Ergodic theory

Ergodic theory is like playing with Legos. You know how to build one kind of cool model and it's so fun to make it over and over again. But what if you want to build something new? You could randomly start putting pieces together, but you might not get exactly what you wanted. Ergodic theory is like creating a set of rules to make sure you build what you want every time!

Imagine you have a big bag of Legos and you're going to build a castle. You can put the pieces together in a certain way to make a cool castle. However, in Ergodic theory, we don't want to create just one castle. We want to create lots and lots of castles!

So, imagine building a bunch of castles with your Legos every single day. You could build them all in different ways and end up with a lot of unique castles, right? But what if you wanted to know what all of them had in common? What if you wanted to find out how they were all built?

Here is where Ergodic theory comes in. The goal is to figure out how similar all of your castles are, even if they look different. This means you can create rules to build in a certain way, as long as everything is the same in the end. By doing this, you can make sure that every castle you build is exactly the same in every way.

In math, we use Ergodic theory to study how different systems work, like how the weather changes or how the stock market moves. By studying these things, we can create rules to understand them better, just like how you can create a system to build a castle every time.