View source for Anosov diffeomorphism

{{Short description|Diffeomorphism that has a hyperbolic structure on the tangent bundle}}
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 Anosov|Dmitri Victorovich Anosov]], who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all).<ref>[[Dmitri Anosov|Dmitri V. Anosov]], ''Geodesic flows on closed Riemannian manifolds with negative curvature'', (1967) Proc. Steklov Inst. Mathematics. '''90'''.</ref>

== Overview ==
Three closely related definitions must be distinguished:
* If a differentiable [[map (mathematics)|map]] ''f'' on ''M'' has a [[Hyperbolic set|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 (mathematics)|flow]] on a manifold splits the tangent bundle into three invariant [[subbundle]]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 (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''<sup>1</sup> topology.

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.

The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still {{As of|2023|lc=on}} has no answer for dimension over 3. The only known examples are [[infranilmanifold]]s, and it is conjectured that they are the only ones.

A sufficient condition for transitivity is that all points are nonwandering: <math> \Omega(f)=M </math>. This in turn holds for codimension-one Anosov diffeomorphisms (i.e., those for which the contracting or the expanding subbundle is one-dimensional)<ref>{{cite journal |last1=Newhouse |first1=Sheldon E. |title=On codimension one Anosov diffeomorphisms |journal=American Journal of Mathematics |date=1970 |volume=92 |pages=761–770 |doi=10.2307/2373372 |url=https://doi.org/10.2307/2373372|url-access=subscription }}</ref> and for codimension one Anosov flows on manifolds of dimension greater than three<ref>{{cite journal |last1=Verjovsky |first1=Alberto |title=Codimension one Anosov flows |journal=Boletín de la Sociedad Matemática Mexicana. Segunda Serie |date=1974 |volume=19 |issue=2 |pages=49–77}}</ref> as well as Anosov flows whose Mather spectrum is contained in two sufficiently thin annuli.<ref>{{cite journal |last1=Brin |first1=M. I. |title=Nonwandering points of Anosov diffeomorphisms |journal=Astérisque |date=1977 |volume=49 |pages=11–18}}</ref> It is not known whether Anosov diffeomorphisms are transitive (except on infranilmanifolds), but Anosov flows need not be topologically transitive.<ref>{{cite journal |last1=Béguin |first1=François |last2=Bonatti |first2=Christian |last3=Yu |first3=Bin |title=Building Anosov flows on 3-manifolds |journal=Geometry & Topology |date=2017 |volume=21 |issue=3 |pages=1837–1930 |doi=10.2140/gt.2017.21.1837 |url=https://doi.org/10.2140/gt.2017.21.1837|arxiv=1408.3951 }}</ref>

Also, it is unknown if every <math>C^1</math> volume-preserving Anosov diffeomorphism is ergodic. Anosov proved it under a <math>C^2</math> assumption. It is also true for <math>C^{1+\alpha}</math> volume-preserving Anosov diffeomorphisms.

For <math>C^2</math> transitive Anosov diffeomorphism <math>f\colon M\to M </math> there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen) <math> \mu_f </math> supported on <math> M </math> such that its basin <math> B(\mu_f)</math> is of full volume, where 

:<math> B(\mu_f)= \left \{x\in M:\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}\to\mu_f \right \}. </math>

==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 [[Poincaré half-plane model]] of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as [[Fuchsian model]]s, 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''&nbsp;=&nbsp;''H''/Γ be a Riemann surface of negative curvature as the quotient of "M" by the action of the group Γ, and let <math>T^1 M</math> be the tangent bundle of unit-length vectors on the manifold ''M'', and let <math>T^1 H</math> 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]] of  a complex [[line bundle]].

===Lie vector fields===
One starts by noting that <math>T^1 H</math> is isomorphic to the [[Lie group]] [[PSL2(R)|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

:<math>J=\begin{pmatrix} 1/2 &0\\ 0&-1/2\\ \end{pmatrix} \qquad X=\begin{pmatrix}0&1\\ 0&0\\ \end{pmatrix} \qquad Y=\begin{pmatrix}0&0\\ 1&0  \end{pmatrix}</math>

which have the algebra

:<math>[J,X]=X \qquad [J,Y] = -Y \qquad [X,Y] = 2J</math>

The [[exponential map (Lie theory)|exponential map]]s

:<math>g_t = \exp(tJ)= \begin{pmatrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{pmatrix} \qquad h^*_t = \exp(tX)=\begin{pmatrix}1&t\\ 0&1\\ \end{pmatrix} \qquad h_t = \exp(tY)= \begin{pmatrix}1&0\\ t&1\\ \end{pmatrix}</math>

define right-invariant [[flow (mathematics)|flow]]s on the manifold of <math>T^1 H = \operatorname{PSL}(2,\R)</math>, and likewise on <math>T^1M</math>. Defining <math>P=T^1H</math> and <math>Q=T^1M</math>, 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 <math>g_t</math> 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 <math>g_t</math> of the geodesic flow. In other words, the spaces ''TP'' and ''TQ'' are split into three one-dimensional spaces, or [[subbundle]]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 of vector bundles|direct sum]] 

:<math>TQ = E^+ \oplus E^0 \oplus E^-</math>

or, at a point <math>g \cdot e = q \in Q</math>, the direct sum

:<math>T_qQ = E_q^+ \oplus E_q^0 \oplus E_q^-</math>

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 <math>E_e^+=Y, E_e^0=J</math> and <math>E_e^-=X</math>. These spaces are each [[subbundle]]s, and are preserved (are invariant) under the action of the  [[geodesic flow]]; that is, under the action of group elements <math>g=g_t</math>.

To compare the lengths of vectors in <math>T_qQ</math> at different points ''q'', one needs a metric. Any [[inner product]] at <math>T_eP=sl(2,\R)</math> extends to a left-invariant [[Riemannian metric]] on ''P'', and thus to a Riemannian metric on ''Q''. The length of a vector <math>v \in E^+_q</math> expands exponentially as exp(t) under the action of <math>g_t</math>. The length of a vector <math>v \in E^-_q</math> shrinks exponentially as exp(-t) under the action of <math>g_t</math>. Vectors in <math>E^0_q</math> are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant,

:<math>g_sg_t=g_tg_s=g_{s+t} </math>

but the other two shrink and expand:

:<math>g_sh^*_t = h^*_{t\exp(-s)}g_s</math>

and 

:<math>g_sh_t = h_{t\exp(s)}g_s </math>

where we recall that a tangent vector in <math>E^+_q</math> is given by the [[derivative]], with respect to ''t'', of the [[curve]] <math>h_t</math>, the setting <math>t=0</math>.

===Geometric interpretation of the Anosov flow===
When acting on the point <math>z=i</math> of the upper half-plane, <math>g_t</math> corresponds to a [[geodesic]] on the upper half plane, passing through the point <math>z=i</math>. The action is the standard [[Möbius transformation]] action of [[SL2(R)|SL(2,'''R''')]] on the upper half-plane, so that 

:<math>g_t \cdot i = \begin{pmatrix} \exp(t/2) & 0 \\  0 & \exp(-t/2) \end{pmatrix} \cdot i = i\exp(t) </math>

A general geodesic is given by

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot i\exp(t) = \frac{ai\exp(t)+b}{ci\exp(t)+d} </math>

with ''a'', ''b'', ''c'' and ''d'' real, with <math>ad-bc=1</math>.  The curves <math>h^*_t</math> and <math>h_t</math> are called '''[[horocycle]]s'''. Horocycles correspond to the motion of the normal vectors of a [[horosphere]] on the upper half-plane.

== See also ==
* [[Ergodic flow]]
* [[Morse–Smale system]]
* [[Pseudo-Anosov map]]

==Notes==
{{reflist}}

== References ==
* {{springer|author= |title=Y-system,U-system, C-system|id=Y/y099010}}
* 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''').)
*{{PlanetMath attribution|title=Anosov diffeomorphism|id=4511}}
* [[Toshikazu Sunada]], ''Magnetic flows on a Riemann surface'', Proc. KAIST Math. Workshop (1993), 93–108.

{{Chaos theory}}

[[Category:Diffeomorphisms]]
[[Category:Dynamical systems]]
[[Category:Hyperbolic geometry]]