Ergodic theory is a branch of mathematics that studies the long-term average behavior of dynamical systems. It focuses on systems that evolve over time, examining how their states change and whether they settle into predictable patterns. This theory is particularly useful in understanding complex systems in various fields, including physics, economics, and biology. One key concept in ergodic theory is the idea of ergodicity, which suggests that, given enough time, the time spent in different states of a system will reflect the overall distribution of those states. This means that the average behavior observed over time can be used to infer properties of the system as a whole.