The space of Anosov diffeomorphisms

TL;DR

This paper explores the topological structure of the space of Anosov diffeomorphisms homotopic to a fixed automorphism on infranilmanifolds, revealing simple topology in 2D and complex, disconnected structures in higher dimensions.

Contribution

It characterizes the homotopy type of the space of Anosov diffeomorphisms on infranilmanifolds, showing simple topology in dimension 2 and rich, disconnected topology in higher dimensions.

Findings

In the 2-torus case, the space is homotopy equivalent to a 2-torus.

For higher dimensions, the space has infinitely many connected components.

The topology of the space varies dramatically with the dimension of the manifold.

Abstract

We consider the space $\X$ of Anosov diffeomorphisms homotopic to a fixed automorphism $L$ of an infranilmanifold $M$. We show that if $M$ is the 2-torus $\mathbb T^2$ then $\X$ is homotopy equivalent to $\mathbb T^2$. In contrast, if dimension of $M$ is large enough, we show that $\X$ is rich in homotopy and has infinitely many connected components.