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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more particularly in the fields of
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
and
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, an Anosov map on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
''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).


Overview

Three closely related definitions must be distinguished: * If a differentiable
map A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
''f'' on ''M'' has a hyperbolic structure on the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
, then it is called an Anosov map. Examples include the
Bernoulli map The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to , 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, ...
, 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 subbundle">flow (mathematics)">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 (spanned by the flow direction), then the flow is called an Anosov flow. A classical example of Anosov diffeomorphism is the Arnold's cat map. Anosov proved that Anosov diffeomorphisms are structurally stable and form an open subset of mappings (flows) with the ''C''1 topology. Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. 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 has no answer for dimension over 3. The only known examples are infranilmanifolds, and it is conjectured that they are the only ones. A sufficient condition for transitivity is that all points are nonwandering: \Omega(f)=M . This in turn holds for codimension-one Anosov diffeomorphisms (i.e., those for which the contracting or the expanding subbundle is one-dimensional) and for codimension one Anosov flows on manifolds of dimension greater than three as well as Anosov flows whose Mather spectrum is contained in two sufficiently thin annuli. It is not known whether Anosov diffeomorphisms are transitive (except on infranilmanifolds), but Anosov flows need not be topologically transitive. Also, it is unknown if every C^1 volume-preserving Anosov diffeomorphism is ergodic. Anosov proved it under a C^2 assumption. It is also true for C^ volume-preserving Anosov diffeomorphisms. For C^2 transitive Anosov diffeomorphism f\colon M\to M there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen) \mu_f supported on M such that its basin B(\mu_f) is of full volume, where : B(\mu_f)= \left \.


Anosov flow on (tangent bundles of) Riemann surfaces

As an example, this section develops the case of the Anosov flow on the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
. This flow can be understood in terms of the flow on the tangent bundle of the
Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with co ...
of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as
Fuchsian model In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface ''R'' as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazar ...
s, that is, as the quotients of the
upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
and a
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or co ...
. For the following, let ''H'' be the upper half-plane; let Γ be a Fuchsian group; let ''M'' = ''H''/Γ be a Riemann surface of negative curvature as the quotient of "M" by the action of the group Γ, and let T^1 M be the tangent bundle of unit-length vectors on the manifold ''M'', and let T^1 H be the tangent bundle of unit-length vectors on ''H''. Note that a bundle of unit-length vectors on a surface is the
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
of a complex
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
.


Lie vector fields

One starts by noting that T^1 H is isomorphic to the
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
PSL(2,R). This group is the group of orientation-preserving
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the upper half-plane. The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of PSL(2,R) is sl(2,R), and is represented by the matrices :J=\begin 1/2 &0\\ 0&-1/2\\ \end \qquad X=\begin0&1\\ 0&0\\ \end \qquad Y=\begin0&0\\ 1&0 \end which have the algebra : ,XX \qquad ,Y= -Y \qquad ,Y= 2J The exponential maps :g_t = \exp(tJ)= \begine^&0\\ 0&e^\\ \end \qquad h^*_t = \exp(tX)=\begin1&t\\ 0&1\\ \end \qquad h_t = \exp(tY)= \begin1&0\\ t&1\\ \end define right-invariant flows on the manifold of T^1 H = \operatorname(2,\R), and likewise on T^1M. Defining P=T^1H and Q=T^1M, 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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
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). More precisely, the tangent bundle ''TQ'' may be written as the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
:TQ = E^+ \oplus E^0 \oplus E^- or, at a point g \cdot e = q \in Q, the direct sum :T_qQ = E_q^+ \oplus E_q^0 \oplus E_q^- 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 E_e^+=Y, E_e^0=J and E_e^-=X. These spaces are each subbundles, and are preserved (are invariant) under the action of the
geodesic flow In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
; that is, under the action of group elements g=g_t. To compare the lengths of vectors in T_qQ at different points ''q'', one needs a metric. Any
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
at T_eP=sl(2,\R) extends to a left-invariant
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
on ''P'', and thus to a Riemannian metric on ''Q''. The length of a vector v \in E^+_q expands exponentially as exp(t) under the action of g_t. The length of a vector v \in E^-_q shrinks exponentially as exp(-t) under the action of g_t. Vectors in E^0_q are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant, :g_sg_t=g_tg_s=g_ but the other two shrink and expand: :g_sh^*_t = h^*_g_s and :g_sh_t = h_g_s where we recall that a tangent vector in E^+_q is given by the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
, with respect to ''t'', of the
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
h_t, the setting t=0.


Geometric interpretation of the Anosov flow

When acting on the point z=i of the upper half-plane, g_t corresponds to a
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
on the upper half plane, passing through the point z=i. The action is the standard
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
action of SL(2,R) on the upper half-plane, so that :g_t \cdot i = \begin \exp(t/2) & 0 \\ 0 & \exp(-t/2) \end \cdot i = i\exp(t) A general geodesic is given by :\begin a & b \\ c & d \end \cdot i\exp(t) = \frac with ''a'', ''b'', ''c'' and ''d'' real, with ad-bc=1. The curves h^*_t and h_t are called
horocycle In hyperbolic geometry, a horocycle ( from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics ( normals) through a point on a horocycle are ...
s. Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane.


See also

*
Ergodic flow In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ ...
*
Morse–Smale system In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic set, hyperbolic periodic orbits and satisfyi ...
*
Pseudo-Anosov map In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a Surface (topology), surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notio ...


Notes


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). ''(Provides an expository introduction to the Anosov flow on'' SL(2,R).) * *
Toshikazu Sunada is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recogni ...
, ''Magnetic flows on a Riemann surface'', Proc. KAIST Math. Workshop (1993), 93–108. {{Chaos theory Diffeomorphisms Dynamical systems Hyperbolic geometry