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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?

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  • $\begingroup$ 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? $\endgroup$ – miosaki Dec 3 '19 at 11:02
  • $\begingroup$ 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 $\endgroup$ – Daniel D. Dec 6 '19 at 4:21
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One important case is when you have a probability measure, in which case if a map has an invariant measure, then it preserves probability. This is the proper setting for the Birkhoff ergodic theorem, which presumably you will soon learn.

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  • $\begingroup$ why one would be interested in preserving probability? can you explain a little more with more real-world intuition, application, motivation, examples? Thanks! $\endgroup$ – miosaki Dec 3 '19 at 11:01

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