## The real decay rate

Let be a uniform expanding map on the 1-torus , be the unique ACIP of which is exponentially mixing. That is, there exists such that for any two Lipschitz functions on , where

be the correlation function.

Let be a diffeomorphism on , be the induced map, and The new correlation function

,

where . Therefore, the two smoothly conjugate systems and have the same mixing rate.

Assuming is close to identity, we see that is also uniformly expanding, and one may derive the mixing rate of independently. However, this new rate may be different (better or worse) from . For example, could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from itself?

In the general case, two expanding maps on are only topologically conjugate (via full shifts). So it is possible that the decay rate varies in the topologically conjugate classes.

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