Musings on Ergodic Theory...

There comes a time in every man's life where he feels that he should know more about ergodic theory than he does, for me that time arrived at 2:30 pm this afternoon while reading Brian Arthur's Increasing Returns and Path Dependence in the Economy. 

First I would like to prove to myself that Cosma Shalizi's assertions in this post are in fact correct.  Specifically he claims that...

"It is not true that non-stationarity is a sufficient condition for non-ergodicity; nor is it a necessary one."

This says that non-stationarity does not imply non-ergodicity.  I want to prove this by contradiction, so I need an example of a non-stationary process that is ergodic.

"It is not true that 'positive destabilizing feedback' implies non-ergodicity."

Again to prove by contradiction, I need an example of an ergodic process that exhibits positive destabilizing feedback 

"It is not true that ergodicity is incompatible with sensitive dependence on initial conditions."

Here I need an example of an ergodic process that exhibits sensitive dependence to initial conditions.  Cosma has already pointed out in his post that chaotic processes will generally serve as an example of an ergodic process with sensitive dependence on initial conditions

"It is not true that ergodicity rules out path-dependence, at least not the canonical form of it exhibited by Arthur's models"

Finally, I will need an example of an ergodic process that also exhibits path-dependence. 

I am currently reading Scott Page's essay on path dependence and I suspect that I will be able find several of the examples that I will need included in the text.

While all of this might seem very far removed from economics, I think understanding all of the above will provide useful constraints on the types of macroeconomic modelling techniques that I should pursue...at least this is my hope.