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 diffeomorphisms were introduced by D. V. Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

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.

• If the map is a diffeomorphism, then it is called an Anosov diffeomorphism.

• If a flow on a manifold splits the tangent bundle into three invariant subbundles, 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. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ...

Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere . 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. A sphere is a perfectly symmetrical geometrical object. ...

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. 2005 is a common year starting on Saturday of the Gregorian calendar. ...

Another famous problem that still remains open 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).

Anosov flow on (tangent bundles of) Riemann surfaces

As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M=HΓ 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 bundle. In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... Curvature is the amount by which a geometric object deviates from being flat. ... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ... In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... 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 group PSL(2,R). This group is the group of orientation-preserving isometries of the upper half-plane. The Lie algebra of PSL(2,R) is sl(2,R), and is represented by the matrices In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

which have the algebra

The exponential maps In differential geometry, the exponential map is the map from (a subset of) the tangent space of a Riemannian manifold M to M itself. ...

define right-invariant flows 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 g_t 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 g_t of the geodesic flow. In other words, the spaces TP and TQ are split into three one-dimensional spaces, or subbundles, 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). This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ...

More precisely, the tangent bundle TQ may be written as the direct sum In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

or, at a point , the direct sum

corresponding to 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 subbundles, and are preserved (are invariant) under the action of the geodesic flow; that is, under the action of group elements g = g_t. In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

To compare the lengths of vectors in T_qQ 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 g_t. The length of a vector shrinks exponentially as exp(-t) under the action of g_t. Vectors in are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant, In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

g_sg_t = g_tg_s = g_{s + t}

but the other two shrink and expand:

and

g_sh_t = h_texp(s)g_s

where we recall that a tangent vector in is given by the derivative, with respect to t, of the curve h_t, the setting t=0. In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

Geometric interpretation of the Anosov flow

When acting on the point z=i of the upper half-plane, g_t corresponds to a geodesic on the upper half plane, passing through the point z=i. The action is the standard Mobius transform action of SL(2,R) on the upper half-plane, so that In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ...

A general geodesic is given by

with a, b, c and d real, with ad-bc=1. The curves and h_t are called horocycles. Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane.

Historical references

• D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, (1967) Proc. Steklov Inst. Maths. 90.

Modern references

• 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).)

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