Connection between ergodicity and Lyapunov exponents

Question

This will be a soft reference question in a sense, as I will not state any rigorous theorems/results. Whenever I happen to read about ergodic systems, be it a specific book, article or a blog post, a lot of times ergodicity seems to appear as a gateway and/or cause for chaos. This is to the point that a novice, such as myself, might accept tacitly that "ergodic system" = "chaotic system". However, chaos is usually characterized as sensitivity to small changes in initial conditions. Perhaps the best concept to quantify precisely this is the Lyapunov exponent. Ergodicity, on the other hand, seems to be more about restrictions on invariance (as in volume preserving systems) or about the connection between long-time and space averages. Under these interpretations, it is not really clear to me why an ergodic system would have to possess chaos. So, are there some well established theorems and results which quantify how chaotic an ergodic system will have to be? Abstract spaces can be hard to work with, so let us furthermore suppose that we have some physical-ish situation with our ambient space being that of $\mathbb{R}^n$ and our space $X\subset\mathbb{R}^n$ being a non-empty compact path-connected set. In this case do we know anything about the Lyapunov exponents and/or how chaotic an ergodic system has to be, perhaps only locally? What if we throw some extra conditions such as mixing into play? Does it change anything?