## The real decay rate

Let $f$ be a $C^2$ uniform expanding map on the 1-torus $\mathbb{T}$, $\mu$ be the unique ACIP of $f,$ which is exponentially mixing. That is, there exists $\lambda\in(0,1)$ such that $|C(\phi,\psi,f^n)|\le C\lambda^n$ for any two Lipschitz functions $\phi,\psi$ on $\mathbb{T}$, where
$C(\phi,\psi,f^n,\mu)=\int \phi\circ f^n\cdot \psi d\mu -\mu(\phi)\cdot\mu(\psi)$ be the correlation function.

Let $h$ be a $C^2$ diffeomorphism on $\mathbb{T}$, $g=h^{-1}fh$ be the induced map, and $h_\ast \nu=\mu.$ The new correlation function

$C(\phi,\psi,g^n,\nu)=\int \phi\circ g^n\cdot \psi d\nu-\nu(\phi)\cdot\nu(\psi)$
$=\int \hat\phi\circ f^n(hx)\cdot \hat\psi(hx) d\nu-\nu(\phi)\cdot\nu(\psi)$
$=\int \hat\phi\circ f^n \cdot \hat\psi d\mu-\mu(\hat\phi)\cdot\mu(\hat\psi)$,

where $\hat\phi=\phi\circ h^{-1}$. Therefore, the two smoothly conjugate systems $(g,\nu)$ and $(f,\mu)$ have the same mixing rate.

Assuming $h$ is close to identity, we see that $g=h^{-1}fh$ is also $C^2$ uniformly expanding, and one may derive the mixing rate of $(g,\nu)$ independently. However, this new rate may be different (better or worse) from $\lambda$. For example, $f$ could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from $(g,\nu)$ itself?

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