Anosov diffeomorphism

In mathematics , Anosov diffeomorphisms , named after Dmitri Viktorovich Anosov , are a well-understood example of chaotic dynamics . On the one hand they show all the typical effects of chaotic behavior, but on the other hand they are easily accessible to a mathematical treatment.

definition

A diffeomorphism of a Riemannian manifold is called Anosov diffeomorphism if it is a continuous, -invariant decomposition ${\displaystyle f\colon M\to M}$ ${\displaystyle M}$ ${\displaystyle f}$

${\displaystyle TM=E^{s}\oplus E^{u}}$

of the tangential bundle are so that or are contracted or expanded by uniformly, d. H. there is with ${\displaystyle TM}$ ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle Df}$ ${\displaystyle 0<\lambda <1,C>0}$

${\displaystyle \|Df^{m}(v^{s})\|\leq C\lambda ^{m}\|v^{s}\|\quad \forall v^{s}\in E^{s},m\in \mathbb {N} }$

${\displaystyle \|Df^{-m}(v^{u})\|\geq C\lambda ^{m}\|v^{u}\|\quad \forall v^{u}\in E^{u},m\in \mathbb {N} }$.

The sub-bundles and are called stable and unstable bundles. ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$

example

→ Main article : Arnold's cat picture

The picture shows how the illustration with the matrix deforms the unit square and how the pieces are rearranged modulo 1. The dashed lines indicate the directions of maximum expansion and compression, they correspond to the eigenvectors of the matrix. ${\displaystyle \left({\begin{matrix}2&1\\1&1\end{matrix}}\right)}$

By

${\displaystyle F(x,y)=(2x+y,x+y)\mod 1}$

or in matrix notation

${\displaystyle F\left({\begin{bmatrix}x\\y\end{bmatrix}}\right)={\begin{bmatrix}2&1\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\mod 1={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\mod 1}$

The defined self-mapping of the torus is an Anosov diffeomorphism: the matrix has two eigenvalues and , the eigenvectors provide a decomposition ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ ${\displaystyle \left({\begin{matrix}2&1\\1&1\end{matrix}}\right)}$ ${\displaystyle \lambda _{1}>1}$ ${\displaystyle \lambda _{2}<1}$

${\displaystyle T_{x}T^{2}=E_{x}^{u}\oplus E_{x}^{s}}$

at each point , with and after the canonical identification ${\displaystyle x\in T^{2}}$ ${\displaystyle E_{x}^{u}}$ ${\displaystyle E_{x}^{s}}$

${\displaystyle T_{x}T^{2}\cong \mathbb {R} ^{2}}$

correspond to the eigenvectors and . The projections of the straight lines parallel to the eigenvectors onto the torus are the stable and unstable manifolds of the mapping. ${\displaystyle \lambda _{1}}$ ${\displaystyle \lambda _{2}}$

existence

A conjecture by Smale says that there are Anosov diffeomorphisms only on manifolds that are homeomorphic to an infranile manifold . On infranil manifolds , Anosov diffeomorphisms are always conjugated to affine (i.e. homomorphism- induced) maps. But there are Anosov diffeomorphisms on manifolds which are only homeomorphic (and not diffeomorphic ) to an infranil manifold . ${\displaystyle G/\Gamma }$ ${\displaystyle G\to G}$

literature

Stephen Smale : Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 1967 747-817 pdf

Individual evidence

1. ^ John Franks: Anosov diffeomorphisms. Proc. Symp. In Pure Math of AMS 14, 61-94, 1968
2. ^ Anthony Manning: There are no new anosov diffeomorphisms on tori. Amer. Jour. of Math. 96, 424-429, 1974
3. ↑ FT Farrell, LE Jones: Anosov diffeomorphisms Constructed from Diff ${\displaystyle \pi _{1}}$ ${\displaystyle (S^{n})}$ , Topology 17, 273-282, 1978