There are many definitions of chaos, but perhaps one of the most common is by Devaney:
Let $X$ be a metric space. A continuous map $f:X\to X$ is said to be chaotic on $X$ if
where, in turn, we usually define sensitive dependence on initial conditions as:
A funtion $f:X \to X$ has sensitive dependence on initial conditions if there exists $\delta >0$ such that, for every $x \in X$ and any neighborhood $N$ of $x$, there exists $y \in N$ and $n \geq 0$ such that $|f^n(x)-f^n(y)| > \delta$.
Some actually define maps to be chaotic when they have a positive Lyapunov exponent. Informally this is a measure of how fast trajectories diverge from each other, and for chaotic maps it is exponentially fast (positive Lyapunov exponent). Almost no matter how you define chaos (sensibly), it seems they all have this property, and it has therefore been called an indication of chaos.
Question: Have I misunderstood something trivial, or how do the ‘usual ingredients’ of chaos (such as the ones in Devaney’s definition) imply an exponential divergence of trajectories? It seems to me that a map could have sensitive dependence without having an exponential divergence of trajectories (?), for example. In short, what in most definitions of chaos implies exponential (as opposed to just a linear) divergence of trajectories? Is this a simple matter, or perhaps a more complicated one?