A diffeomorphism $f:M→M$ is said to be an Anosov diffeomorphism, if it is uniformly hyperbolic on the whole $M$. This means that there exist two invariant bundles, $E^s$ and $E^u$, $TM=E^s⊕E^u$, and constants $c < 1 < C$, $λ < 1 < Λ$, such that

\begin{align}
‖f_*^nv‖&≤Cλ^n‖v‖,&\text{if }&v∈E^s\\
‖f_*^nv‖&≥cΛ^n‖v‖,&\text{if }&v∈E^u.
\end{align}

Anosov diffeomorphisms are known to be structurally stable, and exhibit chaotic behavior.

The simplest example of an Anosov diffeomorphism is the linear map given by the matrix $\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)$ acting on the torus $ℝ^2/ℤ^2$. It turns out that all known Anosov diffeomorphisms, up to a continuous change of coordinates, can be obtained by a similar construction. A conjecture due to Smale says that this is true for all Anosov diffeomorphisms, not only for all currently known.

I will discuss a few approaches to this conjecture, including my joint work with Victor Kleptsyn, and a paper by Andrey Gogolev and Federico Rodriguez Hertz.