# User:Dmitri Anosov/Proposed/Anosov diffeomorphism

The Anosov diffeomorphism (A.d.) is a diffeomorphism $$f$$ of a continuous bounded set $$M\ ;$$ it is informally characterized by a significant inconsistency of all trajectories of $$f^n(x)$$ that belong to the dynamic system with a discrete time $$\lbrace f^n \rbrace$$ (system of iterations of $$f$$) that is born from this set, and with that inconsistency being as great as possible. Using the language of the modern theory of dynamic systems, The A.d. can be defined as a diffeomorphism $$M \to M$$ that has its entire set $$M$$ as a Hyperbolic set (see Hyperbolic Dynamics).

## Definitions

Let us better explain this "technical" definition. Here we discuss the properties of linear approximations to $$f$$ around all possible trajectories. All this is formalized in terms of action of the tangential reflections $$T_xf : T_xM \to T_{f(x)}M$$ ($$f$$ differentials) onto the tangents to $$M$$ spaces $$T_xM$$ (actions of the reflections of $$T_x(f^n)$$). Extended definition of the A.d. states: for every point $$x \in M\ ,$$ $$T_xM$$ is a linear sum $$E^s_x \oplus E^u_x$$ of such two spaces (stable or shrinking and unstable and stretching) that for any Riemann metric $$M$$ have constants $$a, b, \lambda, \mu (a, b > 0, 0 < \lambda < 1, \mu>1$$), and at all natural $$n$$

$\mid T_x(f^n)\xi \mid \le a\lambda^n \mid \xi \mid, \mid T_x(f^{-n})\mid \xi \mid \ge b \mu^n \mid \xi \mid \ ,$ for all $$\xi \in E^s_x$$ $\mid T_x(f^{-n})\eta \mid \le a\lambda^n \mid \eta \mid, \mid T_x(f^n)\mid \eta \mid \ge b \mu^n \mid \eta \mid \ ,$ for all $$\eta \in E^u_x$$

Subspaces $$E^s_x\ ,$$ $$E^u_x$$ are defined uniquely and do not depend on a particular Riemann metric. They constantly depend on the $$x$$ (although such dependency does not necessarily have to be level, no matter how level is the $$f$$). Their dimensions dim $$E^s_x\ ,$$ dim $$E^u_x$$ remain constant. They are layers of some linear subbundles $$E^s_x\ ,$$ $$E^u_x$$ of a tangential exfoliation $$TM\ ,$$ invariable relatively to $$Tf$$ (reflection group $$\lbrace T_xf \rbrace$$) in the sense that

$T_xf(E^s_x)= E^s_{f(x)}, T_xf(E^u_x)= E^u_{f(x)} \ ,$ for all $$x \in M\ ,$$

and $$TM$$ is a Whitney sum $$E^s \oplus E^u\ .$$

## Examples and properties

Figure 1: a–c. Action of $$\tilde{T} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$$ on $$\mathbb{R}^2$$ illustrating the action of $$T$$ on the torus; d. Action of $$T^3$$ on a, magnified.

The simplest example of the A.d.s that demonstrates its basic properties is a hyperbolic automorphism $$T$$ of a two-dimensional torus $$\mathbb{T}^2$$ (it is discussed in [6], ch.7, without a reference to of mentioning of the general theory of the A.d.s. Also see Hyperbolic Dynamics).

The Torus is the result of the factorization of the plane $$\mathbb{R}^2$$ over the integer-valued grid $$\mathbb{Z}^2\ ,$$ and, with such factorization, $$T$$ is formed from a hyperbolic linear reflection (“hyperbolic rotation”) of $$\tilde{T}$$ of this plane with an integer-valued coefficient matrix and a determinant of $$\pm 1$$(the Hyperbolic transformation of $$\tilde{T}$$ means that it does not have its own values $$\lambda$$ with $$\mid \lambda \mid$$ < 1).

The properties of $$\tilde{T}$$ are often illustrated to show how a certain picture is transformed on a plane, for example, the picture of the cat $$\tilde{C}\ .$$ In order to somehow use this picture to demonstrate the properties of $$T\ ,$$ we add the following construction. Imagine that the cat on the picture $$C$$ is drawn on the Torus and the figure $$\tilde{C}\ ,$$ with which we will have to deal directly, covers $$C\ .$$ Let us split $$\mathbb{R}^2$$ into squares $$K_{mn}$$ of type $$m \le x \le m + 1, n \le y \le n + 1 (m, n \in \mathbb{Z})\ .$$ The points of either one of them naturally represent points of the Torus (every side of the square has the same points as the opposite side, and 4 vertices represent the same point). Let’s split the figure $$\tilde{T}\tilde{C}$$ into two parts that are placed in different squares $$K_{mn}$$ and move the part $$\tilde{T}\tilde{C} \cap K_{mn}$$ thus will move onto $$(-m,-n)$$ so that this part will get into $$K_{00}\ .$$ (With the correct projection of $$K_{00} \to \mathbb{T}^2$$ those points of the shifted into the $$K_{00}$$ parts of the figure $$\tilde{T}\tilde{C}$$ that landed on the boundary of the quadrant will naturally attach to each other. The figure on the actual torus can’t be cut. We can only cut the covering figure $$\tilde{T}\tilde{C} \subset R^2\ ,$$ since it can’t be fit into the quadrant $$K_{00}\ ,$$ and we want to show $$\tilde{T}\tilde{C}$$ while still remaining in the $$K_{00}\ .$$)

Even with small $$n\ ,$$ the picture starts to look “striped”, and this makes it clear that as $$n$$ grows bigger, all the points of $$T^nC$$ fall closer to each other on the surface of the Torus. This picture also shows, though not as clearly, that $$T$$ has a property of “mixing”: a certain “amount” of points of $$T^nC$$ that, under the influence of $$T^n$$ fall into a fixed area $$B$$ (i.e., the ratio of the measures $$\mu (T^nC \cap B)/ \mu(C))\ ,$$ as $$n \to \infty \ ,$$ tends toward $$\mu (B)$$ (the measure $$\mu$$ is to have a norm such that $$\mu(\mathbb{T}^2) = 1$$). (Of course, the proof of the last property (especially not just for the regions $$B\ ,$$ $$C$$ of a “simple” form but also for the measurable sets) is beyond the scope of what is presented in the picture.)

With every small (meaning $$C^1$$) disturbance of the A.d. we still have the A.d. This way, if in a variety $$M$$ there is an A.d., it is present in multiple quantities – they now form an open subset in the space Diff1$$(M)$$ of diffeomorphisms $$M \to M$$ with a $$C^1$$-topology. However, the A.d.s do not exist on all the $$M\ .$$ For example, in a case with two-dimensions, the A.d.s only exist on $$\mathbb{T}^2\ .$$ All known examples of $$M$$ that allow an A.d. have an “algebraic origin” – they are homeomorphic to n-dimensional Toruses $$T^n\ ,$$ and, in a more general case, to some infranilsets. (There are examples by A. Farrell and L. Johnes where a homeomorphism of $$M$$ to given “algebraic models” shows exactly the homeomorphism, while there are no diffeomorphisms.) It has been proven (initially by G. Franks and finally by A. Manning) that on such $$M$$ any A.d. is topologically connected with an A.d. of the “algebraic origin” – an A.d. induced by some hyperbolic automorphism of a certain covering, single-connected nilpotent group $$Le$$ (e.g., in the case of $$\mathbb{T}^n$$ – a hyperbolic automorphism of the covering Euclidian space that retains an integer-valued grid.) It is also known (initially G. Franks and finally S. Newhouse) that if $$\dim E^s_x$$ = 1 or $$\dim E^u_x$$ = 1 (“an A.d. of a co-dimension of 1”), then $$M = \mathbb{T}^n\ .$$ The presence of some subbundles in $$TM$$ imposes certain topological restrictions upon $$M$$ (e.g., in the case of a co-dimension of 1 the Euler’s Identity $$\chi (M) = 0\ .$$) There is no doubt, however, that there are bound to be other restrictions, even though so far they are not fully explored. (As an example, in a two-dimensional case the condition $$\chi (M) = 1$$ does not exclude a possibility for the Klein bottle, even though in reality this bottle does not have an A.d. on it. Some new restrictions were found by V.K. Kleptsin and Y. Kudryashov, whose work is to be published in “Groups geometry and dynamics”. For example, it is impossible to have $$M = S^2 \times S^2\ .$$)

There are dynamic systems with an uninterrupted time – analogous A.d.s – Anosov streams (see Hyperbolic dynamics). Over the last quarter of a century more works were dedicated to them than to the actual A.d.s.

Objects that retain one property or another (in this particular case, hyperbolicity) to the maximum can be relatively rare. Yet, often during closer examination of those objects the role of such properties is clearly seen, and this, in turn, helps in studying objects that have the same properties, but they are less pronounced. The A.d.s (along with the analogous streams and Smale horseshoe) have been used as a polygon for a development of a number of ideas of the Hyperbolistic theory for dynamic systems: rephrasing the definitions of hyperbolicity and its revelation with the help of cones or the action of $$Tf$$ onto the cross-section $$TM\ ;$$ structural stability; adapted metric; families of stable and unstable diversities (for an A.d. those are bundles); the role of the Hölder condition; various options of following pseudo-trajectories (both separate and in families); absolute continuum in the bundles; distinctive rigidity; invariable measures (for an A.d. it is their continuum, but some should be examined especially closely); the markovian segmentations; the ergodicity of $$f$$ toward the “outstanding” invariable measures and more powerful statistical properties, including bernouliness; diminution of correlations; connection between entropy and characteristic indicators; asymptotic form of the number $$a_n$$ of periodic trajectories with a period $$n$$ and the $$\theta$$-function – a type of a generating function for $$a_n\ .$$

## History

A.d.s (with its analogous streams) were first introduced under a different name by D. Anosov in 1962 [1]. The immediate reason for that was him finding a proof of a hypothesis regarding structural stability of the automorphisms of a torus and geodesic streams on the closed Riemann manifold of a negative curve. This hypothesis was proposed by S. Smale during his visit to the USSR in the fall of 1961 after he constructed the Smale horseshoe and proved its structural stability. The infinite number of periodic points on the horseshoe contradicted naïve hopes (once even shared by Smale himself) for a "simple construction" of structurally stable systems. The A.d.s also have an infinite number of periodic points and, in addition, some (not all) A.d.s have an invariable measure with a smooth density - this gave even more "acute" contradiction of those hopes; at first, the mathematicians, who did not participate in the designing of the developing hyperbolic theory, treated those A.d.s as another (after the horseshoe) example of unexpected phenomena.

For the "technical apparatus" of the research of the A.d.s (and in general, with evenly-distributed hyperbolic behavior of the trajectories), the most substantial contribution was made by the works of J. Adamar and O. Perrone on the conditional stability. (For some reason, earlier works by A. Lyapunov became more useful in hyperbolic dynamics much later, in the research of the uneven hyperbolicity.)

An overview of the results of the A.d.s and their connection to other aspects of the hyperbolic theory of dynamic systems, along with the bibliography can be found in [2]-[5].

## References

[1] D. Anosov. Roughness of geodesic flows on compact Riemannian manifolds of negative curvature. Dokl. Akad. nauk SSSR, v. 145, 707-709 (1962).

[2] V. Solodov. Topological problems of the theory of dynamical systems. Uspekhi mat. nauk (Russian math. surveys), v. 46, N. 5, 93-114 (1982).

[3] Y. Sinai (ed.) Dynamical systems II. Dynamical systems, ergodic theory and applications. Encyclopaedia math. sci., v. 2. Springer, Berlin etc. qq

[4] D. Anosov (ed.) Dynamical systems IX. Dynamical systems with hyperbolic behaviour. Encyclopaedia math. sci., v. 66. Springer, Berlin etc. (1995).

[5] B. Hassleblatt, A. Katok. Handbook of dynamical systems, v. 1a. Elsevier, Amsterdam etc. (2002).

[6] B. Hasselblatt, A. Katok. A first course in dynamics: with a panorama of recent developments. Cambridge univ. press (2003).