# what is so great about having an invariant measure?

I am a student who just started to learn basic concepts of ergodic theory.

It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But the books I am reading doesn't really convince me why it is good to have invariant measures.

For example, the Gauss map on the unit interval $x \mapsto \{ 1/x \}$ has the invariant measure $1/({1+x})$. What kind of effective results can we prove about the Gauss map using this invariant measure?

• Thanks, I was also interested in the same question, what I would say to a non-mathematician, that why one would be interested to know whether invariant measure exists or not? – miosaki Dec 3 '19 at 11:02
• Don't know but wikipedia mentions the relation between invariant measure and Liouville equation, it seems you can recast the condition of having a invariant measure to having a hamiltonian that satisfy a continuity equation – Daniel D. Dec 6 '19 at 4:21