M=H\G be a Riemann surface of negative curvature, and let T¹M be the tangent bundle of unit-length vectors on the manifold M, and let T¹H be the tangent bundle of unit-length vectors on H. Note that a bundle of unit-length vectors on a surface is a complex line bundleLine bundle Overview In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space.... . Lie vector fields One starts by noting that T¹H is isomorphic to the Lie groupLie group In mathematics, a Lie group is a continuous group, in the sense that the group elements have the topology of a manifold, an... PSL(2,R). This group is the group of orientation-preserving isometries of the upper half-plane. The Lie algebraLie algebra Overview In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... of PSL(2,R) is sl(2,R), and is represented by the matrices which have the algebra The exponential mapExponential map There are two different notions of an exponential map in differential geometry, both of which generalize the ordinary expone... s define right-invariant flowFlow (mathematics) In mathematics, a flow formalizes, in mathematical terms, the general idea of "a variable that depends on time" that occurs ... s on the manifold of T¹H=PSL(2,R), and likewise on T¹M. Defining P=T¹H and Q=T¹M, these flows define vector fields on P and Q, whose vectors lie in TP and TQ. These are just the standard, ordinary Lie vector fields on the manifold of a Lie group, and the presentation above is a standard exposition of a Lie vector field. Anosov flow The connection to the Anosov flow comes from the realization that is the geodesic flow on P and Q. Lie vector fields being (by definition) left invariant under the action of a group element, one has that these fields are left invariant under the specific elements of the geodesic flow. In other words, the spaces TP and TQ are split into three one-dimensional spaces, or subbundleSubbundle In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces ... s, each of which are invariant under the geodesic flow. The final step is to notice that vector fields in one subbundle expand (and expand exponentially), those in another are unchanged, and those in a third shrink (and do so exponentially). More precisely, the tangent bundle TQ may be written as the direct sum or, at a point , the direct sum corresponding to the Lie algebra generators Y, J and X, respectively, carried, by the left action of group element g, from the origin e to the point q. That is, one has , and . These spaces are each subbundleSubbundle In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces ... s, and are preserved (are invariant) under the action of the geodesic flow; that is, under the action of group elements . To compare the lengths of vectors in at different points q, one needs a metric. Any inner product at extends to a left-invariant Riemannian metric on P, and thus to a Riemannian metric on Q. The length of a vector expands exponentially as exp(t) under the action of . The length of a vector shrinks exponentially as exp(-t) under the action of . Vectors in are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant, but the other two shrink and expand: and where we recall that a tangent vector in is given by the derivativeDerivative In mathematics, the derivative is defined as the instantaneous rate of change of a function.... , with respect to t, of the curveCurve Summary In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and con... , the setting t=0. Geometric interpretation of the Anosov flow When acting on the point z=i of the upper half-plane, corresponds to a geodesicGeodesic In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".... on the upper half plane, passing through the point z=i. The action is the standard Mobius transform action of SL(2,R)SL2(R) In mathematics, the special linear group SL2 is the group of all real 2 × 2 matrices with determinant one:... on the upper half-plane, so that A general geodesic is given by with a, b, c and d real, with ad-bc=1. The curves and are called horocyclesHorocycle In hyperbolic geometry, a horocycle is a curve whose normals all converge asymptotically.... . Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane. Further reading D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, (1967) Proc. Steklov Inst. Mathematics. 90. Anthony Manning, Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature, (1991), appearing as Chapter 3 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X (Provides an expository introduction to the Anosov flow on SL(2,R).) abstract from International Conference on Topology and its Applications 2007 at Kyoto