Understanding hyperbolic dynamical systems

Question

I am trying to understand uniformly hyperbolic dynamical systems from the definition given here.

I understand Smale's horseshoe with expansion and contraction that is very clear to see, but I don't see how the derivative gives us this expansion and contraction. How does the derivative express the squeezing and stretching on the original set?

Answer 1

The distance along a path $\gamma\colon[0,1]\to M$ between two points $p,q\in M$ is given by $$ d(p,q)=\int_0^1\|\gamma'(t)\|\,dt. $$ Therefore, for a diffeomorphism $f\colon M\to M$ we have $$ d(f(p),f(q))=\int_0^1\|(f\circ\gamma)'(t)\|\,dt\le\sup\|df\|\int_0^1\|\gamma'(t)\|\,dt=\sup\|df\|d(p,q). $$ In particular, if the derivative contracts, then the length of $\gamma([0,1])$ decreases when we apply $f$.

In other words, lengths decrease when the derivative contracts. Similarly, lengths increase when the derivative expands.

That's the beginning of hyperbolicity theory, which generalizes this simple example in various ways.